Recent zbMATH articles in MSC 65https://www.zbmath.org/atom/cc/652022-01-14T13:23:02.489162ZUnknown authorWerkzeugPrefacehttps://www.zbmath.org/1475.000252022-01-14T13:23:02.489162ZFrom the text: This special issue is devoted to collecting the most recent results for modelling real world problems.Prefacehttps://www.zbmath.org/1475.000282022-01-14T13:23:02.489162ZSummary: A meeting on Approximation Theory (ICMA2016) took place in 2016 at the Schloß (Chateau) Rauischholzhausen near Giessen, Germany, which is the conference center of the Justus-Liebig University at Giessen. This meeting brought together a community of outstanding researchers in this area of mathematics who presented talks about their specific field of work, covering the state-of-the-art in Nonlinear Approximation, Harmonic Analysis and Wavelets, Kernel-Based (radial basis function) Methods, Multivariate Polynomial and Spline Approximation, and Subdivision Methods.
A number of these speakers have contributed their research work to a special volume of the Journal of Approximation Theory (JAT) which we now publish as a collection of papers in a variety of fields.
The meeting was dedicated to the memory of Professor M.J.D. ``Mike'' Powell (1936--2015).Preface. Special Issue dedicated to professor Jie Shen's 60th birthday.https://www.zbmath.org/1475.000302022-01-14T13:23:02.489162Z(no abstract)Introduction to special issue on Monte Carlo methods in statisticshttps://www.zbmath.org/1475.000412022-01-14T13:23:02.489162ZFrom the text: While Monte Carlo methods are used in a wide range of domains which started with particle physics in the 1940's, statistics has a particular connection with these methods in that it both relies on them to handle complex models and validates their convergence by providing assessment tools. Both the bootstrap and the Markov chain Monte Carlo (MCMC) revolutions of the 1980's and 1990's have changed for good the way Monte Carlo methods are perceived by statisticians, moving them from a peripheral tool to an essential component of statistical analysis.Model reduction of complex dynamical systems. Editorial of the special issue corresponding to a workshop held at SDU Odense, Denmark, January 11--13, 2017.https://www.zbmath.org/1475.000892022-01-14T13:23:02.489162ZFrom the text: This special issue of ACOM is dedicated to the workshop ``Model reduction of complex dynamical systems'' (MODRED) that took place January 11--13, 2017, at the University of Southern Denmark, Odense.Special section: 2020 Copper Mountain conferencehttps://www.zbmath.org/1475.001172022-01-14T13:23:02.489162ZFrom the text: The sixteenth Copper Mountain Conference on iterative methods was scheduled to be held March 21--26, 2020, in Copper Mountain, Colorado.Reliable methods of mathematical modelinghttps://www.zbmath.org/1475.001272022-01-14T13:23:02.489162ZFrom the text: [...] the biennial RMMM conferences bring together scientists developing reliable methods for mathematical modeling.
The 9th edition of the conference took place on September 9--13, 2019 at TU Wien, Vienna, Austria. This special issue collects selected works from participants of RMMM 2019 that are related to their presentations.Editorial in memoriam Luc Wuytackhttps://www.zbmath.org/1475.001432022-01-14T13:23:02.489162ZSummary: This foreword to the Virtual Special Issue dedicated to Luc Wuytack contains some historical information related to the founding of Journal CAM as well as to Luc's memory. It also serves as a preface to the collection of articles that have been selected to exemplify the field to which Luc devoted his career and around which the development of the journal CAM started.A foreword to the special issue in honor of Professor Bernardo Cockburn on his 60th birthday: a life time of discontinuous schemingshttps://www.zbmath.org/1475.001452022-01-14T13:23:02.489162ZSummary: We present this special issue of the ``Journal of Scientific Computing'' to celebrate Bernardo Cockburn's sixtieth birthday. The theme of this issue is discontinuous Galerkin methods, a hallmark of Bernardo's distinguished professional career. This foreword provides an informal but rigorous account of what enabled Bernardo's achievements, based on the concluding presentation he gave at the the IMA workshop ``Recent advances and challenges in discontinuous Galerkin methods and related approaches'' on July 1, 2017 which was widely deemed as the best lecture of his career so far.On Gower's inverse matrixhttps://www.zbmath.org/1475.150022022-01-14T13:23:02.489162Z"López-Bonilla, J."https://www.zbmath.org/authors/?q=ai:lopez-bonilla.jose-luis"López-Vázquez, R."https://www.zbmath.org/authors/?q=ai:lopez-vazquez.r"Vidal-Beltrán, S."https://www.zbmath.org/authors/?q=ai:vidal-beltran.sSummary: We show that Faddeev-Sominsky's process allows construct a natural inverse for any square matrix, which is an alternative to the inverse obtained by \textit{J. C. Gower} [Linear Algebra Appl. 31, 61--70 (1980; Zbl 0435.65028)].
For the entire collection see [Zbl 1460.92003].A note on perturbation estimations for spectral projectorshttps://www.zbmath.org/1475.150032022-01-14T13:23:02.489162Z"Song, Chuanning"https://www.zbmath.org/authors/?q=ai:song.chuanning"Wei, Yimin"https://www.zbmath.org/authors/?q=ai:wei.yimin"Xu, Qingxiang"https://www.zbmath.org/authors/?q=ai:xu.qingxiangFor any singular matrix \(A\in \mathbb{C}^{n\times n}\), let \(A^D\), \(A^{\pi}=I_n-AA^{D}\) be its Drazin inverse and spectral projector, respectively. Let \(\|.\|\) be any norm on \(\mathbb{C}^{n\times n}\) and \(\bar{A}\in \mathbb{C}^{n\times n}\) be any stable perturbation of \(A\). In this note, a sufficient condition is given under which \(\|A^{\bar{\pi}}- A^{\pi}\|<1\). As a main result, it is proved that if there exists \(k, l\in \mathbb{N}\) with \(k\geq \operatorname{ind}(\bar{A})\) such that
\[
\delta=\max(\|(A^{D})^l E_{k,l}\|,\|E_{k,l}(A^D)^l\|)<\frac{1}{1+3\|A^{\pi}\|},
\]
then \(\|A^{\bar{\pi}}- A^{\pi}\|<1\).
Reviewer: Kui Ji (Shijiazhuang)Finite dimensional applications of the Dunford-Taylor integralhttps://www.zbmath.org/1475.150052022-01-14T13:23:02.489162Z"Caratelli, Diego"https://www.zbmath.org/authors/?q=ai:caratelli.diego"Palini, Ernesto"https://www.zbmath.org/authors/?q=ai:palini.ernesto"Ricci, Paolo Emilio"https://www.zbmath.org/authors/?q=ai:ricci.paolo-emilioSummary: The Dunford-Taylor integral is used in order to compute the inverse of a non-singular complex matrix. Then the obtained result is applied to derive the solution of some basic analytic problems as the solution of linear algebraic equations, the solution of matrix equations and of initial value problems for a linear system of ordinary differential equations.On the maximal number of Pareto eigenvalues in a matrix of given orderhttps://www.zbmath.org/1475.150062022-01-14T13:23:02.489162Z"Baillon, Jean-Bernard"https://www.zbmath.org/authors/?q=ai:baillon.jean-bernard"Seeger, Alberto"https://www.zbmath.org/authors/?q=ai:seeger.albertoThe article studies the number of Pareto eigenvalues in real \(n\times n\) matrices. A Pareto eigenvalue of a real matrix \(A\) is a real number \(\lambda\) for which there exists a nonzero nonnegative vector \(u\) with \(Au-\lambda u\) nonnegative and \(u^T(Au-\lambda u)=0\). Notice for example that a classic eigenvalue is a Pareto eigenvalue if it admits a nonnegative eigenvector.
The number of such eigenvalues is linked among others to the number of nonisomorphic connected subgraph of a graph, so it finds applications in applicated fields.
Given all the \(n\times n\) real matrices, the possible maximum number of Pareto eigenvalues \(c_n\) is not known (except for \(n=1,2\)), and the existing bounds are very rough. In fact, the relative gap between lower and upper bounds is of the order of \(n\).
This paper improves lower and upper bounds on such numbers by approximately 20\%. These new results are obtained following different ideas coming from heuristics, and the authors manage to prove that \(c_{n+1}/c_n\) is asymptotically \(2\). Moreover, they compute the exact number \(c_3\) for \(3\times 3\) matrices, and state a number of necessary conditions for \(n\times n\) matrices to have \(c_n\) Pareto eigenvalues.
Reviewer: Giovanni Barbarino (Helsinki)Eigenvectors of orthogonally decomposable functionshttps://www.zbmath.org/1475.150072022-01-14T13:23:02.489162Z"Belkin, Mikhail"https://www.zbmath.org/authors/?q=ai:belkin.mikhail"Rademacher, Luis"https://www.zbmath.org/authors/?q=ai:rademacher.luis-a"Voss, James"https://www.zbmath.org/authors/?q=ai:voss.james-eThe authors consider a simple first order generalization of the power method which provides an efficient and easily implementable method for basis recovery. The main result of this article proves that ``the set \(\{\pm e_i \mid i \in [m]\}\) is a complete enumeration of the local maxima of \(|F|\) with respect to the domain \(S^{d-1}\)''. The authors explain that function basis recovery closely resembles the problem of recovering the top eigenvector of a symmetric matrix. Throughout the paper, the authors assume that the function is only approximately orthogonally decomposable, with bounds that are polynomial in the dimension and w.r.t. other relevant parameters, such as perturbation size.
The content of the article is relevant in the field of eigenvalues and eigenvectors. The abstract of the article is clear and to the point, stressing both specific applications and general aspects of the work.
Reviewer: P. Shakila Banu (Erode)Structured strong \(\ell \)-ifications for structured matrix polynomials in the monomial basishttps://www.zbmath.org/1475.150162022-01-14T13:23:02.489162Z"De Terán, Fernando"https://www.zbmath.org/authors/?q=ai:de-teran.fernando"Hernando, Carla"https://www.zbmath.org/authors/?q=ai:hernando.carla"Pérez, Javier"https://www.zbmath.org/authors/?q=ai:perez.javier-jThis paper introduces a family of structured \(\ell\)-ifications, ready to use for all divisors \(\ell\), of the grade \(k\) of the matrix polynomial \(P(\lambda) = \sum\nolimits_{j = 0}^k {P_j}\), with \({P_0},\dots,{P_k} \in \mathbb{F}^{n \times n}\), where \(\mathbb{F}\) is an arbitrary field, such that \(k = (2d + 1)\ell \), for some \(d\), namely \(k\) is the product of \(\ell\) times an odd number. This construction is a special case of the block-minimal bases \(\ell\)-ifications introduced by \textit{F. M. Dopico} et al. [Numer. Math. 140, No. 2, 373--426 (2018; Zbl 1416.65094)]. Further, a special minimal basis is chosen to create structured constructions which lie in the family described by \textit{F. M. Dopico} et al. [Linear Algebra Appl. 562, 163--204 (2019; Zbl 1404.65020)]. These families allow one to recover the minimal indices (as part of the eigenstructure, and not preserved by general \(\ell\)-ifications) of the above mentioned matrix polynomials from the minimal indices of any \(\ell\)-ification in the families. Explicit and very elementary formulas for the minimal indices of the matrix polynomial in terms of the minimal indices of these \(\ell\)-ifications are provided. They are a direct consequence of the general formulas provided in [Dopico et al., 2019, loc. cit.] for the block-minimal bases \(\ell\)-ifications.
Reviewer: Mihail Voicu (Iaşi)On the Hermitian positive definite solution and Newton's method for a nonlinear matrix equationhttps://www.zbmath.org/1475.150192022-01-14T13:23:02.489162Z"Zhang, Juan"https://www.zbmath.org/authors/?q=ai:zhang.juan.4|zhang.juan.2|zhang.juan.1|zhang.juan.3"Li, Shifeng"https://www.zbmath.org/authors/?q=ai:li.shifengSummary: In this paper, necessary and sufficient conditions for the existence of the Hermitian positive definite solution for a nonlinear matrix equation are derived. Then, a sufficient condition for the existence of the unique Hermitian positive definite solution is presented. Further, we propose to use Newton's method to solve this nonlinear matrix equation with some constraints. In addition, we prove the convergence of Newton's method. Finally, we present some numerical examples to illustrate the effectiveness of the derived results.On the estimation of \(x^TA^{-1}x\) for symmetric matriceshttps://www.zbmath.org/1475.150292022-01-14T13:23:02.489162Z"Fika, Paraskevi"https://www.zbmath.org/authors/?q=ai:fika.paraskevi"Mitrouli, Marilena"https://www.zbmath.org/authors/?q=ai:mitrouli.marilena"Turek, Ondrej"https://www.zbmath.org/authors/?q=ai:turek.ondrejLet \(A\) be a symmetric positive definite matrix of order \(n\) and \(x \in \mathbb{R}^n\). The authors study the quadratic form \(x^T A^{-1}x\) without the direct computation of the matrix \(A^{-1}\). They investigate the accuracy of the proposed estimates for \(x^T A^{-1}x\). In particular, they study upper bounds for the absolute error of the estimation, investigate estimates for the bilinear form \(x^T A^{-1}y\) and study a numerical implementation for it.
Reviewer: Erich W. Ellers (Toronto)The inverse eigenvalue problems of tensors with an application in tensor nearness problemshttps://www.zbmath.org/1475.150302022-01-14T13:23:02.489162Z"Liang, Maolin"https://www.zbmath.org/authors/?q=ai:liang.maolin"Dai, Lifang"https://www.zbmath.org/authors/?q=ai:dai.lifang"Zhao, Ruijuan"https://www.zbmath.org/authors/?q=ai:zhao.ruijuanSummary: In this paper, we are concerned with the solution of the inverse eigenvalue problem of tensors. The problem under consideration can be regarded as an extension of the matrix case. Explicitly, for several given eigenpairs of a tensor, it is shown that this problem can be converted into solving the system of tensor equations with Einstein product. Using the Moore-Penrose inverses of tensors, we obtain the sufficient and necessary conditions for the solvability of the tensor inverse eigenvalue problem as well as the general solution. As an application of the obtained results, we address the tensor nearness problem corresponding to the tensor inverse eigenvalue problem, and derived its unique solution under some conditions. In order to solve those problems more easily, we also develop the associated iterative approaches originating from the classical conjugate gradient method. The performed numerical results illustrate the feasibility and effectiveness of the proposed methods.Three classes of copositive-type tensors and tensor complementarity problemshttps://www.zbmath.org/1475.150322022-01-14T13:23:02.489162Z"Zhang, Ting"https://www.zbmath.org/authors/?q=ai:zhang.ting.3|zhang.ting.2|zhang.ting|zhang.ting.1"Huang, Zheng-Hai"https://www.zbmath.org/authors/?q=ai:huang.zheng-hai"Li, Yu-Fan"https://www.zbmath.org/authors/?q=ai:li.yufanSummary: In the field of complementary problems, an important issue is to investigate under what conditions feasibility of the problem can lead to its solvability. For the linear complementarity problem, such an issue has been studied when the matrix involved is a copositive star matrix, a pseudomonotone matrix, or a copositive plus matrix. In this paper, we first introduce the concepts of copositive star tensors, pseudomonotone tensors, and copositive plus tensors, which are natural extensions of copositive star matrices, pseudomonotone matrices, and copositive plus matrices, respectively. We discuss the relationships among these three classes of tensors and give a complete characterization. Then we establish an existence result of solutions to the tensor complementarity problem under the assumption that the tensor involved is one of
these three classes of tensors and an addition condition. Finally we show the equivalence of solvability and feasibility for the tensor complementarity problem with the tensor involved being one of these three classes of tensors.On numerical solution of axisymmetric reaction-diffusion equation and some of its applications to biophysicshttps://www.zbmath.org/1475.300902022-01-14T13:23:02.489162Z"Khatiashvili, N."https://www.zbmath.org/authors/?q=ai:khatiashvili.nino"Komurjishvili, O."https://www.zbmath.org/authors/?q=ai:komurjishvili.omar"Kutchava, Z."https://www.zbmath.org/authors/?q=ai:kutchava.z"Pirumova, K."https://www.zbmath.org/authors/?q=ai:pirumova.kSummary: The numerical solution of the axi-symmetric reaction-diffusion equation is obtained by means of the second order accurate implicit finite difference schemes. The result is applied to the model of oxygen diffusion at the brain capillary.On the numerical treatment of system of partial differential equations connected with the Schrödinger equation and some applications to the particle transport at the cubical latticed nanostructureshttps://www.zbmath.org/1475.300912022-01-14T13:23:02.489162Z"Khatiashvili, N."https://www.zbmath.org/authors/?q=ai:khatiashvili.nino"Shanidze, R."https://www.zbmath.org/authors/?q=ai:shanidze.revaz"Komurjishvili, O."https://www.zbmath.org/authors/?q=ai:komurjishvili.omar"Kutchava, Z."https://www.zbmath.org/authors/?q=ai:kutchava.zSummary: The electron transport in the materials having cubical crystal structures (gold, silver) is considered from the relativistic point of view. The process is modeled by the system of partial differential equations connected with the 3D non-stationary Schrödinger equation with the appropriate initial-boundary conditions. The numerical treatment of this system by means of the implicit finite difference schemes is given. The modulus of the wave function is estimated. The numerical example for the gold nanostructure is considered. For the small time interval this system is reduced to the Fredholm integral equation and then analyzed.Method of boundary integral equations with hypersingular integrals in boundary-value problemshttps://www.zbmath.org/1475.300932022-01-14T13:23:02.489162Z"Setukha, A. V."https://www.zbmath.org/authors/?q=ai:setukha.alexey-vSummary: In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper.Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann spherehttps://www.zbmath.org/1475.300982022-01-14T13:23:02.489162Z"Bobenko, Alexander I."https://www.zbmath.org/authors/?q=ai:bobenko.alexander-ivanovich"Bücking, Ulrike"https://www.zbmath.org/authors/?q=ai:bucking.ulrikeSummary: We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere \(\hat{\mathbb{C}}\). Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.Orthogonal polynomials and linear functionals. An algebraic approach and applicationshttps://www.zbmath.org/1475.330012022-01-14T13:23:02.489162Z"García-Ardila, Juan Carlos"https://www.zbmath.org/authors/?q=ai:garcia-ardila.juan-carlos"Marcellán, Francisco"https://www.zbmath.org/authors/?q=ai:marcellan-espanol.francisco"Marriaga, Misael E."https://www.zbmath.org/authors/?q=ai:marriaga.misael-eQuoting the abstract of this short book in its entirety:
\begin{quote} This book presents an introduction to orthogonal polynomials, with an algebraic flavor, based on linear functionals defining the orthogonality and the Jacobi matrices associated with them. Basic properties of their zeros as well as quadrature rules are discussed. A key point is the analysis of those functionals satisfying Pearson equations (semiclassical case) and the hierarchy based on their class. The book's structure reflects the fact that its content is based on a set of lectures delivered by one of the authors at the first Orthonet Summer School in Seville, Spain in 2016. The presentation of the material is self-contained and will be valuable to students and researchers interested in a novel approach to the study of orthogonal polynomials, focusing on their analytic properties. \end{quote}
This reviewer found that abstract to be quite accurate, although the book is very short for the amount of material that it touches on, making the claim of being ``self-contained'' a little iffy. The authors do point out ``Hoping to make up for this lack of exhaustiveness, we have added a list of references that an interested reader can consult.'' Their literature survey is not comprehensive either, but it does contain useful links.
Indeed the book, short as it is, contains several ideas that were very surprising to me, and details of proofs that I had not seen elsewhere and which I found to be very interesting and valuable. First, the idea of using linear functionals to unify the treatment of orthogonal polynomials was new to me, although I have used orthogonal polynomials for, well, decades. I found the unity that this idea provides quite remarkable. Second, the book contains a very short but very illuminating chapter on potential theory, a classical subject for orthogonal polynomials that I had previously only seen in approximation theory (see e.g. Nick Trefethen's book Approximation Theory and Approximation Practice).
The book is extremely well-written (my only cavil is the use of ``It is clear that'' is a bit too frequent, and indeed when the authors use it in Chapter 3 to claim that a general continued fraction converges, well, this is not even true, much less ``clear''. This error is only of small consequence, however, and is the only one I noticed). The book would make a nice text for an advanced short course in orthogonal polynomials; however, the instructors would have to provide their own exercises for the students.
Reviewer: Rob Corless (London)An approximate wavelets solution to the class of variational problems with fractional orderhttps://www.zbmath.org/1475.340082022-01-14T13:23:02.489162Z"Rayal, Ashish"https://www.zbmath.org/authors/?q=ai:rayal.ashish"Verma, Sag Ram"https://www.zbmath.org/authors/?q=ai:ram-verma.sagSummary: In the present work, a generalized fractional integral operational matrix is derived by using classical Legendre wavelets. Then, a numerical scheme based on this operational matrix and Lagrange multipliers is proposed for solving variational problems with fractional order. This approach has been applied on some illustrative examples. The results obtained for these examples demonstrate that the suggested technique is efficient for solving variational problems with fractional order and gives a very perfect agreement with the exact solution. The results are depicted in graphical maps and data tables. The integral square error, maximum absolute error, and order of convergence have been evaluated to analyze the precision of the suggested method. The present scheme provides better and comparable results with some other existing approaches available in the literature.Some questions of approximate solutions for composite bodies weakened by cracks in the case of antiplanar problems of elasticity theoryhttps://www.zbmath.org/1475.340172022-01-14T13:23:02.489162Z"Papukashvili, A."https://www.zbmath.org/authors/?q=ai:papukashvili.archil|papukashvili.a-r"Gordeziani, D."https://www.zbmath.org/authors/?q=ai:gordeziani.david"Davitashvili, T."https://www.zbmath.org/authors/?q=ai:davitashvili.t-dSummary: Antiplane problems of the theory of elastisity by using the theory of analytical functions are presented in the paper. These problems lead to a system of singular integral equations with immovable singularity with the respected to leap of the tangent stress. The probems of behavior of solutions at the boundary are studied. A singular integral equation containing an immovable singularity is solved by collocation and asymptotic methods. It is shown that the system of the corresponding algebraic equations is solvable for sufficiently big number of the integral division. Experimental convergence of aproximate solutions to the exact one is detected.Convergence rates for discretized Monge-Ampère equations and quantitative stability of optimal transporthttps://www.zbmath.org/1475.351512022-01-14T13:23:02.489162Z"Berman, Robert J."https://www.zbmath.org/authors/?q=ai:berman.robert-jSummary: In recent works -- both experimental and theoretical -- it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge-Ampère equations. This yields \(H^1\)-converge rates, in terms of the corresponding spatial resolution \(h\), of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.A posteriori verification for the sign-change structure of solutions of elliptic partial differential equationshttps://www.zbmath.org/1475.351652022-01-14T13:23:02.489162Z"Tanaka, Kazuaki"https://www.zbmath.org/authors/?q=ai:tanaka.kazuakiSummary: This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution \(u\) and a numerically computed approximate solution \(\hat{u}\), we evaluate the number of sign-changes of \(u\) (the number of nodal domains) and determine the location of zero level-sets of \(u\) (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen-Cahn equation. The nodal line of solutions of this equation represents the interface between two coexisting phases.Local and parallel finite element algorithms for the time-dependent Oseen equationshttps://www.zbmath.org/1475.352282022-01-14T13:23:02.489162Z"Ding, Qi"https://www.zbmath.org/authors/?q=ai:ding.qi"Zheng, Bo"https://www.zbmath.org/authors/?q=ai:zheng.bo.1|zheng.bo"Shang, Yueqiang"https://www.zbmath.org/authors/?q=ai:shang.yueqiangSummary: Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms.Spontaneous periodic orbits in the Navier-Stokes flowhttps://www.zbmath.org/1475.352472022-01-14T13:23:02.489162Z"van den Berg, Jan Bouwe"https://www.zbmath.org/authors/?q=ai:van-den-berg.jan-bouwe"Breden, Maxime"https://www.zbmath.org/authors/?q=ai:breden.maxime"Lessard, Jean-Philippe"https://www.zbmath.org/authors/?q=ai:lessard.jean-philippe"van Veen, Lennaert"https://www.zbmath.org/authors/?q=ai:van-veen.lennaertIn this paper, the authors propose a computer-assisted approach to the proof of the existence of time-periodic solutions of the Navier-Stokes equations corresponding to a time-independent force.
Consider the Navier-Stokes equations on the three-torus \({\mathbb T}^3\) with size length \(2\pi\):
\[
\begin{cases} \partial_t u+(u\cdot\nabla)u-\nu\Delta u+\nabla p=f \\
\nabla\cdot u=0 \end{cases}\quad\text{in }\mathbb{T}^3\times\mathbb{R},\tag{1}
\]
where \(u=u(x,t)\) and \(p=p(x,t)\) denote the fluid velocity and pressure field (scaled by the fluid -- constant -- density), respectively, \(\nu\) is the coefficient of kinematic viscosity, and \(f=f(x)\) is the given external force which is assumed to be independent of time and having zero spatial average. The problem is whether there exist a period \(T\) and a corresponding analytic periodic solution \((u,p)\) of (1) with period \(T\). This solution corresponds to what the authors call ``spontaneous periodic motion'' as it represents a (non-stationary) time-periodic flow of a fluid driven by a time-independent force.
The existence of spontaneous periodic motions of a fluid governed by the Navier-Stokes equations has been investigated in the following works: [\textit{V. I. Yudovich}, J. Appl. Math. Mech. 35, 587--603 (1971; Zbl 0247.76044); translation from Prikl. Mat. Mekh. 35, 638--655 (1971)], [\textit{G. Iooss}, Arch. Ration. Mech. Anal. 47, 301--329 (1972; Zbl 0258.35057)], [\textit{D. D. Joseph} and \textit{D. H. Sattinger}, Arch. Ration. Mech. Anal. 45, 79--109 (1972; Zbl 0239.76057)], and [\textit{G. P. Galdi}, Arch. Ration. Mech. Anal. 222, No. 1, 285--315 (2016; Zbl 1352.35096)]. These papers are concerned with periodic solutions branching off from a steady state undergoing bifurcation. In contrast, the authors of the present paper provide a proof of the existence of spontaneous periodic motions which are not necessarily ``close'' to a bifurcation point of a steady-state solution.
The strategy of the proof consists in three main steps. The first step is to identify a zero finding problem \(\mathcal F(W)=0\) on the Banach space of geometrically decaying Fourier coefficients. The solution \(W\) corresponds to an angular frequency \(\Omega\) and a time-periodic solution \(\omega\) to the vorticity equation with period \(2\pi/\Omega\). In Lemma 2.5, it is then proved that solutions to \(\mathcal F(W)=0\) correspond to time-periodic solutions \((u,p)\) of (1).
As a second step, consider a numerical approximation \(\bar W\) of \(W\), i.e., \(\mathcal F(\bar W)\approx 0\), then the exact zero \(W\) of \(\mathcal F\) will be found as a fixed point of the operator
\[
T:\; W\mapsto W-D\mathcal F(\bar W)^{-1}\mathcal F(W),
\]
in a neighborhood of \(\bar W\). By Banach fixed point theorem, it is enough to show that \(T\) is a contraction in a ball centered at \(\bar W\). To prove the latter, computable estimates of \(\|D\mathcal F(\bar W)^{-1}\|\) are needed. Instead of working with \(D\mathcal F(\bar W)^{-1}\), the authors construct approximations \(\hat A\) and \(A\) of \(D\mathcal F(\bar W)\) and \(D\mathcal F(\bar W)^{-1}\), respectively, and use these operators to find sufficient conditions to ensure that \(T\) is contraction in a ball centered at \(\bar W\) (Theorem 2.15).
The final step is to derive and implement explicit bounds that meet the hypothesis of Theorem 2.15. Given the high computational cost to evaluate such bounds, the authors use the symmetries of the model to reduce the size of the zero finding problem (Theorem 4.23). The implementation of the bounds in the symmetric setting can be found in [\textit{J. B. van den Berg} et al., MATLAB code for ``Spontaneous periodic orbits in the Navier-Stokes flow'' (2019), \url{https://www.math.vu.nl/~janbouwe/code/navierstokes/}]. The results for time-periodic solutions which are homogeneous in one space variable (more precisely, they are independent of the third space variable and their third component is zero) corresponding to the Taylor-Green forcing
\[
f(x)=\left(\begin{matrix} 2\sin x_1\cos x_2 \\
-2\cos x_1 \sin x_2 \\
0 \end{matrix}\right)
\]
are presented.
Reviewer: Giusy Mazzone (Kingston)The Darcy problem with porosity depending exponentially on the pressurehttps://www.zbmath.org/1475.352592022-01-14T13:23:02.489162Z"Birhanu, Zerihun Kinfe"https://www.zbmath.org/authors/?q=ai:birhanu.zerihun-kinfe"Mengesha, Tadele"https://www.zbmath.org/authors/?q=ai:mengesha.tadele"Salgado, Abner J."https://www.zbmath.org/authors/?q=ai:salgado.abner-jSummary: We consider the flow of a viscous incompressible fluid through a porous medium. We allow the permeability of the medium to depend exponentially on the pressure and provide an analysis for this model. We study a splitting formulation where a convection diffusion problem is used to define the permeability, which is then used in a linear Darcy equation. We also study a discretization of this problem, and provide an error analysis for it.Generalized resolvent of the Stokes problem with Navier-type boundary conditionshttps://www.zbmath.org/1475.352802022-01-14T13:23:02.489162Z"Hind, Al Baba"https://www.zbmath.org/authors/?q=ai:hind.al-baba"Jabbour, Antonia"https://www.zbmath.org/authors/?q=ai:jabbour.antoniaSummary: We study in this paper the generalized resolvent of the Stokes problem with Navier-type boundary conditions.
For the entire collection see [Zbl 1448.65007].An implicit semi-linear discretization of a bi-fractional Klein-Gordon-Zakharov system which conserves the total energyhttps://www.zbmath.org/1475.353212022-01-14T13:23:02.489162Z"Martínez, Romeo"https://www.zbmath.org/authors/?q=ai:martinez.romeo"Macías-Díaz, Jorge E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo"Sheng, Qin"https://www.zbmath.org/authors/?q=ai:sheng.qinSummary: In this work, we propose an implicit finite-difference scheme to approximate the solutions of a generalization of the well-known Klein-Gordon-Zakharov system. More precisely, the system considered in this work is an extension to the spatially fractional case of the classical Klein-Gordon-Zakharov model, considering two different orders of differentiation and fractional derivatives of the Riesz type. The numerical model proposed in this work considers fractional-order centered differences to approximate the spatial fractional derivatives. The energy associated to this discrete system is a non-negative invariant, in agreement with the properties of the continuous fractional model. We establish rigorously the existence of solutions using fixed-point arguments and complex matrix properties. To that end, we use the fact that the two difference equations of the discretization are decoupled, which means that the computational implementation is easier than for other numerical models available in the literature. We prove that the method has square consistency in both time and space. In addition, we prove rigorously the stability and the quadratic convergence of the numerical model. As a corollary of stability, we are able to prove the uniqueness of numerical solutions. Finally, we provide some illustrative simulations with a computer implementation of our scheme.Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersionhttps://www.zbmath.org/1475.353222022-01-14T13:23:02.489162Z"Marty, Renaud"https://www.zbmath.org/authors/?q=ai:marty.renaudSummary: We study a Lie splitting scheme for a nonlinear Schrödinger-type equation with random dispersion. The main result is an approximation of the local error. Then we can deduce sharp order estimates, for instance in the case of a white noise dispersion.Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type mediahttps://www.zbmath.org/1475.353302022-01-14T13:23:02.489162Z"Assylbekov, Yernat M."https://www.zbmath.org/authors/?q=ai:assylbekov.yernat-m"Zhou, Ting"https://www.zbmath.org/authors/?q=ai:zhou.tingSummary: In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.A direction preserving discretization for computing phase-space densitieshttps://www.zbmath.org/1475.353402022-01-14T13:23:02.489162Z"Chappell, David"https://www.zbmath.org/authors/?q=ai:chappell.david-j"Crofts, Jonathan J."https://www.zbmath.org/authors/?q=ai:crofts.jonathan-j"Richter, Martin"https://www.zbmath.org/authors/?q=ai:richter.martin"Tanner, Gregor"https://www.zbmath.org/authors/?q=ai:tanner.gregor-kParameter estimation for the forest fire propagation modelhttps://www.zbmath.org/1475.353462022-01-14T13:23:02.489162Z"Ambroz, Martin"https://www.zbmath.org/authors/?q=ai:ambroz.martin"Mikula, Karol"https://www.zbmath.org/authors/?q=ai:mikula.karol"Fraštia, Marek"https://www.zbmath.org/authors/?q=ai:frastia.marek"Marčiš, Marián"https://www.zbmath.org/authors/?q=ai:marcis.marianSummary: This paper first gives a brief overview of the Lagrangian forest fire propagation model [\textit{M. Ambroz} et al., Adv. Comput. Math. 45, No. 2, 1067--1103 (2019; Zbl 1415.65206)], which we apply to grass-field areas. Then, we aim to estimate the optimal model parameters. To achieve this goal, we use data assimilation of the measured data. From the data, we are able to estimate the normal velocity of the fire front (rate of spread), dominant wind direction and selected model parameters. In the data assimilation process, we use the Hausdorff distance as well as the mean Hausdorff distance as a criterion. Moreover, we predict the fire propagation in small time intervals.Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculaturehttps://www.zbmath.org/1475.353522022-01-14T13:23:02.489162Z"Fernández-Romero, A."https://www.zbmath.org/authors/?q=ai:fernandez-romero.a"Guillén-González, F."https://www.zbmath.org/authors/?q=ai:guillen-gonzalez.francisco-m"Suárez, A."https://www.zbmath.org/authors/?q=ai:suarez.almudena|suarez.adan|suarez.armando|suarez.antoine|suarez.abril|suarez.alejandro|suarez.ascander|suarez.anna-laura|suarez.adam-justin|suarez.alberto|suarez.antonio|suarez.alvaroSummary: In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy estimates of the continuous problem.The basal layer of the epidermis: a mathematical model for cell production under a surface density constrainthttps://www.zbmath.org/1475.353532022-01-14T13:23:02.489162Z"Gandolfi, Alberto"https://www.zbmath.org/authors/?q=ai:gandolfi.alberto.1"Iannelli, Mimmo"https://www.zbmath.org/authors/?q=ai:iannelli.mimmo"Marinoschi, Gabriela"https://www.zbmath.org/authors/?q=ai:marinoschi.gabrielaFeatures of solving the direct and inverse scattering problems for two sets of monopole scatterershttps://www.zbmath.org/1475.354122022-01-14T13:23:02.489162Z"Dmitriev, Konstantin V."https://www.zbmath.org/authors/?q=ai:dmitriev.konstantin-v"Rumyantseva, Olga D."https://www.zbmath.org/authors/?q=ai:rumyantseva.olga-dSummary: Research presented in this paper was initiated by the publication [\textit{A. D. Agal'tsov} and \textit{R. G. Novikov}, Russ. Math. Surv. 74, No. 3, 373--386 (2019; Zbl 1440.35051); translation from Usp. Mat. Nauk 74, No. 3, 3--16 (2019)] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another.Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLifthttps://www.zbmath.org/1475.354322022-01-14T13:23:02.489162Z"Gong, Yuxuan"https://www.zbmath.org/authors/?q=ai:gong.yuxuan"Li, Peijun"https://www.zbmath.org/authors/?q=ai:li.peijun.1"Wang, Xu"https://www.zbmath.org/authors/?q=ai:wang.xu.4|wang.xu.5|wang.xu|wang.xu.1|wang.xu.3|wang.xu.2"Xu, Xiang"https://www.zbmath.org/authors/?q=ai:xu.xiangIn this paper, the following initial boundary value problem for the one-dimensional stochastic time fractional diffusion equation is considered:
\begin{align*}
& \partial_t^\alpha u(x,t)- \partial_{xx} u(x,t) = F(t)\dot{W}_x, \quad (x,t) \in (0,1)\times \mathbb{R}_+, \\
& u(x,0) = 0, \quad x \in [0,1], \\
& \partial_x u(0,t) = 0, \quad u(1,t) = 0, \quad t \in \mathbb{R}_+,
\end{align*}
where \( \partial_t^\alpha \) denotes the Caputo fractional derivative of order \( 0 < \alpha < 1 \) with respect to the variable \( t \), and \( F \) is a deterministic function satisfying \( F(0) = 0 \). In addition, \( W_x \) is the spatial Brownian motion satisfying \( \mathbb{E}[W_xW_y] = \min\{x, y\} \) for \( x,y \in (0,1) \), and \( \dot{W}_x \) denotes the formal derivative of \( W_x \) known as the white noise. The authors deduce the Green's function for the following equivalent problem in frequency domain:
\begin{align*}
& \partial_{xx} U(x,\omega) -(\text{i}\omega)^\alpha U(x,\omega) = -\hat{F}(\omega)\dot{W}_x, \quad x \in (0,1), \ \omega \in \mathbb{R}, \\
& \partial_x U(0,\omega) = 0, \quad U(1,\omega) = 0, \quad \omega \in \mathbb{R},
\end{align*}
where \( \hat{F} \) denotes the Fourier transform of the zero extention of \( F \) in \( (-\infty, 0) \). This provides the necessary tools for showing well-posedness of the direct problem. Subsequently, the inverse problem is considered which is to reconstruct the diffusion coefficient \( F \) of the random source from the measured data \( u(0,t) \) for \( t > 0 \). It is shown that the modulus \( \vert \hat{F}(\omega) \vert \) is uniquely and unstable determined by the data \( \mathbb{V}[U(0,\omega)] \). The phase retrieval for the inverse problem is also discussed. The paper concludes with some numerical illustrations that make use of a finite difference method to discretize the problem, and in addition a regularized convex optimization scheme is used as a stabilizer.
Reviewer: Robert Plato (Siegen)Digitally generating true orbits of binary shift chaotic maps and their conjugateshttps://www.zbmath.org/1475.370922022-01-14T13:23:02.489162Z"Öztürk, İsmail"https://www.zbmath.org/authors/?q=ai:ozturk.ismail"Kılıç, Recai"https://www.zbmath.org/authors/?q=ai:kilic.recaiSummary: It is impossible to obtain chaotic behavior using conventional finite precision calculations on a digital platform. All such realizations are eventually periodic. Also, digital calculations of the periodic orbits are often erroneous due to round-off and truncation errors. Because of these errors, periodic orbits quickly diverge from the true orbit and they end up into one of the few cycles that occur for almost all initial conditions. Hence, digital calculations of chaotic systems do not represent the true orbits of the mathematically defined original system. This discrepancy becomes evident in the simulations of the binary shift chaotic maps like Bernoulli map or tent map. Although these systems are perfectly well defined chaotic systems, their digital realizations always converge to zero. In the literature, there are some studies which replace the least significant zero bits by random bits to overcome this problem.
In this paper, we propose the algorithms using this simple method for digitally implementing binary shift chaotic maps. These algorithms are suitable for both software and hardware solutions, and they are also applicable with any random number generator or a repeated bit sequence. According to the type of the random number generator, either true periodic orbits or true chaotic orbits of the map are obtained. Moreover, it is shown that, utilizing topological conjugacies, obtained true orbits of binary shift chaotic maps can be used to calculate true orbits of other maps such as logistic and Chebyshev maps which are normally subject to round-off and truncation errors. The hardware implementations of binary shift chaotic maps, logistic map and Chebyshev maps have been realized on a Field Programmable Gate Array (FPGA) platform using the proposed algorithms.Correction to: ``Kernel-based interpolation at approximate Fekete points''https://www.zbmath.org/1475.410012022-01-14T13:23:02.489162Z"Karvonen, Toni"https://www.zbmath.org/authors/?q=ai:karvonen.toni"Särkkä, Simo"https://www.zbmath.org/authors/?q=ai:sarkka.simo"Tanaka, Ken'ichiro"https://www.zbmath.org/authors/?q=ai:tanaka.kenichiroThis is a paper with corrections of certain formulae in an earlier paper that can be summarized as follows: The interpolation with kernel functions and in particular using radial basis function kernel functions is a very good strategy to obtain useful, accurate approximations to multivariable data and functions. Examples of suitable radial basis functions include the so-called Gaussian kernel and the famous multiquadric function. This works in any dimension, and convergence results within ``native spaces'' are available. As the authors note, these features make radial basis function interpolation very attractive. A caveat when using these approximations is that the condition numbers of the Gram (collocation, interpolation) matrices can be very high. The high condition numbers are usually dealt with by maximizing determinants of collocation (interpolation) matrices by choosing well-adjusted centers.
This is because these condition numbers not only depend on the choice of the radial basis function, but also on the choices of the interpolation points. There are so-called Fekete points and other choices, which are considered in the present paper, that minimize these condition numbers. In this article, approximations to Fekete points are computed, and the results are presented within the context of approximation (error) estimates in the Chebyshev norm and numerical examples for the Gauss kernel. The error estimates use the standard methods with power function estimates. These improvements are possible for the special case \(\phi(r)=\exp(-c^2x^2)\), because the expansions of the power functions are explicitly computed. Although the interesting special case of Gauss kernels is only carried out in the one-dimensional setting, the authors generalize this method by tensor-product formulations of multivariable approximations.
The approximation of the Fekete points is understood in the way the kernels are used: to find the approximation points, expansions of the (radial basis function) kernels are carried out, and the approximations of the points are made by truncating these (orthogonal) expansions. The orthogonality with respect to which those expansions take place are defined via inner products within the reproducing kernel Hilbert space (native space).
In the special case of the univariate Gauss kernel, the computation of those points can be interpreted as solving a convex optimization problem.
Reviewer: Martin D. Buhmann (Gießen)Sample numbers and optimal Lagrange interpolation in Sobolev spaceshttps://www.zbmath.org/1475.410032022-01-14T13:23:02.489162Z"Xu, Guiqiao"https://www.zbmath.org/authors/?q=ai:xu.guiqiao"Wang, Hui"https://www.zbmath.org/authors/?q=ai:wang.hui.6This paper studies approximations in weighted Sobolev spaces with respect to the Chebyshev norm. For this purpose we may choose a space embedded in the continuous functions over a compact domain which is a Banach space, and we wish to approximate functions given only at a finite number of nodes. The sampling numbers which are also computed in this article (this is the main point of the work in the paper) are the smallest errors for algorithms based on this number of knots for the worst errors with respect to the choice of approximants in the unit ball of the mentioned spaces. The sampling numbers are closely related to the well-known theory of the so-called \(n\)-widths.
The authors are interested in optimal recovery from such spaces when the weight functions are integrable on \((-1,1)\). They then choose Chebyshev points of the first kind as interpolation nodes and show that those are optimal if Lagrange interpolation is used. A variation of this theme is the search for the optimal interpolation algorithm if the Chebyshev points are used \textbf{and} the end-points of the unit interval, and this is also investigated by the authors.
Reviewer: Martin D. Buhmann (Gießen)Gaussian functions combined with Kolmogorov's theorem as applied to approximation of functions of several variableshttps://www.zbmath.org/1475.410122022-01-14T13:23:02.489162Z"Chernov, A. V."https://www.zbmath.org/authors/?q=ai:chernov.alexey.1|chernov.andrew-v|chernov.aleksei-vyacheslavovich|chernov.andrei-vladimirovichAuthor's abstract: A special class of approximations of continuous functions of several variables on the unit coordinate cube is investigated. The class is constructed using Kolmogorov's theorem stating that functions of the indicated type can be represented as a finite superposition of continuous single-variable functions and another result on the approximation of such functions by linear combinations of quadratic exponentials (also known as Gaussian functions). The effectiveness of such a representation is based on the author's previously proved assertion that the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions. It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.
Reviewer: Aurelian Bejancu (Safat)Effective numerical evaluation of the double Hilbert transformhttps://www.zbmath.org/1475.420052022-01-14T13:23:02.489162Z"Sun, Xiaoyun"https://www.zbmath.org/authors/?q=ai:sun.xiaoyun"Dang, Pei"https://www.zbmath.org/authors/?q=ai:dang.pei"Leong, Ieng Tak"https://www.zbmath.org/authors/?q=ai:leong.ieng-tak|leong.iengtak"Ku, Min"https://www.zbmath.org/authors/?q=ai:ku.minIn this paper the authors study two new methods to compute the double Hilbert transform of periodic functions. In particular, they study the double trigonometric interpolation and, for it, they deduce a quadrature formula of trigonometric interpolation type for the double Hilbert transform and they apply it on the 2-torus. Some numerical examples are given. The calculations to be performed give a convergence under certain restrictions.
Reviewer: Antonio López-Carmona (Granada)Haar and Shannon wavelet expansions with explicit coefficients of the Takagi functionhttps://www.zbmath.org/1475.420492022-01-14T13:23:02.489162Z"Fukuda, Naohiro"https://www.zbmath.org/authors/?q=ai:fukuda.naohiro"Kinoshita, Tamotu"https://www.zbmath.org/authors/?q=ai:kinoshita.tamotu"Suzuki, Toshio"https://www.zbmath.org/authors/?q=ai:suzuki.toshioThe blancmange curve or the Takagi curve is the attractor of a system of two iterated functions. The Takagi (or blancmange) function is continuous, 1-periodic and nowhere differentiable. The paper presents the explicit computations of the wavelet coefficients of the Takagi function using the Haar wavelet and the Shannon wavelet.
Reviewer: Françoise Bastin (Liège)The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functionshttps://www.zbmath.org/1475.440062022-01-14T13:23:02.489162Z"di Dio, Philipp J."https://www.zbmath.org/authors/?q=ai:di-dio.philipp-j"Kummer, Mario"https://www.zbmath.org/authors/?q=ai:kummer.mario-denisThe paper under review is dedicated to some questions in the context of truncated moment problems, especially to the concept of Carathéodory number. We recall that the Carathéodory number is the minimal number \(N\) with the property that every moment sequence with a fixed truncation is the sum of \(N\) Dirac measures. The precise value of this number is not known in general but, nevertheless, evaluations in some important particular cases can be given. For the case of algebraic varieties with small gaps (that is, not all monomial are present), the bounds for the Carathéodory number, previously studied by other authors, are improved by the authors. In fact, they provide explicit lower lower and upper bounds on algebraic varieties, \(\mathbb R^n\), and \([0,1]^n\), also treating moment problems with small gaps. In particular, results concerning Hankel matrices are recaptured by the authors.
Reviewer: Florian-Horia Vasilescu (Villeneuve d'Ascq)Separability of the kernel function in an integral formulation for the anisotropic radiative transfer equationhttps://www.zbmath.org/1475.450022022-01-14T13:23:02.489162Z"Ren, Kui"https://www.zbmath.org/authors/?q=ai:ren.kui"Zhao, Hongkai"https://www.zbmath.org/authors/?q=ai:zhao.hongkai"Zhong, Yimin"https://www.zbmath.org/authors/?q=ai:zhong.yiminDirac integral equations for dielectric and plasmonic scatteringhttps://www.zbmath.org/1475.450032022-01-14T13:23:02.489162Z"Helsing, Johan"https://www.zbmath.org/authors/?q=ai:helsing.johan"Rosén, Andreas"https://www.zbmath.org/authors/?q=ai:rosen.andreasSummary: A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper.Nonlinear algorithms approach to split common solution problemshttps://www.zbmath.org/1475.470562022-01-14T13:23:02.489162Z"He, Zhenhua"https://www.zbmath.org/authors/?q=ai:he.zhenhua"Du, Wei-Shih"https://www.zbmath.org/authors/?q=ai:du.wei-shihSummary: In this paper, we introduce some new iterative algorithms for the split common solution problems for equilibrium problems and fixed point problems of nonlinear mappings. Some examples illustrating our results are also given.Space mapping for optimal control of a nonisothermal tube drawing processhttps://www.zbmath.org/1475.490022022-01-14T13:23:02.489162Z"Butt, Azhar Iqbal Kashif"https://www.zbmath.org/authors/?q=ai:butt.azhar-iqbal-kashif"Mumtaz, Kinza"https://www.zbmath.org/authors/?q=ai:mumtaz.kinza"Reséndiz-Flores, Edgar O."https://www.zbmath.org/authors/?q=ai:resendiz-flores.edgar-omarSummary: We implement space mapping technique for the first time to optimize geometry of a glass tube during its production process. The strategy is to align the optimizer of the coarse model (isothermal tube drawing) with the fine model (nonisothermal tube drawing) through space mapping technique in order to find an optimal solution of a control problem. Adjoint variable approach is used to find optimizer of the coarse model. Numerical results obtained through aggressive space mapping (ASM) algorithm are presented and discussed.Optimal control of the principal coefficient in a scalar wave equationhttps://www.zbmath.org/1475.490032022-01-14T13:23:02.489162Z"Clason, Christian"https://www.zbmath.org/authors/?q=ai:clason.christian"Kunisch, Karl"https://www.zbmath.org/authors/?q=ai:kunisch.karl"Trautmann, Philip"https://www.zbmath.org/authors/?q=ai:trautmann.philipSummary: We consider optimal control of the scalar wave equation where the control enters as a coefficient in the principal part. Adding a total variation penalty allows showing existence of optimal controls, which requires continuity results for the coefficient-to-solution mapping for discontinuous coefficients. We additionally consider a so-called \textit{multi-bang} penalty that promotes controls taking on values pointwise almost everywhere from a specified discrete set. Under additional assumptions on the data, we derive an improved regularity result for the state, leading to optimality conditions that can be interpreted in an appropriate pointwise fashion. The numerical solution makes use of a stabilized finite element method and a nonlinear primal-dual proximal splitting algorithm.A quasi-Monte Carlo method for optimal control under uncertaintyhttps://www.zbmath.org/1475.490042022-01-14T13:23:02.489162Z"Guth, Philipp A."https://www.zbmath.org/authors/?q=ai:guth.philipp-a"Kaarnioja, Vesa"https://www.zbmath.org/authors/?q=ai:kaarnioja.vesa"Kuo, Frances Y."https://www.zbmath.org/authors/?q=ai:kuo.frances-y"Schillings, Claudia"https://www.zbmath.org/authors/?q=ai:schillings.claudia"Sloan, Ian H."https://www.zbmath.org/authors/?q=ai:sloan.ian-hThe authors consider an optimal control problem in the presence of uncertainty: the target function is the solution of an elliptic partial differential equation, steered by a control function and with a random field as input coefficient. They present a specially designed quasi-Monte Carlo method to approximate the expected values with respect to the uncertainty. Their method provides error bounds for the approximation of the stochastic integral, which do not depend on the number of uncertain variables. It results in faster convergence rates compared to Monte Carlo methods in the case of smooth integrands. The nonnegative quadrature weights preserve the convexity structure of the optimal control problem. The random field is, in principle, infinite-dimensional, and in practice, might need a large finite number of terms for accurate approximation. The novelty lies in using and analyzing a specially designed quasi-Monte Carlo method to approximate the possibly high-dimensional integrals with respect to the stochastic variables. The authors present a gradient descent algorithm to solve the optimal control problem for the case without control constraints and a projected variant of the algorithm for the problem with control constraints. Moreover, they present error estimates and convergence rates for the dimension truncation and the finite element discretization together with confirming numerical experiments.
Reviewer: Patrícia Nunes da Silva (Rio de Janeiro)Optimal control of geometric partial differential equationshttps://www.zbmath.org/1475.490052022-01-14T13:23:02.489162Z"Hintermüller, Michael"https://www.zbmath.org/authors/?q=ai:hintermuller.michael"Keil, Tobias"https://www.zbmath.org/authors/?q=ai:keil.tobiasSummary: Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of non-degenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint.
For the entire collection see [Zbl 1458.35003].A posteriori error estimates for parabolic optimal control problems with controls acting on lower dimensional manifoldshttps://www.zbmath.org/1475.490062022-01-14T13:23:02.489162Z"Manohar, Ram"https://www.zbmath.org/authors/?q=ai:manohar.ram-p"Sinha, Rajen Kumar"https://www.zbmath.org/authors/?q=ai:sinha.rajen-kumarSummary: This article concerns a posteriori error estimates for the fully discrete finite element approximation to the optimal control problem governed by parabolic partial differential equations where the control is acting on lower dimensional manifolds. The manifold considered in this paper involves either a point, or a curve or a surface which is lying completely in the space domain. Further, the manifold is assumed to be either time independent or evolved with the time. The space discretization consists of piecewise linear and continuous finite elements for the state and co-state variables and the piecewise constant functions are employed to approximate the control variable. Moreover, the time derivative is approximated by using the backward Euler scheme. We derive a posteriori error estimates for the various dimensions of the manifold. Our numerical results exhibit the effectiveness of the derived error estimators.Finite element approximations of parabolic optimal control problem with measure data in timehttps://www.zbmath.org/1475.490072022-01-14T13:23:02.489162Z"Shakya, Pratibha"https://www.zbmath.org/authors/?q=ai:shakya.pratibha"Sinha, Rajen Kumar"https://www.zbmath.org/authors/?q=ai:sinha.rajen-kumarSummary: The purpose of this paper is to study the a priori error analysis of finite element method for parabolic optimal control problem with measure data in a bounded convex domain. The solution of the state equation of this kind of problem exhibits low regularity which introduces some difficulties for both theory and numerics of finite element method. We first prove the existence, uniqueness and regularity results for the solutions of control problem under low regularity assumption on the state variable. For numerical approximations, we use continuous piecewise linear functions for the state and co-state variables, and piecewise constant functions for the control variable. Both spatially discrete and fully discrete finite element approximations of the control problem are analyzed. We derive a priori error estimates of order \(\mathcal{O} (h)\) for the state, co-state and control variables for the spatially discrete problem. A time discretization scheme based on implicit backward-Euler method is applied to obtain error estimates of order \(\mathcal{O} (h+k^{1/2})\) for the state, co-state and control variables. Numerical results are presented to illustrate our theoretical findings.Approximate optimal controls via instanton expansion for low temperature free energy computationhttps://www.zbmath.org/1475.490272022-01-14T13:23:02.489162Z"Ferré, Grégoire"https://www.zbmath.org/authors/?q=ai:ferre.gregoire"Grafke, Tobias"https://www.zbmath.org/authors/?q=ai:grafke.tobiasStability of efficient solutions to set optimization problemshttps://www.zbmath.org/1475.490282022-01-14T13:23:02.489162Z"Anh, L. Q."https://www.zbmath.org/authors/?q=ai:anh.lam-quoc"Duy, T. Q."https://www.zbmath.org/authors/?q=ai:duy.tran-quoc"Hien, D. V."https://www.zbmath.org/authors/?q=ai:hien.dinh-vinhThe authors consider set optimization problems in real topological Hausdorff spaces as well as the Painleve-Kuratowski convergence of Pareto minimal elements. In particular, they study stability properties of solution sets of corresponding perturbed set optimization problems. Finally, they introduce a compact convergence concept in order to study the internal stability of solution sets.
Reviewer: Jan-Joachim Rückmann (Bergen)On the linear convergence rates of exchange and continuous methods for total variation minimizationhttps://www.zbmath.org/1475.490322022-01-14T13:23:02.489162Z"Flinth, Axel"https://www.zbmath.org/authors/?q=ai:flinth.axel"de Gournay, Frédéric"https://www.zbmath.org/authors/?q=ai:de-gournay.frederic"Weiss, Pierre"https://www.zbmath.org/authors/?q=ai:weiss.pierreSummary: We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space \(\mathcal{M}(\varOmega )\) of Radon measures on a subset \(\varOmega\) of \(\mathbb{R}^d\). Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.Numerical analysis of an optimal control problem governed by the stationary Navier-Stokes equations with global velocity-constrainedhttps://www.zbmath.org/1475.490332022-01-14T13:23:02.489162Z"Niu, Haifeng"https://www.zbmath.org/authors/?q=ai:niu.haifeng"Yang, Danping"https://www.zbmath.org/authors/?q=ai:yang.danping"Zhou, Jianwei"https://www.zbmath.org/authors/?q=ai:zhou.jianweiSummary: A state-constrained optimal control problem governed by the stationary Navier-Stokes equations is studied. Finite element approximation is constructed, the optimal-order \textit{a priori} \(\mathbf{H}^1\)-norm and \(\mathbf{L}^2\)-norm error estimates are given, for which the optimal state is a nonsingular solution of the Navier-Stokes equations to the optimal control.Shape and topology optimizationhttps://www.zbmath.org/1475.490482022-01-14T13:23:02.489162Z"Allaire, Grégoire"https://www.zbmath.org/authors/?q=ai:allaire.gregoire"Dapogny, Charles"https://www.zbmath.org/authors/?q=ai:dapogny.charles"Jouve, François"https://www.zbmath.org/authors/?q=ai:jouve.francoisSummary: This chapter is an introduction to shape and topology optimization, with a particular emphasis on the method of Hadamard for appraising the sensitivity of quantities of interest with respect to the domain, and on the level set method for the numerical representation of shapes and their evolutions. At the theoretical level, the method of Hadamard considers variations of a shape as ``small'' deformations of its boundary; this results in a mathematically convenient and versatile notion of differentiation with respect to the domain, which has historically often been associated with ``body-fitted'' geometric optimization methods. At the numerical level, the level set method features an implicit description of the shape, which arises as the negative subdomain of an auxiliary ``level set function''. This type of representation is well-known to be very efficient when it comes to describing dramatic evolutions of domains (including topological changes). The combination of these two ingredients is an ideal approach for optimizing both the geometry and the topology of shapes, and two related implementation frameworks are presented. The first and oldest one is a Eulerian shape capturing method, using a fixed mesh of a working domain in which the optimal shape is sought. The second and newest one is a Lagrangian shape tracking method, where the shape is exactly meshed at each iteration of the optimization process. In both cases, the level set algorithm is instrumental in updating the shapes, allowing for dramatic deformations between the iterations of the process, and even for topological changes. Most of our applicative examples stem from structural mechanics although some other physical contexts are briefly exemplified. Other topology optimization methods, like density-based algorithms or phase-field methods are also presented, at a lesser level of details, for comparison purposes.
For the entire collection see [Zbl 1458.35003].First and second order shape optimization based on restricted mesh deformationshttps://www.zbmath.org/1475.490502022-01-14T13:23:02.489162Z"Etling, Tommy"https://www.zbmath.org/authors/?q=ai:etling.tommy"Herzog, Roland"https://www.zbmath.org/authors/?q=ai:herzog.roland"Loayza, Estefanía"https://www.zbmath.org/authors/?q=ai:loayza.estefania"Wachsmuth, Gerd"https://www.zbmath.org/authors/?q=ai:wachsmuth.gerdLet \(D\) be an open subset of \(\mathbb{R}^{n}\). The problem under consideration is
\[
\text{minimize}\int_{\Omega }udx\text{ subject to }\Omega \subset D\text{ is quasi-open,}-\bigtriangleup u=f\text{ in }\Omega \text{, }u=0\text{ on } \partial \Omega \text{.}
\]
Solving this problem numerically by a finite element method has the drawback of yielding meshes with very degenerate cells. When applying any optimization algorithm that uses the gradient of the objective function of the discretized problem, this degeneration phenomenon ends up producing descent directions that are inappropiate for the continuous problem. To avoid this drawback, after observing that, in the continuous case, only normal forces on the boundary of the current domain contribute to the gradient, for the discretized problem the authors propose to project the gradient onto the subspace of perturbation fields generated by normal forces. They present some numerical examples showing that, by their approach, gradient or Newton-like methods can solve the problem to high accuracy.
Reviewer: Juan-Enrique Martínez-Legaz (Barcelona)Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measurehttps://www.zbmath.org/1475.490552022-01-14T13:23:02.489162Z"de Gournay, Frédéric"https://www.zbmath.org/authors/?q=ai:de-gournay.frederic"Kahn, Jonas"https://www.zbmath.org/authors/?q=ai:kahn.jonas"Lebrat, Léo"https://www.zbmath.org/authors/?q=ai:lebrat.leoThe paper focuses on the second-order differentiability of the optimal transport problem of the cost function \(c(y,x)\) between the measures \(\nu\) and \(\mu=\sum_{i=1}^n m^{i}\delta_{z^i}\), with respect to the parameters of the latter discrete measure. Taking advantage of the dual formulation of this problem, the main goal of the article is to show that it is possible to differentiate with respect to \(z=(z_i)_{i=1}^n\), \(z_i\in \mathbb{R}^d\), under rather general assumptions on the cost function \(c\) and the measure \(\nu\). In particular, the result applies when \(c(y,x)=\frac{1}{2}\| y-x\|_2^2\) and \(\nu\) admits a \(C^0(\Omega)\cap W^{1,1}(\Omega)\) density with respect to the Lebesgue measure on a bounded, convex, Lipschitz set \(\Omega\subset \mathbb{R}^d\).
In the final part of the paper, the authors show an application of their theoretical results providing numerical illustrations in the blue noise and the stippling problems.
A similar study, under slightly different assumptions, is performed in [\textit{J. Kitagawa} et al., J. Eur. Math. Soc. (JEMS) 21, No. 9, 2603--2651 (2019; Zbl 1439.49053)], which kind of complement the present paper.
Reviewer: Nicolò De Ponti (Trieste)Graph Riemann hypothesis and Ihara zeta function of nonregular Ramanujan graph generated by \(p\)-adic chaoshttps://www.zbmath.org/1475.600152022-01-14T13:23:02.489162Z"Naito, Koichiro"https://www.zbmath.org/authors/?q=ai:naito.koichiroSummary: In our previous papers, applying chaotic properties of the \(p\)-adic dynamical system given by the \(p\)-adic logistic map, we constructed a new pseudorandom number generator. In this paper, using the sequences of these pseudorandom numbers given by this generator, we construct some pseudorandom adjacency matrices and their graphs. Since the regular Ramanujan graph satisfies the graph Riemann hypothesis, we numerically investigate our pseudorandom nonregular graphs by calculating the distributions of poles of the Ihara zeta functions, which are obtained by substituting our pseudorandom adjacency matrices into the Ihara determinant formula.Submartingale property of set-valued stochastic integration associated with Poisson process and related integral equations on Banach spaceshttps://www.zbmath.org/1475.600992022-01-14T13:23:02.489162Z"Zhang, Jinping"https://www.zbmath.org/authors/?q=ai:zhang.jinping"Mitoma, Itaru"https://www.zbmath.org/authors/?q=ai:mitoma.itaru"Okazaki, Yoshiaki"https://www.zbmath.org/authors/?q=ai:okazaki.yoshiakiSummary: In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the main result: the integral of a set-valued stochastic process with respect to the compensated Poisson measure is a set-valued submartingale but not a martingale unless the integrand degenerates into a single-valued process. Secondly we study the strong solution to the set-valued stochastic integral equation, which includes a set-valued drift, a single-valued diffusion driven by a Brownian motion and the set-valued jump driven by a Poisson process.Data informed solution estimation for forward-backward stochastic differential equationshttps://www.zbmath.org/1475.601002022-01-14T13:23:02.489162Z"Bao, Feng"https://www.zbmath.org/authors/?q=ai:bao.feng"Cao, Yanzhao"https://www.zbmath.org/authors/?q=ai:cao.yanzhao"Yong, Jiongmin"https://www.zbmath.org/authors/?q=ai:yong.jiongminPolynomial stability of highly non-linear time-changed stochastic differential equationshttps://www.zbmath.org/1475.601062022-01-14T13:23:02.489162Z"Liu, Wei"https://www.zbmath.org/authors/?q=ai:liu.wei.1|liu.wei.6|liu.wei|liu.wei.8|liu.wei.3|liu.wei.5|liu.wei.2|liu.wei.7|liu.wei.9Summary: The moment stability of a class of time-changed stochastic differential equations (SDEs) is studied. The coefficients of the time-changed SDEs are allowed to grow super-linearly. The decay rate is proved to be polynomial for certain kinds of subordinators, which is significantly different from the classical SDEs driven by Brownian motion.Approximation of BSDEs with super-linearly growing generators by Euler's polygonal line method: a simple proof of the existencehttps://www.zbmath.org/1475.601072022-01-14T13:23:02.489162Z"Li, Yunzhang"https://www.zbmath.org/authors/?q=ai:li.yunzhang.1|li.yunzhang"Tang, Shanjian"https://www.zbmath.org/authors/?q=ai:tang.shanjianSummary: This paper develops the Euler's polygonal line method for the backward stochastic differential equations (BSDEs) with super-linearly growing generators. The generators are allowed to be super-linearly growing in the first unknown variable \(y\) and sub-quadratic growing in the second unknown variable \(z\) when the monotonicity condition is satisfied. The convergence rate of the Euler approximation is derived, which also provides a simple proof for the existence of the solution to the monotone BSDEs. The proof is very simple and short, without involving the conventional techniques of truncating and smoothing on the generators.Numerical solutions of stochastic PDEs driven by arbitrary type of noisehttps://www.zbmath.org/1475.601122022-01-14T13:23:02.489162Z"Chen, Tianheng"https://www.zbmath.org/authors/?q=ai:chen.tianheng"Rozovskii, Boris"https://www.zbmath.org/authors/?q=ai:rozovskii.boris-l"Shu, Chi-Wang"https://www.zbmath.org/authors/?q=ai:shu.chi-wangSummary: So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Lévy noise. Recently, \textit{R. Mikulevicius} and the second author [Stoch. Partial Differ. Equ., Anal. Comput. 4, No. 2, 319--360 (2016; Zbl 1342.60082)] proposed a distribution-free Skorokhod-Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.On the weak convergence rate of an exponential Euler scheme for SDEs governed by coefficients with superlinear growthhttps://www.zbmath.org/1475.601302022-01-14T13:23:02.489162Z"Bossy, Mireille"https://www.zbmath.org/authors/?q=ai:bossy.mireille"Jabir, Jean-François"https://www.zbmath.org/authors/?q=ai:jabir.jean-francois"Martínez, Kerlyns"https://www.zbmath.org/authors/?q=ai:martinez.kerlynsSummary: We consider the problem of designing robust numerical integration scheme of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as \(x^{\alpha}\), with \(\alpha >1\). We propose an (semi-explicit) exponential-Euler scheme for which we obtain a theoretical convergence rate for the weak error. To this aim, we analyze the \(C^{1,4}\) regularity of the solution of the associated backward Kolmogorov PDE using its Feynman-Kac representation and the flow derivative of the involved processes. Under some suitable hypotheses on the parameters of the model, we prove a rate of weak convergence of order one for the proposed exponential Euler scheme, and illustrate it with some numerical experiments.Continuous-discrete smoothing of diffusionshttps://www.zbmath.org/1475.601502022-01-14T13:23:02.489162Z"Mider, Marcin"https://www.zbmath.org/authors/?q=ai:mider.marcin"Schauer, Moritz"https://www.zbmath.org/authors/?q=ai:schauer.moritz"van der Meulen, Frank"https://www.zbmath.org/authors/?q=ai:van-der-meulen.frank-hSummary: Suppose \(X\) is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the \textit{Backward Filtering Forward Guiding (BFFG) algorithm}. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in [\textit{M. Schauer} et al., Bernoulli 23, No. 4A, 2917--2950 (2017; Zbl 1415.65022)]. We illustrate its efficiency in a number of challenging problems.Data fusion using factor analysis and low-rank matrix completionhttps://www.zbmath.org/1475.620142022-01-14T13:23:02.489162Z"Ahfock, Daniel"https://www.zbmath.org/authors/?q=ai:ahfock.daniel-c"Pyne, Saumyadipta"https://www.zbmath.org/authors/?q=ai:pyne.saumyadipta"McLachlan, Geoffrey J."https://www.zbmath.org/authors/?q=ai:mclachlan.geoffrey-johnSummary: Data fusion involves the integration of multiple related datasets. The statistical file-matching problem is a canonical data fusion problem in multivariate analysis, where the objective is to characterise the joint distribution of a set of variables when only strict subsets of marginal distributions have been observed. Estimation of the covariance matrix of the full set of variables is challenging given the missing-data pattern. Factor analysis models use lower-dimensional latent variables in the data-generating process, and this introduces low-rank components in the complete-data matrix and the population covariance matrix. The low-rank structure of the factor analysis model can be exploited to estimate the full covariance matrix from incomplete data via low-rank matrix completion. We prove the identifiability of the factor analysis model in the statistical file-matching problem under conditions on the number of factors and the number of shared variables over the observed marginal subsets. Additionally, we provide an EM algorithm for parameter estimation. On several real datasets, the factor model gives smaller reconstruction errors in file-matching problems than the common approaches for low-rank matrix completion.Efficient importance sampling for large sums of independent and identically distributed random variableshttps://www.zbmath.org/1475.620182022-01-14T13:23:02.489162Z"Ben Rached, Nadhir"https://www.zbmath.org/authors/?q=ai:ben-rached.nadhir"Haji-Ali, Abdul-Lateef"https://www.zbmath.org/authors/?q=ai:haji-ali.abdul-lateef"Rubino, Gerardo"https://www.zbmath.org/authors/?q=ai:rubino.gerardo"Tempone, Raúl"https://www.zbmath.org/authors/?q=ai:tempone.raul-fSummary: We discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., \( \mathbb{P}(\sum_{i=1}^N{X_i} \le \gamma)\), via importance sampling (IS). We are particularly interested in the rare event regime when \(N\) is large and/or \(\gamma\) is small. The exponential twisting is a popular technique for similar problems that, in most cases, compares favorably to other estimators. However, it has some limitations: (i) It assumes the knowledge of the moment-generating function of \(X_i\) and (ii) sampling under the new IS PDF is not straightforward and might be expensive. The aim of this work is to propose an alternative IS PDF that approximately yields, for certain classes of distributions and in the rare event regime, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. The first class includes distributions whose probability density functions (PDFs) are asymptotically equivalent, as \(x \rightarrow 0\), to \(bx^p \), for \(p>-1\) and \(b>0\). For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the rare event regime corresponding to small values of \(\gamma\) and/or large values of \(N\), the same performance of the estimator based on the use of the exponential twisting technique. In the second class, we consider the Log-normal setting, whose PDF at zero vanishes faster than any polynomial, and we show numerically that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting IS PDF. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large \(N\) and/or small \(\gamma \).A piecewise deterministic Monte Carlo method for diffusion bridgeshttps://www.zbmath.org/1475.620192022-01-14T13:23:02.489162Z"Bierkens, Joris"https://www.zbmath.org/authors/?q=ai:bierkens.joris"Grazzi, Sebastiano"https://www.zbmath.org/authors/?q=ai:grazzi.sebastiano"van der Meulen, Frank"https://www.zbmath.org/authors/?q=ai:van-der-meulen.frank-h"Schauer, Moritz"https://www.zbmath.org/authors/?q=ai:schauer.moritzSummary: We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy-Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber-Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of the \textit{fully local} algorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by the \textit{subsampling} technique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.Accelerating sequential Monte Carlo with surrogate likelihoodshttps://www.zbmath.org/1475.620202022-01-14T13:23:02.489162Z"Bon, Joshua J."https://www.zbmath.org/authors/?q=ai:bon.joshua-j"Lee, Anthony"https://www.zbmath.org/authors/?q=ai:lee.anthony-j-t"Drovandi, Christopher"https://www.zbmath.org/authors/?q=ai:drovandi.christopher-cSummary: Delayed-acceptance is a technique for reducing computational effort for Bayesian models with expensive likelihoods. Using a delayed-acceptance kernel for Markov chain Monte Carlo can reduce the number of expensive likelihoods evaluations required to approximate a posterior expectation. Delayed-acceptance uses a surrogate, or approximate, likelihood to avoid evaluation of the expensive likelihood when possible. Within the sequential Monte Carlo framework, we utilise the history of the sampler to adaptively tune the surrogate likelihood to yield better approximations of the expensive likelihood and use a surrogate first annealing schedule to further increase computational efficiency. Moreover, we propose a framework for optimising computation time whilst avoiding particle degeneracy, which encapsulates existing strategies in the literature. Overall, we develop a novel algorithm for computationally efficient SMC with expensive likelihood functions. The method is applied to static Bayesian models, which we demonstrate on toy and real examples.Quantifying uncertainty with a derivative tracking SDE model and application to wind power forecast datahttps://www.zbmath.org/1475.620212022-01-14T13:23:02.489162Z"Caballero, Renzo"https://www.zbmath.org/authors/?q=ai:caballero.renzo"Kebaier, Ahmed"https://www.zbmath.org/authors/?q=ai:kebaier.ahmed"Scavino, Marco"https://www.zbmath.org/authors/?q=ai:scavino.marco"Tempone, Raúl"https://www.zbmath.org/authors/?q=ai:tempone.raul-fSummary: We develop a data-driven methodology based on parametric Itô's Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of forecast errors, including the uncertainty of the forecast at time zero. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent diffusion term. Proofs of the existence, strong uniqueness, and boundedness of the SDE solutions are shown by imposing conditions on the time-varying mean-reversion parameter. We develop the structure of the drift term based on sound mathematical theory. A truncation procedure regularizes the prediction function to ensure that the trajectories do not reach the boundaries almost surely in a finite time. Inference based on approximate likelihood, constructed through the moment-matching technique both in the original forecast error space and in the Lamperti space, is performed through numerical optimization procedures. We propose a fixed-point likelihood optimization approach in the Lamperti space. Another novel contribution is the characterization of the uncertainty of the forecast at time zero, which turns out to be crucial in practice. We extend the model specification by considering the length of the unknown time interval preceding the first time a forecast is provided through an additional parameter in the density of the initial transition. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply our SDE framework to model historical Uruguayan normalized wind power production and forecast data between April and December 2019. Sharp empirical confidence bands of wind power production forecast error are obtained for the best-selected model.Ensemble sampler for infinite-dimensional inverse problemshttps://www.zbmath.org/1475.620242022-01-14T13:23:02.489162Z"Coullon, Jeremie"https://www.zbmath.org/authors/?q=ai:coullon.jeremie"Webber, Robert J."https://www.zbmath.org/authors/?q=ai:webber.robert-jSummary: We introduce a new Markov chain Monte Carlo (MCMC) sampler for infinite-dimensional inverse problems. Our new sampler is based on the affine invariant ensemble sampler, which uses interacting walkers to adapt to the covariance structure of the target distribution. We extend this ensemble sampler for the first time to infinite-dimensional function spaces, yielding a highly efficient gradient-free MCMC algorithm. Because our new ensemble sampler does not require gradients or posterior covariance estimates, it is simple to implement and broadly applicable.Efficient stochastic optimisation by unadjusted Langevin Monte Carlo. Application to maximum marginal likelihood and empirical Bayesian estimationhttps://www.zbmath.org/1475.620262022-01-14T13:23:02.489162Z"De Bortoli, Valentin"https://www.zbmath.org/authors/?q=ai:de-bortoli.valentin"Durmus, Alain"https://www.zbmath.org/authors/?q=ai:durmus.alain"Pereyra, Marcelo"https://www.zbmath.org/authors/?q=ai:pereyra.marcelo"Vidal, Ana F."https://www.zbmath.org/authors/?q=ai:vidal.ana-fSummary: Stochastic approximation methods play a central role in maximum likelihood estimation problems involving intractable likelihood functions, such as marginal likelihoods arising in problems with missing or incomplete data, and in parametric empirical Bayesian estimation. Combined with Markov chain Monte Carlo algorithms, these stochastic optimisation methods have been successfully applied to a wide range of problems in science and industry. However, this strategy scales poorly to large problems because of methodological and theoretical difficulties related to using high-dimensional Markov chain Monte Carlo algorithms within a stochastic approximation scheme. This paper proposes to address these difficulties by using unadjusted Langevin algorithms to construct the stochastic approximation. This leads to a highly efficient stochastic optimisation methodology with favourable convergence properties that can be quantified explicitly and easily checked. The proposed methodology is demonstrated with three experiments, including a challenging application to statistical audio analysis and a sparse Bayesian logistic regression with random effects problem.Fast incremental expectation maximization for finite-sum optimization: nonasymptotic convergencehttps://www.zbmath.org/1475.620322022-01-14T13:23:02.489162Z"Fort, G."https://www.zbmath.org/authors/?q=ai:fort.gersende"Gach, P."https://www.zbmath.org/authors/?q=ai:gach.p"Moulines, E."https://www.zbmath.org/authors/?q=ai:moulines.ericSummary: Fast incremental expectation maximization (FIEM) is a version of the EM framework for large datasets. In this paper, we first recast FIEM and other incremental EM type algorithms in the \textit{Stochastic Approximation within EM} framework. Then, we provide nonasymptotic bounds for the convergence in expectation as a function of the number of examples \(n\) and of the maximal number of iterations \(K_{\max}\). We propose two strategies for achieving an \(\epsilon\)-approximate stationary point, respectively with \(K_{\max}= O(n^{2/3}/\epsilon)\) and \(K_{\max}= O(\sqrt{n}/\epsilon^{3/2})\), both strategies relying on a random termination rule before \(K_{\max}\) and on a constant step size in the Stochastic Approximation step. Our bounds provide some improvements on the literature. First, they allow \(K_{\max}\) to scale as \(\sqrt{n}\) which is better than \(n^{2/3}\) which was the best rate obtained so far; it is at the cost of a larger dependence upon the tolerance \(\epsilon\), thus making this control relevant for small to medium accuracy with respect to the number of examples \(n\). Second, for the \(n^{2/3}\)-rate, the numerical illustrations show that thanks to an optimized choice of the step size and of the bounds in terms of quantities characterizing the optimization problem at hand, our results design a less conservative choice of the step size and provide a better control of the convergence in expectation.Ensemble slice sampling. Parallel, black-box and gradient-free inference for correlated \& multimodal distributionshttps://www.zbmath.org/1475.620402022-01-14T13:23:02.489162Z"Karamanis, Minas"https://www.zbmath.org/authors/?q=ai:karamanis.minas"Beutler, Florian"https://www.zbmath.org/authors/?q=ai:beutler.florianSummary: Slice sampling has emerged as a powerful Markov Chain Monte Carlo algorithm that adapts to the characteristics of the target distribution with minimal hand-tuning. However, Slice Sampling's performance is highly sensitive to the user-specified initial length scale hyperparameter and the method generally struggles with poorly scaled or strongly correlated distributions. This paper introduces Ensemble Slice Sampling (ESS), a new class of algorithms that bypasses such difficulties by adaptively tuning the initial length scale and utilising an ensemble of parallel walkers in order to efficiently handle strong correlations between parameters. These affine-invariant algorithms are trivial to construct, require no hand-tuning, and can easily be implemented in parallel computing environments. Empirical tests show that Ensemble Slice Sampling can improve efficiency by more than an order of magnitude compared to conventional MCMC methods on a broad range of highly correlated target distributions. In cases of strongly multimodal target distributions, Ensemble Slice Sampling can sample efficiently even in high dimensions. We argue that the parallel, black-box and gradient-free nature of the method renders it ideal for use in scientific fields such as physics, astrophysics and cosmology which are dominated by a wide variety of computationally expensive and non-differentiable models.A parallel algorithm for ridge-penalized estimation of the multivariate exponential family from data of mixed typeshttps://www.zbmath.org/1475.620442022-01-14T13:23:02.489162Z"Laman Trip, Diederik S."https://www.zbmath.org/authors/?q=ai:laman-trip.diederik-s"van Wieringen, Wessel N."https://www.zbmath.org/authors/?q=ai:van-wieringen.wessel-nSummary: Computationally efficient evaluation of penalized estimators of multivariate exponential family distributions is sought. These distributions encompass among others Markov random fields with variates of mixed type (e.g., binary and continuous) as special case of interest. The model parameter is estimated by maximization of the pseudo-likelihood augmented with a convex penalty. The estimator is shown to be consistent. With a world of multi-core computers in mind, a computationally efficient parallel Newton-Raphson algorithm is presented for numerical evaluation of the estimator alongside conditions for its convergence. Parallelization comprises the division of the parameter vector into subvectors that are estimated simultaneously and subsequently aggregated to form an estimate of the original parameter. This approach may also enable efficient numerical evaluation of other high-dimensional estimators. The performance of the proposed estimator and algorithm are evaluated and compared in a simulation study. Finally, the presented methodology is applied to data of an integrative omics study.Control variate selection for Monte Carlo integrationhttps://www.zbmath.org/1475.620462022-01-14T13:23:02.489162Z"Leluc, Rémi"https://www.zbmath.org/authors/?q=ai:leluc.remi"Portier, François"https://www.zbmath.org/authors/?q=ai:portier.francois"Segers, Johan"https://www.zbmath.org/authors/?q=ai:segers.johanSummary: Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model with the integrand as response and the control variates as covariates. Even without special knowledge on the integrand, significant efficiency gains can be obtained if the control variate space is sufficiently large. Incorporating a large number of control variates in the ordinary least squares procedure may however result in (i) a certain instability of the ordinary least squares estimator and (ii) a possibly prohibitive computation time. Regularizing the ordinary least squares estimator by preselecting appropriate control variates via the Lasso turns out to increase the accuracy without additional computational cost. The findings in the numerical experiment are confirmed by concentration inequalities for the integration error.A Metropolis-class sampler for targets with non-convex supporthttps://www.zbmath.org/1475.620512022-01-14T13:23:02.489162Z"Moriarty, John"https://www.zbmath.org/authors/?q=ai:moriarty.john"Vogrinc, Jure"https://www.zbmath.org/authors/?q=ai:vogrinc.jure"Zocca, Alessandro"https://www.zbmath.org/authors/?q=ai:zocca.alessandroSummary: We aim to improve upon the exploration of the general-purpose random walk Metropolis algorithm when the target has non-convex support \(A\subset{\mathbb{R}}^d\), by reusing proposals in \(A^c\) which would otherwise be rejected. The algorithm is Metropolis-class and under standard conditions the chain satisfies a strong law of large numbers and central limit theorem. Theoretical and numerical evidence of improved performance relative to random walk Metropolis are provided. Issues of implementation are discussed and numerical examples, including applications to global optimisation and rare event sampling, are presented.Empirically driven orthonormal bases for functional data analysishttps://www.zbmath.org/1475.620522022-01-14T13:23:02.489162Z"Nassar, Hiba"https://www.zbmath.org/authors/?q=ai:nassar.hiba"Podgórski, Krzysztof"https://www.zbmath.org/authors/?q=ai:podgorski.krzysztofSummary: In implementations of the functional data methods, the effect of the initial choice of an orthonormal basis has not been properly studied. Typically, several standard bases such as Fourier, wavelets, splines, etc. are considered to transform observed functional data and a choice is made without any formal criteria indicating which of the bases is preferable for the initial transformation of the data. In an attempt to address this issue, we propose a strictly data-driven method of orthonormal basis selection. The method uses B-splines and utilizes recently introduced efficient orthornormal bases called the splinets. The algorithm learns from the data in the machine learning style to efficiently place knots. The optimality criterion is based on the average (per functional data point) mean square error and is utilized both in the learning algorithms and in comparison studies. The latter indicate efficiency that could be used to analyze responses to a complex physical system.
For the entire collection see [Zbl 1471.65009].Implicitly adaptive importance samplinghttps://www.zbmath.org/1475.620532022-01-14T13:23:02.489162Z"Paananen, Topi"https://www.zbmath.org/authors/?q=ai:paananen.topi"Piironen, Juho"https://www.zbmath.org/authors/?q=ai:piironen.juho"Bürkner, Paul-Christian"https://www.zbmath.org/authors/?q=ai:burkner.paul-christian"Vehtari, Aki"https://www.zbmath.org/authors/?q=ai:vehtari.akiSummary: Adaptive importance sampling is a class of techniques for finding good proposal distributions for importance sampling. Often the proposal distributions are standard probability distributions whose parameters are adapted based on the mismatch between the current proposal and a target distribution. In this work, we present an implicit adaptive importance sampling method that applies to complicated distributions which are not available in closed form. The method iteratively matches the moments of a set of Monte Carlo draws to weighted moments based on importance weights. We apply the method to Bayesian leave-one-out cross-validation and show that it performs better than many existing parametric adaptive importance sampling methods while being computationally inexpensive.Semivariogram methods for modeling Whittle-Matérn priors in Bayesian inverse problemshttps://www.zbmath.org/1475.621202022-01-14T13:23:02.489162Z"Brown, Richard D."https://www.zbmath.org/authors/?q=ai:brown.richard-d"Bardsley, Johnathan M."https://www.zbmath.org/authors/?q=ai:bardsley.johnathan-m"Cui, Tiangang"https://www.zbmath.org/authors/?q=ai:cui.tiangangOn a transform for modeling skewnesshttps://www.zbmath.org/1475.621222022-01-14T13:23:02.489162Z"Kang, Li"https://www.zbmath.org/authors/?q=ai:kang.li"Damien, Paul"https://www.zbmath.org/authors/?q=ai:damien.paul"Walker, Stephen"https://www.zbmath.org/authors/?q=ai:walker.stephen-gSummary: In many applications, data exhibit skewness and in this paper we present a new family of density functions modeling skewness based on a transformation, analogous to those of location and scale. Here we note that location will always refer to mode. Hence, in order to model data to include shape, we need only to find a family of densities exhibiting a variety of shapes, since we can obtain the other three properties via the transformations. The chosen class of densities with the variety of shape is, we argue, the simplest available. Illustrations including regression and time series models are given.Is there an analog of Nesterov acceleration for gradient-based MCMC?https://www.zbmath.org/1475.621232022-01-14T13:23:02.489162Z"Ma, Yi-An"https://www.zbmath.org/authors/?q=ai:ma.yian"Chatterji, Niladri S."https://www.zbmath.org/authors/?q=ai:chatterji.niladri-s"Cheng, Xiang"https://www.zbmath.org/authors/?q=ai:cheng.xiang"Flammarion, Nicolas"https://www.zbmath.org/authors/?q=ai:flammarion.nicolas"Bartlett, Peter L."https://www.zbmath.org/authors/?q=ai:bartlett.peter-l"Jordan, Michael I."https://www.zbmath.org/authors/?q=ai:jordan.michael-irwinIn continuous optimization problems, Nesterov's accelerated gradient descent method accelerates the convergence speed of the simple gradient descent method. This paper considers an analogue of such an acceleration method in Markov chain Monte Carlo (MCMC) methods, which in general have a major issue in convergence speed. As in the accelerated gradient descent for optimization, by introducing a momentum variable, the authors derive the stochastic differential equation for the accelerated dynamics in the space of probability distributions with the Kullback-Leibler divergence to the target distribution as the objective functional. Analyzing the convergence rate of the continuous accelerated dynamics and the discretization error, the authors show that the derived accelerated sampling algorithm improves the dimension and accuracy dependence of the unadjusted Langevin algorithm, which corresponds to the unadjusted gradient decent optimization. Thus, this paper provides a solid theoretical foundation for practical accelerated MCMC methods.
Reviewer: Kazuho Watanabe (Toyohashi)Estimation of semiparametric models with errors following a scale mixture of Gaussian distributionshttps://www.zbmath.org/1475.621372022-01-14T13:23:02.489162Z"Taddeo, Marcelo M."https://www.zbmath.org/authors/?q=ai:taddeo.marcelo-m"Morettin, Pedro A."https://www.zbmath.org/authors/?q=ai:morettin.pedro-albertoSummary: In this paper, we consider a semiparametric regression model where the error follows a scale mixture of Gaussian distributions. The purpose is to estimate the target function which is assumed to belong to some class of functions using the EM algorithm and approximations via \(P\)-splines and \(B\)-splines. We illustrate the proposed methodology through several simulation studies. Other forms of function approximation are also studied, namely Fourier and wavelet expansions.Bias reduction in kernel density estimationhttps://www.zbmath.org/1475.621402022-01-14T13:23:02.489162Z"Slaoui, Yousri"https://www.zbmath.org/authors/?q=ai:slaoui.yousriSummary: In this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen-Rosenblatt's density estimator and \textit{A. Mokkadem} et al. [J. Stat. Plann. Inference 139, No. 7, 2459--2478 (2009; Zbl 1160.62077)] is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic \textit{MISE} (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.Frame-constrained total variation regularization for white noise regressionhttps://www.zbmath.org/1475.621452022-01-14T13:23:02.489162Z"Del Álamo, Miguel"https://www.zbmath.org/authors/?q=ai:del-alamo.miguel"Li, Housen"https://www.zbmath.org/authors/?q=ai:li.housen"Munk, Axel"https://www.zbmath.org/authors/?q=ai:munk.axelThis research introduces a family of estimators in the Gaussian white noise model, obtained by minimization of the bounded variation seminorm under a constraint on the frame coefficients of the residuals. The proposed estimates attain the minimax optimal rate of convergence in any dimension up to logarithmic factors. One extension investigated in the present research is the utilization of the proposed model to the nonparametric regression setting with discretely sampled data. The results of research are in principle extendable to inverse problems, non-Gaussian noise models and stochastic different equations based models.
Reviewer: Dimitrios Bagkavos (Ioannina)Functional additive quantile regressionhttps://www.zbmath.org/1475.621502022-01-14T13:23:02.489162Z"Zhang, Yingying"https://www.zbmath.org/authors/?q=ai:zhang.yingying.4|zhang.yingying.2|zhang.yingying.1|zhang.yingying|zhang.yingying.3"Lian, Heng"https://www.zbmath.org/authors/?q=ai:lian.heng"Li, Guodong"https://www.zbmath.org/authors/?q=ai:li.guodong"Zhu, Zhongyi"https://www.zbmath.org/authors/?q=ai:zhu.zhongyiSummary: We investigate a functional additive quantile regression that models the conditional quantile of a scalar response based on the nonparametric effects of a functional predictor. We model the nonparametric effects of the principal component scores as additive components, which are approximated by B-splines. We select the relevant components using a nonconvex smoothly clipped absolute deviation (SCAD) penalty. We establish that, when the relevant components are known, the convergence rate of the estimator using the estimated principal component scores is the same as that using the true scores. We also show that the estimator based on relevant components is a local solution of the SCAD penalized quantile regression problem. The practical performance of the proposed method is illustrated using simulation studies and an empirical application to corn yield data.Gaussian approximation of general non-parametric posterior distributionshttps://www.zbmath.org/1475.621572022-01-14T13:23:02.489162Z"Shang, Zuofeng"https://www.zbmath.org/authors/?q=ai:shang.zuofeng"Cheng, Guang"https://www.zbmath.org/authors/?q=ai:cheng.guangSummary: In a general class of Bayesian non-parametric models, we prove that the posterior distribution can be asymptotically approximated by a Gaussian process (GP). Our results apply to non-parametric exponential family that contains both Gaussian and non-Gaussian regression and also hold for both efficient (root-\(n)\) and inefficient (non-root-\(n)\) estimations. Our general approximation theorem does not rely on posterior conjugacy and can be verified in a class of GP priors that has a smoothing spline interpretation. In particular, the limiting posterior measure becomes prior free under a Bayesian version of `under-smoothing' condition. Finally, we apply our approximation theorem to examine the asymptotic frequentist properties of Bayesian procedures such as credible regions and credible intervals.Near-optimal large-scale k-medoids clusteringhttps://www.zbmath.org/1475.621962022-01-14T13:23:02.489162Z"Ushakov, Anton V."https://www.zbmath.org/authors/?q=ai:ushakov.anton-vladimirovich"Vasilyev, Igor"https://www.zbmath.org/authors/?q=ai:vasilyev.igor-leonidovich|vasilev.igor-lSummary: The k-medoids (k-median) problem is one of the best known unsupervised clustering problems. Due to its complexity, finding high-quality solutions for huge-scale datasets remains extremely challenging. The application of many approaches finding optimal or quality solutions is limited to only small and medium-size instances. On the other hand, many parallel, distributed algorithms that can handle huge-scale datasets usually provide very poor solutions. In this paper, we develop a first parallel, distributed primal-dual heuristic algorithm for the k-medoids problem. Its main component is a very efficient parallel subgradient column generation that solves a Lagrangian dual problem and finds a tight bound on solution quality. High-quality solutions are then produced by a parallel core selection technique. We considerably reduce computational burden and memory load by employing a nearest neighbor strategy to approximate the dissimilarity matrix. We demonstrate that our algorithm finds very close to optimal solutions, confirmed by the tightness of dual bounds, of instances that are much larger than those considered in the literature to date. Our experiments include clustering large-scale collections of face images into several thousand of clusters. We show that our approach outperforms parallel improved versions of the most popular k-medoids clustering algorithms, achieving nearly linear parallel speedup.Bayesian approach to the logistic regression model using Markov chain Monte Carlo methodshttps://www.zbmath.org/1475.622132022-01-14T13:23:02.489162Z"Díaz González, Lucio"https://www.zbmath.org/authors/?q=ai:diaz-gonzalez.lucio"Covarrubias Melgar, Dante"https://www.zbmath.org/authors/?q=ai:covarrubias-melgar.dante"Sistachs Vega, Vivian"https://www.zbmath.org/authors/?q=ai:sistachs-vega.vivian-del-rosarioSummary: The Logistic Regression is a highly used model on different areas of science where the response variable of studied problems is binary. This model can be studied under the Bayesian approach, nonetheless calculations might be complicated even using computation methods, is for that reason we use MCMC methods, which are iterative methods to obtain an approximation of the posterior distribution of model parameters. We use the R software to obtain the distribution with the MCMC method. On this work we implement the logistic regression through a simulation study under the Bayesian Approach applied to the cognitive health state on elders in the state of Guerrero.Numerical analysis and optimization. Contributions presented at the 5th international conference on numerical analysis and optimization (NAO-V), Muscat, Oman, January 6--9, 2020https://www.zbmath.org/1475.650012022-01-14T13:23:02.489162ZPublisher's description: This book gathers selected, peer-reviewed contributions presented at the Fifth International Conference on Numerical Analysis and Optimization (NAO-V), which was held at Sultan Qaboos University, Oman, on January 6--9, 2020. Each chapter reports on developments in key fields, such as numerical analysis, numerical optimization, numerical linear algebra, numerical differential equations, optimal control, approximation theory, applied mathematics, derivative-free optimization methods, programming models, and challenging applications that frequently arise in statistics, econometrics, finance, physics, medicine, biology, engineering and industry.
Many real-world, complex problems can be formulated as optimization tasks, and can be characterized further as large scale, unconstrained, constrained, non-convex, nondifferentiable or discontinuous, and therefore require adequate computational methods, algorithms and software tools. These same tools are often employed by researchers working in current IT hot topics, such as big data, optimization and other complex numerical algorithms in the cloud, devising special techniques for supercomputing systems. This interdisciplinary view permeates the work included in this volume.
The NAO conference series is held every three years at Sultan Qaboos University, with the aim of bringing together a group of international experts and presenting novel and advanced applications to facilitate interdisciplinary studies among pure scientific and applied knowledge. It is a venue where prominent scientists gather to share innovative ideas and know-how relating to new scientific methodologies, to promote scientific exchange, to discuss possible future cooperations, and to promote the mobility of local and young researchers.
The articles of this volume will not be indexed individually. For the preceding conference see [Zbl 1402.65005].On solving stochastic differential equationshttps://www.zbmath.org/1475.650022022-01-14T13:23:02.489162Z"Ermakov, Sergej M."https://www.zbmath.org/authors/?q=ai:ermakov.sergei-mikhailovich"Pogosian, Anna A."https://www.zbmath.org/authors/?q=ai:pogosian.anna-aSummary: This paper proposes a new approach to solving Ito stochastic differential equations. It is based on the well-known Monte Carlo methods for solving integral equations (Neumann-Ulam scheme, Markov chain Monte Carlo). The estimates of the solution for a wide class of equations do not have a bias, which distinguishes them from estimates based on difference approximations (Euler, Milstein methods, etc.).Approximation method with stochastic local iterated function systemshttps://www.zbmath.org/1475.650032022-01-14T13:23:02.489162Z"Soós, Anna"https://www.zbmath.org/authors/?q=ai:soos.anna"Somogyi, Ildikó"https://www.zbmath.org/authors/?q=ai:somogyi.ildikoSummary: The methods of real data interpolation can be generalized with fractal interpolation. These fractal interpolation functions can be constructed with the so-called iterated function systems. Local iterated function systems are an important generalization of the classical iterated function systems. In order to obtain new approximation methods this methods can be combined with classical interpolation methods. In this paper we focus on the study of the stochastic local fractal interpolation function in the case of a random data set.
For the entire collection see [Zbl 1471.65009].Convergence rates for matrix P-greedy variantshttps://www.zbmath.org/1475.650042022-01-14T13:23:02.489162Z"Wittwar, Dominik"https://www.zbmath.org/authors/?q=ai:wittwar.dominik"Haasdonk, Bernard"https://www.zbmath.org/authors/?q=ai:haasdonk.bernardSummary: When using kernel interpolation techniques for constructing a surrogate model from given data, the choice of interpolation points is crucial for the quality of the surrogate. When dealing with vector-valued target functions which are approximated by matrix-valued kernel models, the selection problem is further complicated as not only the choice of points but also the directions in which the data is projected must be determined.
We thus propose variants of Matrix P-greedy algorithms that enable us to iteratively select suitable sets of point-direction pairs with which the approximation space is enriched. We show that the selected pairs result in quasi-optimal convergence rates. Experimentally, we investigate the approximation quality of the different variants.
For the entire collection see [Zbl 1471.65009].Parameter choices for fast harmonic spline approximationhttps://www.zbmath.org/1475.650052022-01-14T13:23:02.489162Z"Gutting, Martin"https://www.zbmath.org/authors/?q=ai:gutting.martinSummary: The approximation by harmonic trial functions allows the construction of the solution of boundary value problems in geoscience where the boundary is often the known surface of the Earth itself. Using harmonic splines such a solution can be approximated from discrete data on the surface. Due to their localizing properties regional modeling or the improvement of a global model in a part of the Earth's surface is possible with splines.
Fast multipole methods have been developed for some cases of the occurring kernels to obtain a fast matrix-vector multiplication. The main idea of the fast multipole algorithm consists of a hierarchical decomposition of the computational domain into cubes and a kernel approximation for the more distant points. This reduces the numerical effort of the matrix-vector multiplication from quadratic to linear in reference to the number of points for a prescribed accuracy of the kernel approximation. In combination with an iterative solver this provides a fast
computation of the spline coefficients.
The application of the fast multipole method to spline approximation which also allows the treatment of noisy data requires the choice of a smoothing parameter. We summarize several methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level.
For the entire collection see [Zbl 1396.86001].Integro cubic splines on non-uniform grids and their propertieshttps://www.zbmath.org/1475.650062022-01-14T13:23:02.489162Z"Zhanlav, T."https://www.zbmath.org/authors/?q=ai:zhanlav.tugal"Mijiddorj, R."https://www.zbmath.org/authors/?q=ai:mijiddorj.renchin-ochirSummary: Integro cubic splines on a non-uniform grid using the integral values of an unknown function are constructed. We establish a consistency relation for integro cubic spline and derive a local integro cubic spline on non-uniform partitions. Approximation and convexity properties of the local integro cubic splines are also studied.Exponential convergence of the deep neural network approximation for analytic functionshttps://www.zbmath.org/1475.650072022-01-14T13:23:02.489162Z"E, Weinan"https://www.zbmath.org/authors/?q=ai:e.weinan"Wang, Qingcan"https://www.zbmath.org/authors/?q=ai:wang.qingcanSummary: We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.The 8T-LE partition applied to the barycentric division of a 3-D cubehttps://www.zbmath.org/1475.650082022-01-14T13:23:02.489162Z"Padrón, Miguel A."https://www.zbmath.org/authors/?q=ai:padron.miguel-a"Plaza, Ángel"https://www.zbmath.org/authors/?q=ai:plaza.angelSummary: The barycentric partition of a 3D-cube into tetrahedra is carried out by adding a new node to the body at the centroid point and then, new nodes are progressively added to the centroids of faces and edges. This procedure generates three types of tetrahedra in every single step called, Sommerville tetrahedron number 3 (ST3), \textit{isosceles trirectangular} tetrahedron and \textit{regular right-type} tetrahedron. We are interested in studying the number of similarity classes generated when the 8T-LE partition is applied to these tetrahedra.
For the entire collection see [Zbl 1471.65009].Degenerations of NURBS curves while all of weights approaching infinityhttps://www.zbmath.org/1475.650092022-01-14T13:23:02.489162Z"Zhang, Yue"https://www.zbmath.org/authors/?q=ai:zhang.yue"Zhu, Chun-Gang"https://www.zbmath.org/authors/?q=ai:zhu.chungangSummary: Non-Uniform Rational B-Spline (NURBS) is the multidisciplinary topic in Mathematics, Computer Science, and Engineering. The NURBS curves together with their geometric properties are widely used in Computer Aided Design, Computer Aided Geometric Design, and Computational Mathematics. When a single weight approaches infinity, the limit of a NURBS curve tends to the corresponding control point. In this paper, a kind of control structure of a NURBS curve, called regular control curve, is defined. We prove that the limit of the NURBS curve is exactly its regular control curve when all of weights approach infinity, where each weight is multiplied by a certain one-parameter function tending to infinity, different for each control point. Moreover, some representative examples are presented to show this property and indicate its application for shape deformation.A new recursive formula for integration of polynomial over simplexhttps://www.zbmath.org/1475.650102022-01-14T13:23:02.489162Z"Lin, ShaoZhong"https://www.zbmath.org/authors/?q=ai:lin.shaozhong"Xie, ZhiQiang"https://www.zbmath.org/authors/?q=ai:xie.zhiqiangSummary: The simplex integration is a convenient method for the integration over complex domains. It automatically represents the ordinary integral over an arbitrary polyhedron as the algebraic sum of integrals over the oriented simplexes. An existing recursive formula for the integration of monomials over simplex, which was deduced based on special operations of matrices and was presented by the first author of this paper, has significant advantages: not only the computation amount is small, but also the integrals of all the lower order monomials are obtained while computing the integral of the highest order monomial. The extension to a polynomial can be obtained by the linearity of integrals. In this paper, the derivation of the existing recursive formula is detailed. A new recursive formula is emphatically proposed to simplify the existing recursive formula and further increase the speed of computation. The code for computer implementation is also presented. Examples show that the accuracy and efficiency of the recursive formula are higher than numerical integration.On the question of good conditionality of unsaturated quadrature formulashttps://www.zbmath.org/1475.650112022-01-14T13:23:02.489162Z"Belykh, Vladimir Nikitich"https://www.zbmath.org/authors/?q=ai:belykh.vladimir-nikitichSummary: A sufficient sign of good conditionality (resistance to rounding errors) of unsaturated quadrature formulas with a weight function from the Lebesgue space \(L_p\), \(1< p < \infty\) on a nite segment is indicated.Optimization of nodes of composite quadrature formulas in the presence of a boundary layerhttps://www.zbmath.org/1475.650122022-01-14T13:23:02.489162Z"Zadorin, Nikita Alexandrovich"https://www.zbmath.org/authors/?q=ai:zadorin.nikita-alexandrovichSummary: The problem of numerical integration of a function of one variable with large gradients in the region of the exponential boundary layer is studied. The problem is that the use of composite quadrature formulas on a uniform grid with decreasing of the small parameter value leads to significant errors, regardless of the number of nodes of the basic quadrature formula. In the paper it is proposed to choose nodes based on the composite quadrature formula error minimizing. Basic quadrature formula applied between grid nodes, takes into account the cases of the Newton-Cotes and Gauss formulas. It is proved that the minimum error is achieved on the Bakhvalov mesh, and the error of the quadrature formula becomes uniform in a small parameter.Directed evaluationhttps://www.zbmath.org/1475.650132022-01-14T13:23:02.489162Z"van der Hoeven, Joris"https://www.zbmath.org/authors/?q=ai:van-der-hoeven.joris"Lecerf, Grégoire"https://www.zbmath.org/authors/?q=ai:lecerf.gregoireSummary: Let \(\mathbb{K}\) be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure of \(\mathbb{K}\) is to compute modulo its minimal polynomial. The determination of a minimal polynomial from an arbitrary annihilator requires an algorithm for polynomial factorization over \(\mathbb{K}\). Unfortunately, such algorithms do not exist over generic effective fields. They do exist over fields that are explicitly generated over their prime sub-field, but they are often expensive. The dynamic evaluation paradigm, introduced by Duval and collaborators in the eighties, offers an alternative algorithmic solution for computations in the algebraic closure of \(\mathbb{K}\). This approach does not require an algorithm for polynomial factorization, but it still suffers from a non-trivial overhead due to suboptimal recomputations. For the first time, we design another paradigm, called directed evaluation, which combines the conceptual advantages of dynamic evaluation with a good worst case complexity bound.On uniform approximation of Cauchy type integrals on closed contours of integrationhttps://www.zbmath.org/1475.650142022-01-14T13:23:02.489162Z"Sanikidze, J."https://www.zbmath.org/authors/?q=ai:sanikidze.dzemal-g"Ninidze, K."https://www.zbmath.org/authors/?q=ai:ninidze.k-rSummary: Certain quadrature processes are considered for integrals with kernels \((t-z)^{-1}\), \((t-z)^{-2}\) along piece-wise smooth closed contours, bounding finite or infinite domain \(D\) involving \(z\). Uniform estimates are given for the corresponding remainder terms namely for the case of arbitrary closeness of \(z\) to the boundary of the domain.Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equationshttps://www.zbmath.org/1475.650152022-01-14T13:23:02.489162Z"Lu, Xin"https://www.zbmath.org/authors/?q=ai:lu.xin.1"Fang, Zhi-Wei"https://www.zbmath.org/authors/?q=ai:fang.zhiwei"Sun, Hai-Wei"https://www.zbmath.org/authors/?q=ai:sun.haiweiSummary: We study the sine-transform-based splitting preconditioning technique for the linear systems arising in the numerical discretization of time-dependent one dimensional and two dimensional Riesz space fractional diffusion equations. Those linear systems are Toeplitz-like. By making use of diagonal-plus-Toeplitz splitting iteration technique, a sine-transform-based splitting preconditioner is proposed to accelerate the convergence rate efficiently when the Krylov subspace method is implemented. Theoretically, we prove that the spectrum of the preconditioned matrix of the proposed method is clustering around 1. In practical computations, by the fast sine transform the computational complexity at each time level can be done in \({{\mathcal{O}}}(n\log n)\) operations where \(n\) is the matrix size. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomialshttps://www.zbmath.org/1475.650162022-01-14T13:23:02.489162Z"Ramos, Luis García"https://www.zbmath.org/authors/?q=ai:garcia-ramos.luis"Sète, Olivier"https://www.zbmath.org/authors/?q=ai:sete.olivier"Nabben, Reinhard"https://www.zbmath.org/authors/?q=ai:nabben.reinhardSummary: We introduce a new polynomial preconditioner for solving the discretized Helmholtz equation preconditioned with the complex shifted Laplace (CSL) operator. We exploit the localization of the spectrum of the CSL-preconditioned system to approximately enclose the eigenvalues by a non-convex `bratwurst' set. On this set, we expand the function \(1/z\) into a Faber series. Truncating the series gives a polynomial, which we apply to the Helmholtz matrix preconditioned by the shifted Laplacian to obtain a new preconditioner, the Faber preconditioner. We prove that the Faber preconditioner is nonsingular for degrees one and two of the truncated series. Our numerical experiments (for problems with constant and varying wavenumber) show that the Faber preconditioner reduces the number of GMRES iterations.An efficient iterative approach to large sparse nonlinear systems with non-Hermitian Jacobian matriceshttps://www.zbmath.org/1475.650172022-01-14T13:23:02.489162Z"Chen, Min-Hong"https://www.zbmath.org/authors/?q=ai:chen.minhong"Wu, Qing-Biao"https://www.zbmath.org/authors/?q=ai:wu.qingbiao"Gao, Qin"https://www.zbmath.org/authors/?q=ai:gao.qin"Lin, Rong-Fei"https://www.zbmath.org/authors/?q=ai:lin.rongfeiSummary: Inner-outer iterative methods for large sparse non-Hermitian nonlinear systems are considered. Using the ideas of modified generalised Hermitian and skew Hermitian methods and double-parameter GHSS method, we develop a double-parameter modified generalised Hermitian and skew Hermitian method (DMGHSS) for linear non-Hermitian systems. Using this method as the inner iterations and the modified Newton method as the outer iterations, we introduce modified Newton-DMGHSS methods for large sparse non-Hermitian nonlinear systems. The convergence of the methods is studied. Numerical results demonstrate the efficacy of the methods.An analysis of diagonal and incomplete Cholesky preconditioners for singularly perturbed problems on layer-adapted mesheshttps://www.zbmath.org/1475.650182022-01-14T13:23:02.489162Z"Nhan, Thái Anh"https://www.zbmath.org/authors/?q=ai:nhan.thai-anh"Madden, Niall"https://www.zbmath.org/authors/?q=ai:madden.niallSummary: We investigate the solution of linear systems of equations that arise when singularly perturbed reaction-diffusion partial differential equations are solved using a standard finite difference method on layer adapted grids. It is known that there are difficulties in solving such systems by direct methods when the perturbation parameter, \( \varepsilon \), is small [\textit{S. Maclachlan} and \textit{N. Madden}, SIAM J. Sci. Comput. 35, No. 5, A2225--A2254 (2013; Zbl 1290.65097)]. Therefore, iterative methods are natural choices. However, we show that, in two dimensions, the condition number of the coefficient matrix grows unboundedly when \(\varepsilon\) tends to zero, and so unpreconditioned iterative schemes, such as the conjugate gradient algorithm, perform poorly with respect to \(\varepsilon\). We provide a careful analysis of diagonal and incomplete Cholesky preconditioning methods, and show that the condition number of the preconditioned linear system is independent of the perturbation parameter. We demonstrate numerically the surprising fact that these schemes are more efficient when \(\varepsilon\) is small, than when \(\varepsilon\) is \(\mathcal{O}(1)\). Furthermore, our analysis shows that when the singularly perturbed problem features no corner layers, an incomplete Cholesky preconditioner performs extremely well when \(\varepsilon \ll 1\). We provide numerical evidence that our findings extend to three-dimensional problems.A multi-power and multi-splitting inner-outer iteration for PageRank computationhttps://www.zbmath.org/1475.650192022-01-14T13:23:02.489162Z"Pu, Bing-Yuan"https://www.zbmath.org/authors/?q=ai:pu.bingyuan"Wen, Chun"https://www.zbmath.org/authors/?q=ai:wen.chun"Hu, Qian-Ying"https://www.zbmath.org/authors/?q=ai:hu.qianyingSummary: As an effective and possible method for computing PageRank problem, the inner-outer (IO) iteration has attracted wide interest in the past few years since it was first proposed by \textit{D. F. Gleich} et al. [SIAM J. Sci. Comput. 32, No. 1, 349--371 (2010; Zbl 1209.65043)]. In this paper, we present a variant of the IO iteration, which is based on multi-step power and multi-step splitting and is denoted by MPMIO. The description and convergence are discussed in detail. Numerical examples are given to illustrate the effectiveness of the proposed method.Note on analytic solutions of a class of extended constrained matrix maximization problemshttps://www.zbmath.org/1475.650202022-01-14T13:23:02.489162Z"Xu, Wei-wei"https://www.zbmath.org/authors/?q=ai:xu.wei-weiSummary: This note is devoted to dealing with flaws that existed in a recent paper [\textit{W.-w. Xu} et al., ibid. 372, Article ID 124989, 8 p. (2020; Zbl 1433.65060)]. We give corrections of the original (3.9) and (3.12) in Theorem 3.2 of [loc. cit.].Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrixhttps://www.zbmath.org/1475.650212022-01-14T13:23:02.489162Z"Li, Fan-Liang"https://www.zbmath.org/authors/?q=ai:li.fanliangSummary: Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.On the regularization effect of stochastic gradient descent applied to least-squareshttps://www.zbmath.org/1475.650222022-01-14T13:23:02.489162Z"Steinerberger, Stefan"https://www.zbmath.org/authors/?q=ai:steinerberger.stefanSummary: We study the behavior of the stochastic gradient descent method applied to \(\|Ax -b \|_2^2 \rightarrow \min\) for invertible matrices \(A \in \mathbb{R}^{n \times n} \). We show that there is an explicit constant \(c_A\) depending (mildly) on \(A\) such that
\[
\mathbb{E}\left\| Ax_{k+1}-b\right\|^2_2 \leq \left(1 + \frac{c_A}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_2 - \frac{2}{\|A\|_F^2} \left\|A^T A (x_k - x)\right\|^2_2.
\]
This is a curious inequality since the last term involves one additional matrix multiplication applied to the error \(x_k - x\) compared to the remaining terms: if the projection of \(x_k - x\) onto the subspace of singular vectors corresponding to large singular values is large, then the stochastic gradient descent method leads to a fast regularization. For symmetric matrices, this inequality has an extension to higher-order Sobolev spaces. This explains a (known) regularization phenomenon: an energy cascade from large singular values to small singular values acts as a regularizer.Numerical method for solving inverse source problem for Poisson equationhttps://www.zbmath.org/1475.650232022-01-14T13:23:02.489162Z"Benyoucef, Abir"https://www.zbmath.org/authors/?q=ai:benyoucef.abir"Alem, Leïla"https://www.zbmath.org/authors/?q=ai:alem.leila"Chorfi, Lahcène"https://www.zbmath.org/authors/?q=ai:chorfi.lahceneA simplified L-curve method as error estimatorhttps://www.zbmath.org/1475.650242022-01-14T13:23:02.489162Z"Kindermann, Stefan"https://www.zbmath.org/authors/?q=ai:kindermann.stefan"Raik, Kemal"https://www.zbmath.org/authors/?q=ai:raik.kemalSummary: The L-curve method is a well-known heuristic method for choosing the regularization parameter for ill-posed problems by selecting it according to the maximal curvature of the L-curve. In this article, we propose a simplified version that replaces the curvature essentially by the derivative of the parameterization on the \(y\)-axis. This method shows a similar behaviour to the original L-curve method, but unlike the latter, it may serve as an error estimator under typical conditions. Thus, we can accordingly prove convergence for the simplified L-curve method.Computing the matrix fractional power with the double exponential formulahttps://www.zbmath.org/1475.650252022-01-14T13:23:02.489162Z"Tatsuoka, Fuminori"https://www.zbmath.org/authors/?q=ai:tatsuoka.fuminori"Sogabe, Tomohiro"https://www.zbmath.org/authors/?q=ai:sogabe.tomohiro"Miyatake, Yuto"https://www.zbmath.org/authors/?q=ai:miyatake.yuto"Kemmochi, Tomoya"https://www.zbmath.org/authors/?q=ai:kemmochi.tomoya"Zhang, Shao-Liang"https://www.zbmath.org/authors/?q=ai:zhang.shao-liangSummary: Two quadrature-based algorithms for computing the matrix fractional power \(A^\alpha\) are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed into an infinite interval. Therefore, it is necessary to truncate the infinite interval to an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of \(A^\alpha \), is proposed. Then, two algorithms are presented-one where \(A^\alpha\) is computed with a fixed number of abscissa points and one with \(A^\alpha\) computed adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than Gaussian quadrature when \(A\) is ill-conditioned and \(\alpha\) is a non-unit fraction. Numerical results show that our algorithms achieve the required accuracy and are faster than other algorithms in several situations.A stable matrix version of the fast multipole method: stabilization strategies and exampleshttps://www.zbmath.org/1475.650262022-01-14T13:23:02.489162Z"Cai, Difeng"https://www.zbmath.org/authors/?q=ai:cai.difeng"Xia, Jianlin"https://www.zbmath.org/authors/?q=ai:xia.jianlinSummary: The fast multipole method (FMM) is an efficient method for evaluating matrix-vector products related to certain discretized kernel functions. The method involves an underlying FMM matrix given by a sequence of smaller matrices (called generators for convenience). Although there has been extensive work in designing and applying FMM techniques, the stability of the FMM and the stable FMM matrix factorization have rarely been studied. In this work, we propose techniques that lead to stable operations with FMM matrices. One key objective is to give stabilization strategies that can be used to provide low-rank approximations and translation relations in the FMM satisfying some stability requirements. The standard Taylor expansions used in FMMs yield basis generators susceptible to instability. Here, we introduce some scaling factors to control the relevant norms of the generators and give a rigorous analysis of the bounds of the entrywise magnitudes. The second objective is to use the one-dimensional case as an example to provide an intuitive construction of FMM matrices satisfying some stability conditions and then convert an FMM matrix into a hierarchically semiseparable (HSS) form that admits stable ULV-type factorizations. This bridges the gap between the FMM and stable FMM matrix factorizations. The HSS construction is done analytically and does not require expensive algebraic compression. Relevant stability studies are given, which show that the resulting matrix forms are suitable for stable operations. Note that the essential stabilization ideas are also applicable to higher dimensions. Extensive numerical tests are given to illustrate the reliability and accuracy.Modified Newton-PHSS method for solving nonlinear systems with positive definite Jacobian matriceshttps://www.zbmath.org/1475.650272022-01-14T13:23:02.489162Z"Ariani, Dona"https://www.zbmath.org/authors/?q=ai:ariani.dona"Xiao, Xiao-Yong"https://www.zbmath.org/authors/?q=ai:xiao.xiaoyongSummary: By making use of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration as the inner solver for the modified Newton method, we establish the modified Newton-PHSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Moreover, the local convergence theorem is proved under proper conditions. Numerical results are given to show its feasibility and effectiveness. In addition, the advantages of the modified Newton-PHSS method over some other methods are shown by solving two systems of nonlinear equations.Evolutionary algorithm with multiobjective optimization technique for solving nonlinear equation systemshttps://www.zbmath.org/1475.650282022-01-14T13:23:02.489162Z"Gao, Weifeng"https://www.zbmath.org/authors/?q=ai:gao.weifeng"Luo, Yuting"https://www.zbmath.org/authors/?q=ai:luo.yuting"Xu, Jingwei"https://www.zbmath.org/authors/?q=ai:xu.jingwei"Zhu, Shengqi"https://www.zbmath.org/authors/?q=ai:zhu.shengqiSummary: The challenge of solving nonlinear equation systems is how to locate multiple optimal solutions simultaneously in a single run. To address this issue, this paper proposes a novel algorithm by combining a diversity indicator, multi-objective optimization technique, and clustering technique. Firstly, a diversity indicator is designed to maintain the diversity of the population. Then, a K-means clustering-based selection strategy is introduced to locate the promising solutions. Finally, the local search is used to accelerate the convergence of population. The experimental results on 30 nonlinear equation systems show that the proposed algorithm is better than six state-of-the-art algorithms in terms of convergence rate and success rate.Scaling of the steady-state load flow equations for multi-carrier energy systemshttps://www.zbmath.org/1475.650292022-01-14T13:23:02.489162Z"Markensteijn, A. S."https://www.zbmath.org/authors/?q=ai:markensteijn.a-s"Romate, J. E."https://www.zbmath.org/authors/?q=ai:romate.j-e"Vuik, C."https://www.zbmath.org/authors/?q=ai:vuik.cSummary: Coupling single-carrier networks (SCNs) into multi-carrier energy systems (MESs) has recently become more important. Steady-state load flow analysis of energy systems leads to a system of nonlinear equations, which is usually solved using the Newton-Raphson method (NR). Due to various physical scales within a SCN, and between different SCNs in a MES, scaling might be needed to solve the nonlinear system. In single-carrier electrical networks, per unit scaling is commonly used. However, in the gas and heat networks, various ways of scaling or no scaling are used. This paper presents a per unit system and matrix scaling for load flow models for a MES consisting of gas, electricity, and heat. The effect of scaling on NR is analyzed. A small example MES is used to demonstrate the two scaling methods. This paper shows that the per unit system and matrix scaling are equivalent, assuming infinite precision. In finite precision, the example shows that the NR iterations are slightly different for the two scaling methods. For this example, both scaling methods show the same convergence behavior of NR in finite precision.
For the entire collection see [Zbl 1471.65009].Integral semi-discrete scheme for a Kirchhoff type abstract equation with the general nonlinearityhttps://www.zbmath.org/1475.650302022-01-14T13:23:02.489162Z"Rogava, J."https://www.zbmath.org/authors/?q=ai:rogava.jemal-l"Tsiklauri, M."https://www.zbmath.org/authors/?q=ai:tsiklauri.mikheilSummary: Cauchy problem for a Kirchhoff-type abstract equation is considered with the general nonlinearity and self-adjoint positive definite operator, which is more than or equal to the square of the operator in the nonlinear term. Kirchhoff type equation for a beam represents a particular case of this equation. For the stated problem, the semi-discrete scheme is constructed, where for approximation of the term containing the gradient, the integral averaging is used. Stability of the scheme is proved and the error of the approximate solution is estimated.High order accuracy splitting formulas for cosine operator function and their applicationshttps://www.zbmath.org/1475.650312022-01-14T13:23:02.489162Z"Rogava, J."https://www.zbmath.org/authors/?q=ai:rogava.jemal-l"Tsiklauri, M."https://www.zbmath.org/authors/?q=ai:tsiklauri.mikheilSummary: In the present work the high order accuracy rational splitting for cosine operator function is constructed. On the basis of this formula, the fourth order of accuracy decomposition scheme for homogeneous abstract hyperbolic equation with operator \(A\) is constructed. This operator is a self-adjoint, positive definite operator and is represented as a sum of the same type operators. Error of approximate solution is estimated. In the work a method for constructing any order accuracy splitting formula for cosine operator function is also introduced.Convergence analysis for single point Newton-type iterative schemeshttps://www.zbmath.org/1475.650322022-01-14T13:23:02.489162Z"Argyros, Ioannis K."https://www.zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"George, Santhosh"https://www.zbmath.org/authors/?q=ai:george.santhoshSummary: The aim of this article is to present a convergence analysis for single point Newton-type schemes for solving equations with Banach space valued operators. The equations contain a non-differentiable part. Although the convergence conditions are very general, they are weaker than the corresponding ones in earlier works leading to a finer convergence analysis in both the local as well as the semi-local convergence analysis. Therefore, the applicability of these iterative schemes is extended.A general iterative method for split common fixed point problem and variational inclusion problemhttps://www.zbmath.org/1475.650332022-01-14T13:23:02.489162Z"Eslamian, Mohammad"https://www.zbmath.org/authors/?q=ai:eslamian.mohammadSummary: In this paper we consider a new general iterative method for solving the split common fixed point problem and variational inclusion problem. It entails finding a point which belongs to the set of common fixed points of a finite family of demimetric mappings and the common zero point set of a family of monotone operators in a Hilbert space such that its image under a linear transformation belongs to the set of fixed points of a demimetric mapping in a uniformly convex and smooth Banach space in the image space. Strong convergence theorem is established under some suitable conditions. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.Using majorizing sequences for the semi-local convergence of a high-order and multipoint iterative method along with stability analysishttps://www.zbmath.org/1475.650342022-01-14T13:23:02.489162Z"Moccari, Mandana"https://www.zbmath.org/authors/?q=ai:moccari.mandana"Lotfi, Taher"https://www.zbmath.org/authors/?q=ai:lotfi.taherSummary: This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, \(k\)-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to clarify the given theory.Derivative free regularization method for nonlinear ill-posed equations in Hilbert scaleshttps://www.zbmath.org/1475.650352022-01-14T13:23:02.489162Z"George, Santhosh"https://www.zbmath.org/authors/?q=ai:george.santhosh"Kanagaraj, K."https://www.zbmath.org/authors/?q=ai:kanagaraj.kSummary: In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also, we consider the adaptive parameter choice strategy proposed by \textit{S. Pereverzev} and \textit{E. Schock} [SIAM J. Numer. Anal. 43, No. 5, 2060--2076 (2005; Zbl 1103.65058)] for choosing the regularization parameter. Finally, we apply the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.Low-rank matrix completion using nuclear norm minimization and facial reductionhttps://www.zbmath.org/1475.650362022-01-14T13:23:02.489162Z"Huang, Shimeng"https://www.zbmath.org/authors/?q=ai:huang.shimeng"Wolkowicz, Henry"https://www.zbmath.org/authors/?q=ai:wolkowicz.henrySummary: Minimization of the nuclear norm, \(\mathbf{NNM}\), is often used as a surrogate (convex relaxation) for finding the minimum rank completion (recovery) of a \textit{partial matrix}. The minimum nuclear norm problem can be solved as a trace minimization semidefinite programming problem, \(\mathbf{SDP}\). Interior point algorithms are the current methods of choice for this class of problems. This means that it is difficult to: solve large scale problems; exploit sparsity; and get high accuracy solutions. The \(\mathbf{SDP}\) and its dual are regular in the sense that they both satisfy strict feasibility. In this paper we take advantage of the structure of low rank solutions in the \(\mathbf{SDP}\) embedding. We show that even though strict feasibility holds, the facial reduction framework used for problems where strict feasibility fails can be successfully applied to \textit{generically} obtain a proper face that contains all minimum low rank solutions for the original completion problem. This can dramatically reduce the size of the final \(\mathbf{NNM}\) problem, while simultaneously guaranteeing a low-rank solution. This can be compared to identifying part of the active set in general nonlinear programming problems. In the case that the graph of the sampled matrix has sufficient bicliques, we get a low rank solution independent of any nuclear norm minimization. We include numerical tests for both exact and noisy cases. We illustrate that our approach yields lower ranks and higher accuracy than obtained from just the \(\mathbf{NNM}\) approach.Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problemshttps://www.zbmath.org/1475.650372022-01-14T13:23:02.489162Z"Attia, Ahmed"https://www.zbmath.org/authors/?q=ai:attia.ahmed-f|attia.ahmed-hamed"Alexanderian, Alen"https://www.zbmath.org/authors/?q=ai:alexanderian.alen"Saibaba, Arvind K."https://www.zbmath.org/authors/?q=ai:saibaba.arvind-krishnaCG variants for general-form regularization with an application to low-field MRIhttps://www.zbmath.org/1475.650382022-01-14T13:23:02.489162Z"de Leeuw den Bouter, M. L."https://www.zbmath.org/authors/?q=ai:de-leeuw-den-bouter.m-l"van Gijzen, M. B."https://www.zbmath.org/authors/?q=ai:van-gijzen.martin-b"Remis, R. F."https://www.zbmath.org/authors/?q=ai:remis.robert-f|remis.rob-fSummary: In an earlier paper, we generalized the CGME (Conjugate Gradient Minimal Error) algorithm to the \(\ell_2\)-regularized weighted least-squares problem. Here, we use this Generalized CGME method to reconstruct images from actual signals measured using a low-field MRI scanner. We analyze the convergence of both GCGME and the classical Generalized Conjugate Gradient Least Squares (GCGLS) method for the simple case when a Laplace operator is used as a regularizer and indicate when GCGME is to be preferred in terms of convergence speed. We also consider a more complicated \(\ell_1\)-penalty in a compressed sensing framework.
For the entire collection see [Zbl 1471.65009].Augmented Lagrangians, box constrained QP and extensionshttps://www.zbmath.org/1475.650392022-01-14T13:23:02.489162Z"Fletcher, Roger"https://www.zbmath.org/authors/?q=ai:fletcher.rogerSummary: A new method is described for the solution of the box constrained convex quadratic programming (QP) problems, based on modifications of Rockafellar's augmented Lagrangian function. A reduced function is defined by eliminating the multiplier parameter vector in a novel way. A minimizer of this function is shown to solve box-constrained QP. The function is convex and \(C^1\) piecewise quadratic, and under mild conditions it is strictly convex. Thus a Newton iteration is readily devised and easily implemented. The same Newton iteration is already well known and frequently used as a consequence of semi-smooth Newton theory, but the development here, especially the globalization scheme, is thought to be new. A nonmonotonic line search can be used to guarantee finite convergence, but often this is not required. Some important issues in regard to the line search are discussed. Successful numerical results are presented on a wide selection of large dimension test problems from various areas of application and from the CUTE test set. Some very large problems are solved in a reasonable time. Further, this method is extended to solve box constrained QPs that include a few linear equations. Encouraging numerical evidence is presented on a range of practical problems of up to 10\({}^8\) variables from various sources. Issues relating to guaranteed convergence are discussed.A limited memory \(q\)-BFGS algorithm for unconstrained optimization problemshttps://www.zbmath.org/1475.650402022-01-14T13:23:02.489162Z"Lai, Kin Keung"https://www.zbmath.org/authors/?q=ai:lai.kinkeung"Mishra, Shashi Kant"https://www.zbmath.org/authors/?q=ai:mishra.shashi-kant"Panda, Geetanjali"https://www.zbmath.org/authors/?q=ai:panda.geetanjali"Chakraborty, Suvra Kanti"https://www.zbmath.org/authors/?q=ai:chakraborty.suvra-kanti"Samei, Mohammad Esmael"https://www.zbmath.org/authors/?q=ai:samei.mohammad-esmael"Ram, Bhagwat"https://www.zbmath.org/authors/?q=ai:ram.bhagwatSummary: A limited memory \(q\)-BFGS (Broyden-Fletcher-Goldfarb-Shanno) method is presented for solving unconstrained optimization problems. It is derived from a modified BFGS-type update using \(q\)-derivative (quantum derivative). The use of Jackson's derivative is an effective mechanism for escaping from local minima. The \(q\)-gradient method is complemented to generate the parameter \(q\) for computing the step length in such a way that the search process gradually shifts from global in the beginning to almost local search in the end. Further, the global convergence is established under Armijo-Wolfe conditions even if the objective function is not convex. The numerical experiments show that proposed method is potentially efficient.Two nonmonotone trust region algorithms based on an improved Newton methodhttps://www.zbmath.org/1475.650412022-01-14T13:23:02.489162Z"Niri, T. Dehghan"https://www.zbmath.org/authors/?q=ai:niri.t-dehghan"Heydari, M."https://www.zbmath.org/authors/?q=ai:heydari.mehdi|heydari.mohammad-mehdi|heydari.mohammad-hossien|heydari.maysam|heydari.majeed|heydari.masoud|heydari.mohammad-taghi|heydari.maryam|heydari.mohammadhossein|heydari.mahdi|heydari.mojgan"Hosseini, M. M."https://www.zbmath.org/authors/?q=ai:hosseini.mohammad-mehdi|hosseini.mohamad-mehdi|hosseini.mohammad-mahdiSummary: In this paper, we present two improved regularized Newton methods for minimizing nonconvex functions with the trust region method. The fundamental idea of this method is based on the combination of nonmonotone Armijo-type line search proposed by \textit{N.-Z. Gu} and \textit{J.-T. Mo} [Comput. Math. Appl. 55, No. 9, 2158--2172 (2008; Zbl 1183.90387)] and the traditional trust region method. Under some standard assumptions, we analyze the global convergence property for the new methods. Primary numerical results are reported. The obtained results confirm the higher efficiency of the modified algorithms compared to the other presented algorithms.A sharp convergence rate for a model equation of the asynchronous stochastic gradient descenthttps://www.zbmath.org/1475.650422022-01-14T13:23:02.489162Z"Zhu, Yuhua"https://www.zbmath.org/authors/?q=ai:zhu.yuhua"Ying, Lexing"https://www.zbmath.org/authors/?q=ai:ying.lexingSummary: We give a sharp convergence rate for the asynchronous stochastic gradient descent (ASGD) algorithms when the loss function is a perturbed quadratic function based on the stochastic modified equations introduced in [\textit{J. An} et al., Inf. Inference 9, No. 4, 851--873 (2020; Zbl 07382208), Preprint \url{arXiv:1805.08244}]. We prove that when the number of local workers is larger than the expected staleness, then ASGD is more efficient than stochastic gradient descent. Our theoretical result also suggests that longer delays result in slower convergence rate. Besides, the learning rate cannot be smaller than a threshold inversely proportional to the expected staleness.Fixed-point algorithms for frequency estimation and structured low rank approximationhttps://www.zbmath.org/1475.650432022-01-14T13:23:02.489162Z"Andersson, Fredrik"https://www.zbmath.org/authors/?q=ai:andersson.fredrik-k"Carlsson, Marcus"https://www.zbmath.org/authors/?q=ai:carlsson.marcusSummary: We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, i.e., frequency estimation. For the basic formulation of the fixed-point algorithm we show that it converges to the solution of a related minimization problem, namely the one obtained by replacing the original objective function with its convex envelope and keeping the structured matrix constraint unchanged.
It often happens that this solution agrees with the solution to the original minimization problem, and we provide a simple criterion for when this is true. We also provide more general fixed-point algorithms that can be used to treat the problems of making weighted approximations by sums of exponentials given equally or unequally spaced sampling. We apply the method to the case of missing data, although the above mentioned convergence results do not hold in this case. However, it turns out that the method often gives perfect reconstruction (up to machine precision) in such cases. We also discuss multidimensional extensions, and illustrate how the proposed algorithms can be used to recover sums of exponentials in several variables, but when samples are available only along a curve.Galerkin spectral methods for an elliptic optimal control problem with \(L^2\)-norm state constrainthttps://www.zbmath.org/1475.650442022-01-14T13:23:02.489162Z"Lin, Xiuxiu"https://www.zbmath.org/authors/?q=ai:lin.xiuxiu"Chen, Yanping"https://www.zbmath.org/authors/?q=ai:chen.yanping.2"Huang, Yunqing"https://www.zbmath.org/authors/?q=ai:huang.yunqingSummary: In this paper, an optimal control problem governed by elliptic equations with \(L^2\)-norm constraint for state variable is developed. Firstly, the optimality conditions of the optimal control problem are derived, and the optimal control problem is approximated by the Galerkin spectral methods. Similarly, the optimality conditions of the discrete problem are also obtained. Then, some important lemmas are proved to obtain \textit{a priori} error estimates of the coupled state and control approximation rigorously. Moreover, \textit{a posteriori} error estimates are also established for the optimal control problem carefully. Finally, based on the projection gradient algorithm, some numerical experiments are presented to confirm our analytical findings. It is proved that the exponential convergence rate can be achieved.Efficient 3D shape registration by using distance maps and stochastic gradient descent methodhttps://www.zbmath.org/1475.650452022-01-14T13:23:02.489162Z"Okock, Polycarp Omondi"https://www.zbmath.org/authors/?q=ai:okock.polycarp-omondi"Urbán, Jozef"https://www.zbmath.org/authors/?q=ai:urban.jozef"Mikula, Karol"https://www.zbmath.org/authors/?q=ai:mikula.karolSummary: This paper presents an efficient 3D shape registration by using distance maps and stochastic gradient descent method. The proposed algorithm aims to find the optimal affine transformation parameters (translation, scaling and rotation) that maps two distance maps to each other. These distance maps represent the shapes as an interface and we apply level sets methods to calculate the signed distance to these interfaces. To maximize the similarity between the two distance maps, we apply sum of squared difference (SSD) optimization and gradient descent methods to minimize it. To address the shortcomings of the standard gradient descent method, i.e., many iterations to compute the minimum, we implemented the stochastic gradient descent method. The outcome of these two methods are compared to show the advantages of using stochastic gradient descent method. In addition, we implement computational optimization's such as parallelization to speed up the registration process.EBDF-type methods based on the linear barycentric rational interpolants for stiff IVPshttps://www.zbmath.org/1475.650462022-01-14T13:23:02.489162Z"Esmaeelzadeh, Zahra"https://www.zbmath.org/authors/?q=ai:esmaeelzadeh.zahra"Abdi, Ali"https://www.zbmath.org/authors/?q=ai:abdi.ali"Hojjati, Gholamreza"https://www.zbmath.org/authors/?q=ai:hojjati.gholamrezaSummary: Linear barycentric rational interpolants, instead of customary polynomial interpolants, have been recently used to design the efficient numerical integrators for ODEs. In this way, the BDF-type methods based on these interpolants have been introduced as a general class of the methods in this type with better accuracy and stability properties. In this paper, we introduce an extension of them equipped to super-future point technique. The order of convergence and stability of the proposed methods are discussed and confirmed by some given numerical experiments.A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equationhttps://www.zbmath.org/1475.650472022-01-14T13:23:02.489162Z"Obaidat, Saleem"https://www.zbmath.org/authors/?q=ai:obaidat.saleem-a"Butt, Rizwan"https://www.zbmath.org/authors/?q=ai:butt.rizwanSummary: In this article, we have developed an implicit symmetric four-step method of sixth algebraic order with vanished phase-lag and its first derivative. The error and stability analysis of this method are investigated, and its efficiency is tested by solving efficiently the one-dimensional time-independent Schrödinger's equation. The method performance is compared with other methods in the literature. It is found that for this problem the new method performs better than the compared methods.Modified iteration method for numerical solution of nonlinear differential equations arising in science and engineeringhttps://www.zbmath.org/1475.650482022-01-14T13:23:02.489162Z"Pathak, Maheshwar"https://www.zbmath.org/authors/?q=ai:pathak.maheshwar"Joshi, Pratibha"https://www.zbmath.org/authors/?q=ai:joshi.pratibhaPreconditioners for all-at-once system from the fractional mobile/immobile advection-diffusion modelhttps://www.zbmath.org/1475.650492022-01-14T13:23:02.489162Z"Zhao, Yong-Liang"https://www.zbmath.org/authors/?q=ai:zhao.yongliang"Gu, Xian-Ming"https://www.zbmath.org/authors/?q=ai:gu.xian-ming"Li, Meng"https://www.zbmath.org/authors/?q=ai:li.meng"Jian, Huan-Yan"https://www.zbmath.org/authors/?q=ai:jian.huanyanSummary: The all-at-once system arising from fractional mobile/immobile advection-diffusion equations is studied. Firstly, the finite difference method with \(L1\) formula is employed to discretize it. The resulting implicit scheme is a time-stepping scheme, which is not suitable for parallel computing. Based on this scheme, an all-at-once system is established, which will be suitable for parallel computing. Secondly, according to the block lower triangular Toeplitz structure of the all-at-once system, both the block bi-diagonal preconditioner and block stair preconditioner are designed. Finally, numerical examples are presented to show the performances of the proposed preconditioners.Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methodshttps://www.zbmath.org/1475.650502022-01-14T13:23:02.489162Z"Gottlieb, Sigal"https://www.zbmath.org/authors/?q=ai:gottlieb.sigal"Grant, Zachary J."https://www.zbmath.org/authors/?q=ai:grant.zachary-j"Isherwood, Leah"https://www.zbmath.org/authors/?q=ai:isherwood.leahSummary: Strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-innerproduct stability properties to be satisfied. Unlike the case for \(L_2\) linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor Runge-Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge-Kutta methods where the transformed problem is evolved with an explicit SSP Runge-Kutta method with non-decreasing abscissas was recently established. However, these methods typically have smaller SSP coefficients (and therefore a smaller allowable time-step) than the optimal SSP Runge-Kutta methods, which often have some decreasing abscissas. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge-Kutta methods where the Runge-Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some cases the integrating factor approach with explicit SSP Runge-Kutta methods with non-decreasing abscissas performs nearly as well, if not better, than with explicit SSP Runge-Kutta methods with downwinding. In conclusion, while the downwinding approach can be rigorously shown to guarantee the SSP property for a larger timestep, in practice using the integrating factor approach by including downwinding as needed with optimal explicit SSP Runge-Kutta methods does not necessarily provide significant benefit over using explicit SSP Runge-Kutta methods with non-decreasing abscissas.Optimized low-dispersion and low-dissipation two-derivative Runge-Kutta method for wave equationshttps://www.zbmath.org/1475.650512022-01-14T13:23:02.489162Z"Krivovichev, Gerasim V."https://www.zbmath.org/authors/?q=ai:krivovichev.gerasim-vSummary: The paper is devoted to the optimization of the explicit two-derivative sixth-order Runge-Kutta method in order to obtain low dissipation and dispersion errors. The method is dependent on two free parameters, used for the optimization. The optimized method demonstrates the lowest dispersion error in comparison with other widely-used high-order Runge-Kutta methods for hyperbolic problems. The efficiency of the method is demonstrated on the solutions of problems for five linear and nonlinear partial differential equations by the method-of-lines. Spatial derivatives are discretized by finite differences and Petrov-Galerkin approximations. The work-precision and error -- CPU time plots, as the dependence of CPU time on space grid resolution, are considered. Also, the optimized method compared with other two-derivative methods. In most examples, the optimized method has better properties, especially for the cases of computations with adaptive stepsize control at high accuracy. The lowest CPU time takes place for the optimized method, especially in the cases of fine space grids.Well-balanced and asymptotic preserving IMEX-peer methodshttps://www.zbmath.org/1475.650522022-01-14T13:23:02.489162Z"Schneider, Moritz"https://www.zbmath.org/authors/?q=ai:schneider.moritz"Lang, Jens"https://www.zbmath.org/authors/?q=ai:lang.jensSummary: Peer methods are a comprehensive class of time integrators offering numerous degrees of freedom in their coefficient matrices that can be used to ensure advantageous properties, e.g. A-stability or super-convergence. In this paper, we show that implicit-explicit (IMEX) Peer methods are well-balanced and asymptotic preserving by construction without additional constraints on the coefficients. These properties are relevant when solving (the space discretisation of) hyperbolic systems of balance laws, for example. Numerical examples confirm the theoretical results and illustrate the potential of IMEX-Peer methods.
For the entire collection see [Zbl 1471.65009].A unified study on superconvergence analysis of Galerkin FEM for singularly perturbed systems of multiscale naturehttps://www.zbmath.org/1475.650532022-01-14T13:23:02.489162Z"Singh, Maneesh Kumar"https://www.zbmath.org/authors/?q=ai:singh.maneesh-kumar"Singh, Gautam"https://www.zbmath.org/authors/?q=ai:singh.gautam-b"Natesan, Srinivasan"https://www.zbmath.org/authors/?q=ai:natesan.srinivasanSummary: We discuss the superconvergence analysis of the Galerkin finite element method for the singularly perturbed coupled system of both reaction-diffusion and convection-diffusion types. The superconvergence study is carried out by using linear finite element, and it is shown to be second-order (up to a logarithmic factor) uniformly convergent in the suitable discrete energy norm. We have conducted some numerical experiments for the system of reaction-diffusion and system of convection-diffusion models, which validate the theoretical results.Numerical solution for multi-dimensional Riesz fractional nonlinear reaction-diffusion equation by exponential Runge-Kutta methodhttps://www.zbmath.org/1475.650542022-01-14T13:23:02.489162Z"Zhang, Lu"https://www.zbmath.org/authors/?q=ai:zhang.lu"Sun, Hai-Wei"https://www.zbmath.org/authors/?q=ai:sun.haiweiSummary: A spatial discretization of the Riesz fractional nonlinear reaction-diffusion equation by the fractional centered difference scheme leads to a system of ordinary differential equations, in which the resulting coefficient matrix possesses the symmetric block Toeplitz structure. An exponential Runge-Kutta method is employed to solve such a system of ordinary differential equations. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the symmetric positive definiteness of the coefficient matrix guarantees the fast approximation by the shift-invert Lanczos method. Numerical results are given to demonstrate the efficiency of the proposed method.Wavelet B-splines bases on the interval for solving boundary value problemshttps://www.zbmath.org/1475.650552022-01-14T13:23:02.489162Z"Calderón, Lucila"https://www.zbmath.org/authors/?q=ai:calderon.lucila"Martín, María T."https://www.zbmath.org/authors/?q=ai:martin.maria-teresa"Vampa, Victoria"https://www.zbmath.org/authors/?q=ai:vampa.victoriaSummary: The use of multiresolution techniques and wavelets has become increa-singly popular in the development of numerical schemes for the solution of differential equations. Wavelet's properties make them useful for developing hierarchical solutions to many engineering problems. They are well localized, oscillatory functions which provide a basis of the space of functions on the real line. We show the construction of derivative-orthogonal B-spline wavelets on the interval which have simple structure and provide sparse and well-conditioned matrices when they are used for solving differential equations with the wavelet-Galerkin method.
For the entire collection see [Zbl 1464.65005].Comparative study on difference schemes for singularly perturbed boundary turning point problems with Robin boundary conditionshttps://www.zbmath.org/1475.650562022-01-14T13:23:02.489162Z"Janani Jayalakshmi, G."https://www.zbmath.org/authors/?q=ai:janani-jayalakshmi.g"Tamilselvan, Ayyadurai"https://www.zbmath.org/authors/?q=ai:tamilselvan.ayyaduraiSummary: In this paper, singularly perturbed convection diffusion boundary turning point problem with Robin boundary conditions is considered. Upwind difference and hybrid difference schemes on Shishkin mesh are proposed to solve this problem. The error bounds for the numerical approximation are established for both the schemes. Numerical results are compared for both the schemes to illustrate the corresponding error bounds established.A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shiftshttps://www.zbmath.org/1475.650572022-01-14T13:23:02.489162Z"Ranjan, Rakesh"https://www.zbmath.org/authors/?q=ai:ranjan.rakesh"Prasad, Hari Shankar"https://www.zbmath.org/authors/?q=ai:prasad.hari-shankarSummary: A boundary-value problem for singularly perturbed second order differential-difference equation with both the negative (delay) and positive (advance) shifts is considered and a new exponentially fitted three term finite difference scheme is proposed for the numerical approximation of its solution. An approximate version of the considered problem is constructed by the use of Taylor's series expansion procedure first, and then, a new three term recurrence relationship is derived by applying the finite difference approximation techniques on it. Furthermore, an exponential fitting factor is introduced in the derived new scheme using the theory of singular perturbations and an efficient 'discrete invariant imbedding algorithm' is used to solve the resulting tridiagonal system of equations. Convergence of the method is analyzed. Numerical experiments performed on several test examples; presentation of the computational results in terms of maximum absolute errors and the comparison of the results with some other method results, show the applicability and efficiency of the proposed scheme. Graphs plotted for the solution with varying shifts show the effect of small shifts on the boundary layer behavior of the solution. Both the theoretical and numerical analysis of the method reveals that the method is able to produce uniformly convergent solutions with quadratic convergence rate.Study of the NIPG method for two-parameter singular perturbation problems on several layer adapted gridshttps://www.zbmath.org/1475.650582022-01-14T13:23:02.489162Z"Singh, Gautam"https://www.zbmath.org/authors/?q=ai:singh.gautam-b"Natesan, Srinivasan"https://www.zbmath.org/authors/?q=ai:natesan.srinivasanSummary: In this paper, we apply the non-symmetric interior penalty Galerkin (NIPG) method to obtain the numerical solution of two-parameter singularly perturbed convection-diffusion-reaction boundary-value problems. In order to discretize the domain, here, we use the layer-adapted piecewise-uniform Shishkin mesh, the Bakhvalov mesh and the exponentially-graded mesh. We establish a superconvergence result of the NIPG method, that is, the proposed method is parameter-uniformly convergent with the order almost \((k+1)\) on the Shishkin mesh and \((k+1)\) on the Bakhvalov mesh and on the exponentially graded mesh in the energy norm, where \(k\) is the order of the polynomials. Numerical results comparing the three different types of meshes are presented at the end of the article supporting the theoretical error estimates.A note on the convergence of fuzzy transformed finite difference methodshttps://www.zbmath.org/1475.650592022-01-14T13:23:02.489162Z"Verma, Amit K."https://www.zbmath.org/authors/?q=ai:verma.amit-kumar"Kayenat, Sheerin"https://www.zbmath.org/authors/?q=ai:kayenat.sheerin"Jha, Gopal Jee"https://www.zbmath.org/authors/?q=ai:jha.gopal-jeeSummary: In this paper, we develop numerical methods based on the fuzzy transform methods (FTMs). In this approach we apply fuzzy transforms on discrete version of the derivatives and use it to derive FTMs. We also establish convergence of the proposed FTMs. To test the efficiency of the proposed FTMs, we apply the FTM schemes on the second order nonlinear singular boundary value problems and fourth order BVPs. We allow the source term of the differential equation to have jump discontinuity and study the effect of jump on FTMs and finite difference methods. The work shows that FTMs are better for both class of BVPs considered in this paper, having singularity, nonlinearity and jump discontinuity.SDFEM for singularly perturbed boundary-value problems with two parametershttps://www.zbmath.org/1475.650602022-01-14T13:23:02.489162Z"Avijit, D."https://www.zbmath.org/authors/?q=ai:avijit.d"Natesan, S."https://www.zbmath.org/authors/?q=ai:natesan.srinivasanSummary: In this article, we study the convergence of the streamline-diffusion finite element method (SDFEM) for singularly perturbed boundary-value problem with two parameters. We prove that the SDFEM is uniformly convergent in the discrete SD-norm, of second-order on the Shishkin mesh and almost second-order on the Duran-Shishkin mesh. Numerical results are presented to support the theoretical error estimates.Numerical treatment of Gray-Scott model with operator splitting methodhttps://www.zbmath.org/1475.650612022-01-14T13:23:02.489162Z"Karaagac, Berat"https://www.zbmath.org/authors/?q=ai:karaagac.beratSummary: This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, ``strang splitting'' idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms \(L_2 \) and \(L_\infty \). Numerical results arising from the simulation experiments are also presented.Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference schemehttps://www.zbmath.org/1475.650622022-01-14T13:23:02.489162Z"Aghdam, Yones Esmaeelzade"https://www.zbmath.org/authors/?q=ai:aghdam.yones-esmaeelzade"Safdari, Hamid"https://www.zbmath.org/authors/?q=ai:safdari.hamid"Azari, Yaqub"https://www.zbmath.org/authors/?q=ai:azari.yaqub"Jafari, Hossein"https://www.zbmath.org/authors/?q=ai:jafari.hossein"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order \(\mathcal{O}(\delta \tau^2) \) is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature.On a new spatial discretization for a regularized 3D compressible isothermal Navier-Stokes-Cahn-Hilliard system of equations with boundary conditionshttps://www.zbmath.org/1475.650632022-01-14T13:23:02.489162Z"Balashov, Vladislav"https://www.zbmath.org/authors/?q=ai:balashov.vladislav"Zlotnik, Alexander"https://www.zbmath.org/authors/?q=ai:zlotnik.alexander-aA novel spatial finite-difference discretization is constructed for a regularized 3D Navier-Stokes-Cahn-Hilliard system. Such a system describes flows of a viscous compressible isothermal two-component two-phase fluid with surface effects, where the potential body force is taken into consideration. In the discretization procedure, the main sought functions are defined on the same mesh, and an original approximation of the solid wall boundary conditions is proposed. This discretization has the total energy dissipativity property, which eliminates spurious currents. The discrete total mass and component mass conservation laws hold, and the discretization is also well-balanced for the equilibrium solutions. The non-convex part of the Helmholtz free energy is taken in a special logarithmic form, in order to guarantee that the concentration remains within a physically meaningful interval. The results of numerical 3D simulations are presented, together with a discussion on the role of the relaxation parameter.
Reviewer: Kanakadurga Sivakumar (Chennai)Mathematical models and numerical methods for spinor Bose-Einstein condensates.https://www.zbmath.org/1475.650642022-01-14T13:23:02.489162Z"Bao, Weizhu"https://www.zbmath.org/authors/?q=ai:bao.weizhu"Cai, Yongyong"https://www.zbmath.org/authors/?q=ai:cai.yongyongSummary: In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-\(F\) (\(F \geq 3\)) BEC are discussed.Lax-Wendroff approximate Taylor methods with fast and optimized weighted essentially non-oscillatory reconstructionshttps://www.zbmath.org/1475.650652022-01-14T13:23:02.489162Z"Carrillo, H."https://www.zbmath.org/authors/?q=ai:carrillo.humberto"Parés, C."https://www.zbmath.org/authors/?q=ai:pares-madronal.carlos"Zorío, D."https://www.zbmath.org/authors/?q=ai:zorio.davidIn this paper, new families of shock-capturing high-order numerical methods are introduced using fast WENO (FOWENO) reconstructions and approximate Taylor methods. Reconstruction of WENO operators are based on the combination of fast smooth indicators, that coincide with the original smooth indicators in the third-order case, and the computation of optimal weights that preserve the accuracy of the reconstructions close to critical point regardless of their order. Moreover, two different implementations of the approximate Taylor methods are considered: Lax-Wendroff Approximate Taylor and Compact Approximate Taylor methods. These methods are compared between them and against standard WENO implementations for several test problems ranging from scalar linear one-dimensional problems to nonlinear systems of conservation laws in three dimensions.
Reviewer: Bülent Karasözen (Ankara)Generalized SAV approaches for gradient systemshttps://www.zbmath.org/1475.650662022-01-14T13:23:02.489162Z"Cheng, Qing"https://www.zbmath.org/authors/?q=ai:cheng.qing"Liu, Chun"https://www.zbmath.org/authors/?q=ai:liu.chun"Shen, Jie"https://www.zbmath.org/authors/?q=ai:shen.jieSummary: We propose in this paper three generalized auxiliary scalar variable (G-SAV) approaches for developing, efficient energy stable numerical schemes for gradient systems. The first two G-SAV approaches allow a range of functions in the definition of the SAV variable, furthermore, the second G-SAV approach only requires the total free energy to be bounded from below as opposed to the requirement that the nonlinear part of the free energy to be bounded from below. On the other hand, the third G-SAV approach is unconditionally energy stable with respect to the original free energy as opposed to a modified energy. Ample numerical results for various gradient systems are presented to validate the effectiveness and accuracy of the proposed G-SAV approaches.A transformed stochastic Euler scheme for multidimensional transmission PDEhttps://www.zbmath.org/1475.650672022-01-14T13:23:02.489162Z"Étoré, Pierre"https://www.zbmath.org/authors/?q=ai:etore.pierre"Martinez, Miguel"https://www.zbmath.org/authors/?q=ai:martinez.miguelSummary: In this paper we consider multi-dimensional Partial Differential Equations (PDE) of parabolic type in divergence form. The coefficient matrix of the divergence operator is assumed to be discontinuous along some smooth interface. At this interface, the solution of the PDE presents a compatibility transmission condition of its co-normal derivatives (multi-dimensional diffraction problem). We prove an existence and uniqueness result for the solution and study its properties. In particular, we provide new estimates for the partial derivatives of the solution in the classical sense. We then construct a low complexity numerical Monte Carlo stochastic Euler scheme to approximate the solution of the PDE of interest. Using the afore mentioned estimates, we prove a convergence rate for our stochastic numerical method when the initial condition belongs to some iterated domain of the divergence form operator. Finally, we compare our results to classical deterministic numerical approximations and illustrate the accuracy of our method.An efficient energy-preserving method for the two-dimensional fractional Schrödinger equationhttps://www.zbmath.org/1475.650682022-01-14T13:23:02.489162Z"Fu, Yayun"https://www.zbmath.org/authors/?q=ai:fu.yayun"Xu, Zhuangzhi"https://www.zbmath.org/authors/?q=ai:xu.zhuangzhi"Cai, Wenjun"https://www.zbmath.org/authors/?q=ai:cai.wenjun"Wang, Yushun"https://www.zbmath.org/authors/?q=ai:wang.yushunSummary: In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrödinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations.A fourth-order numerical scheme for singularly perturbed delay parabolic problem arising in population dynamicshttps://www.zbmath.org/1475.650692022-01-14T13:23:02.489162Z"Govindarao, L."https://www.zbmath.org/authors/?q=ai:govindarao.l"Mohapatra, J."https://www.zbmath.org/authors/?q=ai:mohapatra.jeet|mohapatra.jugal"Das, A."https://www.zbmath.org/authors/?q=ai:das.amrita|das.arabindo|das.anita|das.aloke-k|das.a-shoba|das.arijit|das.anjana|das.anish|das.arnab|das.ajoy-k-r|das.ananda-swarup|das.abhimanyu|das.a-g|das.anjan-kumar|das.ankush|das.atanu|das.amartya|das.amitabh|das.abhijit|das.alaka|das.ashmita|das.arpit|das.amal-k|das.arpita|das.ajit-kumar|das.anadijibam|das.aruneema|das.a-d|das.anindya-bijoy|das.abhishek|das.arghya|das.anupam|das.aditya-praksh|das.arup-kumar|das.a-n|das.amitabha|das.anjan-kr|das.ariyam|das.ajay|das.ashok-kumar|das.arun-kumar|das.arkaprovo|das.ashok-kr|das.asit-kumar|das.alok|das.avijit|das.abhik-kumar|das.amaresh|das.amiya|das.arpan|das.avinandan|das.arkabrata|kumar-das.ashish|das.anusuya|das.ashish|das.aparna|das.anil-kuman|das.asha|das.apurva-kumar|das.amit-kumar|das.ajoy-kanti|das.arindam|das.angsuman|das.aritra|das.aveek|das.anadijiban|das.antariksha|das.ajanta|das.asok-k|das.arpita.1|das.anup|das.apurba.1|das.ashis-kumar|das.ananta-lal|das.amrit|das.akshat|das.aparajita|das.atin|das.apurba|das.amarendra|das.asim-kumar|das.ajoy-kumar|das.ananya|das.anirban|das.ananga-kumar|das.anukul-chandra|das.ashoke|das.arup-ratan|das.abhiman|das.aniruddha|das.amar-k|das.a-a|das.arabinda-k|das.amlan|das.arundhatiSummary: This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the implicit Euler scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second-order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth-order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method.A fast multi grid algorithm for 2D diffeomorphic image registration modelhttps://www.zbmath.org/1475.650702022-01-14T13:23:02.489162Z"Han, Huan"https://www.zbmath.org/authors/?q=ai:han.huan"Wang, Andong"https://www.zbmath.org/authors/?q=ai:wang.andongSummary: A 2D diffeomorphic image registration model is proposed to eliminate mesh folding by the first author and \textit{Z. Wang} [SIAM J. Imaging Sci. 13, No. 3, 1240--1271 (2020; Zbl 1451.65126)]. To solve the 2D diffeomorphic model, a diffeomorphic fractional-order image registration algorithm (DFIRA for short) is proposed in Han and Wang (2020). DFIRA achieves a satisfactory image registration result but it costs too much CPU time. To accelerate DFIRA, we propose a fast multi grid algorithm for 2D diffeomorphic image registration model in this paper. This algorithm achieves a satisfactory image registration result by using only one-tenth CPU time of DFIRA. At the same time, no mesh folding occurs in proposed algorithm. Furthermore, convergence analysis of the proposed algorithm is also presented. Moreover, numerical tests are also performed to show that the proposed algorithm is competitive compared with some other algorithms.On a nonlinear energy-conserving scalar auxiliary variable (SAV) model for Riesz space-fractional hyperbolic equationshttps://www.zbmath.org/1475.650712022-01-14T13:23:02.489162Z"Hendy, Ahmed S."https://www.zbmath.org/authors/?q=ai:hendy.ahmed-s"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardoSummary: In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar auxiliary variable (SAV) technique, and the expression of the energy is obtained using the new scalar variable. The new differential system is discretized then following the SAV approach. The proposed scheme is a nonlinear implicit method which has an associated discrete energy, and we prove that the discrete model is also conservative. The present work is the first report in which the SAV method is used to design nonlinear conservative numerical method to solve a Hamiltonian space-fractional wave equations.Two linearized schemes for time fractional nonlinear wave equations with fourth-order derivativehttps://www.zbmath.org/1475.650722022-01-14T13:23:02.489162Z"Huang, Jianfei"https://www.zbmath.org/authors/?q=ai:huang.jianfei"Qiao, Zhi"https://www.zbmath.org/authors/?q=ai:qiao.zhi"Zhang, Jingna"https://www.zbmath.org/authors/?q=ai:zhang.jingna"Arshad, Sadia"https://www.zbmath.org/authors/?q=ai:arshad.sadia"Tang, Yifa"https://www.zbmath.org/authors/?q=ai:tang.yifaSummary: In this paper, two linearized schemes for time fractional nonlinear wave equations (TFNWEs) with the space fourth-order derivative are proposed and analyzed. To reduce the smoothness requirement in time, the considered TFNWEs are equivalently transformed into their partial integro-differential forms by the Riemann-Liouville integral. Then, the first scheme is constructed by using piecewise rectangular formulas in time and the fourth-order approximation in space. And, this scheme can be fast evaluated by the sum-of-exponentials technique. The second scheme is developed by using the Crank-Nicolson technique combined with the second-order convolution quadrature formula. By the energy method, the convergence and unconditional stability of the proposed schemes are proved rigorously. Finally, numerical experiments are given to support our theoretical results.A third-order WENO scheme based on exponential polynomials for Hamilton-Jacobi equationshttps://www.zbmath.org/1475.650732022-01-14T13:23:02.489162Z"Kim, Chang Ho"https://www.zbmath.org/authors/?q=ai:kim.changho"Ha, Youngsoo"https://www.zbmath.org/authors/?q=ai:ha.youngsoo"Yang, Hyoseon"https://www.zbmath.org/authors/?q=ai:yang.hyoseon"Yoon, Jungho"https://www.zbmath.org/authors/?q=ai:yoon.junghoSummary: In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.Efficient numerical scheme for the anisotropic modified phase-field crystal model with a strong nonlinear vacancy potentialhttps://www.zbmath.org/1475.650742022-01-14T13:23:02.489162Z"Li, Qi"https://www.zbmath.org/authors/?q=ai:li.qi|li.qi.1"Yang, Xiaofeng"https://www.zbmath.org/authors/?q=ai:yang.xiaofeng"Mei, Liquan"https://www.zbmath.org/authors/?q=ai:mei.liquanSummary: In this paper, we consider numerical approximations for the anisotropic modified phasefield crystal model with a strong nonlinear vacancy potential, which describes microscopic phenomena involving atomic hopping and vacancy diffusion. The model is a nonlinear damped wave equation that includes an anisotropic Laplacian and a strong nonlinear vacancy term. To develop an easy to implement time marching scheme with unconditional energy stability, we combine the multiple scalar auxiliary variable (MSAV) approach with stabilization technique for achieving an efficient and linear numerical scheme, in which two new scalar auxiliary variables are introduced to reformulate the model and a linear stabilization term is added to enhance the stability and keep the required accuracy while using the large time steps. The scheme leads to decoupled linear equations with constant coefficients at each time step, and its unique solvability and unconditional energy stability are proved. Various numerical experiments are performed to show the accuracy, stability, and efficiency of the proposed scheme.Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumpshttps://www.zbmath.org/1475.650752022-01-14T13:23:02.489162Z"Omelyan, Igor"https://www.zbmath.org/authors/?q=ai:omelyan.igor-p"Kozitsky, Yuri"https://www.zbmath.org/authors/?q=ai:kozitsky.yuri-v"Pilorz, Krzysztof"https://www.zbmath.org/authors/?q=ai:pilorz.krzysztofThe purpose of this paper is development of an efficient algorithm to solve the repulsion-jump coalescence kinetic equation in a high-dimensional space. Different techniques including time-space discretization (composite Simpson and trapezoidal rules, Runge-Kutta methods, and adjustable system-size schemes), periodic, Dirichlet, and asymptotic boundary conditions are combined for studying infinite systems on the basis of their finite samples. The algorithm is applied to population dynamics simulations of one-dimensional systems with various initial spatially inhomogeneous densities and forms of the jump, coalescence, and repulsion kernels. A comprehensive discussion of the obtained numerical results is provided for some specific choices of the model parameters and initial densities.
Reviewer: Bülent Karasözen (Ankara)Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systemshttps://www.zbmath.org/1475.650762022-01-14T13:23:02.489162Z"Owolabi, Kolade M."https://www.zbmath.org/authors/?q=ai:owolabi.kolade-matthew"Atangana, Abdon"https://www.zbmath.org/authors/?q=ai:atangana.abdon"Gómez-Aguilar, Jose Francisco"https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-franciscoSummary: A recently proposed numerical scheme for solving nonlinear ordinary differential equations with integer and non-integer Liouville-Caputo derivative is applied to three systems with chaotic solutions. The Adams-Bashforth scheme involving Lagrange interpolation and the fundamental theorem of fractional calculus. We provide the existence and uniqueness of solutions, also the convergence result is stated. The proposed method is applied to several examples that are shown to have unique solutions. The scheme converges to the classical Adams-Bashforth method when the fractional orders of the derivatives converge to integers.Linearly implicit GARK schemeshttps://www.zbmath.org/1475.650772022-01-14T13:23:02.489162Z"Sandu, Adrian"https://www.zbmath.org/authors/?q=ai:sandu.adrian"Günther, Michael"https://www.zbmath.org/authors/?q=ai:gunther.michael"Roberts, Steven"https://www.zbmath.org/authors/?q=ai:roberts.steven-pThe paper is concerned with the numerical solution of ODE initial value problems modelling ``multiphysics'' systems which describe multiple processes with different dynamical characteristics acting simultaneously. The authors construct linearly implicit multimethods with possibly different Rosenbrock methods applied to each of the processes involved. This approach generalizes previous multimethods in various ways. Some numerical experiments illustrate the method.
Reviewer: Michael Plum (Karlsruhe)Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equationshttps://www.zbmath.org/1475.650782022-01-14T13:23:02.489162Z"Seeger, Benjamin"https://www.zbmath.org/authors/?q=ai:seeger.benjaminSummary: We develop a method for constructing convergent approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations. Our results apply to explicit finite difference schemes and Trotter-Kato splitting formulas, and error estimates are found for schemes approximating solutions of stochastic Hamilton-Jacobi equations.
For the entire collection see [Zbl 1471.65009].Banded preconditioners for Riesz space fractional diffusion equationshttps://www.zbmath.org/1475.650792022-01-14T13:23:02.489162Z"She, Zi-Hang"https://www.zbmath.org/authors/?q=ai:she.zihang"Lao, Cheng-Xue"https://www.zbmath.org/authors/?q=ai:lao.cheng-xue"Yang, Hong"https://www.zbmath.org/authors/?q=ai:yang.hong"Lin, Fu-Rong"https://www.zbmath.org/authors/?q=ai:lin.furong|lin.fu-rongIn this paper, numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations are considered. Crank-Nicolson technique is applied to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. They transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. They prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners.
In my opinion, this work is interesting and makes some progress in fast algorithm for solving fractional partial differential equation.
Reviewer: Qifeng Zhang (Hangzhou)The convergence of projection difference method for quasilinear parabolic problem in conditions of generalized solvabilityhttps://www.zbmath.org/1475.650802022-01-14T13:23:02.489162Z"Sotnikov, D. S."https://www.zbmath.org/authors/?q=ai:sotnikov.denis-sSummary: The Cauchy problem for quasilinear parabolic problem in Hilbert space is resolved approximately by the projection difference method. The convergence of the approximate solution to the exact solution in conditions of generalized solvability is established in energy norm with the rate of convergence with respect to time. The rate of convergence with respect to space with exact to order of approximation is obtained for subspaces of the finite element type.Efficient numerical algorithms for Riesz-space fractional partial differential equations based on finite difference/operational matrixhttps://www.zbmath.org/1475.650812022-01-14T13:23:02.489162Z"Srivastava, Nikhil"https://www.zbmath.org/authors/?q=ai:srivastava.nikhil"Singh, Aman"https://www.zbmath.org/authors/?q=ai:singh.aman"Kumar, Yashveer"https://www.zbmath.org/authors/?q=ai:kumar.yashveer"Singh, Vineet Kumar"https://www.zbmath.org/authors/?q=ai:singh.vineet-kumarThe paper is concerned with initial-boundary value problems for linear PDEs with first order time derivative and fractional space derivatives of Riesz type on a bounded interval. Depending on whether one or (a sum of) two such fractional derivatives of appropriate orders occur, the equation is interpreted as a ``Riesz space fractional diffusion equation'' or a ``Riesz space fractional advection-dispersion equation'', respectively. Numerical methods are proposed to compute approximations to both types of problems, using a finite difference scheme based on the ``matrix transform method'' for spatial discretizations and an ``operational matrix method'' for time discretization. Questions of convergence order and stability are discussed, and some numerical experiments are carried out for illustration.
Reviewer: Michael Plum (Karlsruhe)Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operatorhttps://www.zbmath.org/1475.650822022-01-14T13:23:02.489162Z"Vabishchevich, Petr N."https://www.zbmath.org/authors/?q=ai:vabishchevich.petr-nSummary: In this paper we study the numerical approximation of the solution of a Cauchy problem for a first-order-in-time differential equation involving a fractional power of a self-adjoint positive operator. One popular approach for the approximation of fractional powers of such operators is based on rational approximations. The purpose of this work is to construct special approximations in time so that the solution at a new time level is produced by solving a set of standard problems involving the self-adjoint positive operator rather than its fractional power. Stable splitting schemes with weight parameters are proposed for the additive representation of the rational approximation of the fractional power of the operator. Finally, numerical results for a two-dimensional non-stationary problem with a fractional power of the Laplace operator are also presented.A lattice Boltzmann model for \((2+1)\)-dimensional solitary and periodic waves of the Calogero-Bogoyavlenskii-Schiff equationhttps://www.zbmath.org/1475.650832022-01-14T13:23:02.489162Z"Wang, Huimin"https://www.zbmath.org/authors/?q=ai:wang.huiminSummary: A lattice Boltzmann model is constructed to simulate the solitary and periodic wave solutions of the Calogero-Bogoyavlenskii-Schiff equation. Numerical simulations of the corresponding solitary and periodic waves show the efficiency of the method and a good computational accuracy.High-order conservative schemes for the space fractional nonlinear Schrödinger equationhttps://www.zbmath.org/1475.650842022-01-14T13:23:02.489162Z"Wang, Junjie"https://www.zbmath.org/authors/?q=ai:wang.junjieSummary: In the paper, the high-order conservative schemes are presented for space fractional nonlinear Schrödinger equation. First, we give two class high-order difference schemes for fractional Risze derivative by compact difference method and extrapolating method, and show the convergence analysis of the two methods. Then, we apply high-order conservative difference schemes in space direction, and Crank-Nicolson, linearly implicit and relaxation schemes in time direction to solve fractional nonlinear Schrödinger equation. Moreover, we show that the arising schemes are uniquely solvable and approximate solutions converge to the exact solution at the rate \(O(\tau^2+h^4)\), and preserve the mass and energy conservation laws. Finally, we given numerical experiments to show the efficiency of the conservative finite difference schemes.A new fourth-order energy dissipative difference method for high-dimensional nonlinear fractional generalized wave equationshttps://www.zbmath.org/1475.650852022-01-14T13:23:02.489162Z"Xie, Jianqiang"https://www.zbmath.org/authors/?q=ai:xie.jianqiang"Zhang, Zhiyue"https://www.zbmath.org/authors/?q=ai:zhang.zhiyue"Liang, Dong"https://www.zbmath.org/authors/?q=ai:liang.dongSummary: In this paper, a new energy dissipative fourth-order difference scheme for the high-dimensional nonlinear fractional generalized wave equations is constructed. Then, the discrete energy dissipation property of the system is exhibited in detail. Next, we prove that the proposed scheme is uniquely solvable. By the discrete energy method, it is shown that the proposed scheme achieves the optimal convergence rate of \(\mathcal{O}(\Delta t^2+h_x^4+h_y^4)\) in the discrete \(L^2\)-norm, and is unconditionally stable. Besides, the presented convergence analysis is unconditional for the time step size in terms of space mesh sizes. Lastly, some numerical results are given to illustrate the physical effects of the nonzero damping terms and support our theoretical analysis.Numerical investigation to the effect of initial guess for phase-field modelshttps://www.zbmath.org/1475.650862022-01-14T13:23:02.489162Z"Yoon, Sungha"https://www.zbmath.org/authors/?q=ai:yoon.sungha"Wang, Jian"https://www.zbmath.org/authors/?q=ai:wang.jian.9"Lee, Chaeyoung"https://www.zbmath.org/authors/?q=ai:lee.chaeyoung"Yang, Junxiang"https://www.zbmath.org/authors/?q=ai:yang.junxiang"Park, Jintae"https://www.zbmath.org/authors/?q=ai:park.jintae"Kim, Hyundong"https://www.zbmath.org/authors/?q=ai:kim.hyundong"Kim, Junseok"https://www.zbmath.org/authors/?q=ai:kim.junseokSummary: The construction of relevant initial conditions in the phase-field models for interfacial problems is discussed. If the model is supposed to have a local equilibrium at the interface, it must be based on a local distance function. However, since for the Cartesian coordinates non-uniform boundaries occur, the initial conditions have to be corrected in order to match the actual phenomena. We discuss the volume correction method, image initialisation, non-overlapping multi component concentration, etc. The methods presented can be used in the initial guess constructions for various phase-field models.High-order efficient numerical method for solving a generalized fractional Oldroyd-B fluid modelhttps://www.zbmath.org/1475.650872022-01-14T13:23:02.489162Z"Yu, Bo"https://www.zbmath.org/authors/?q=ai:yu.bo|yu.bo.2|yu.bo.1Summary: This paper investigates the high-order efficient numerical method with the corresponding stability and convergence analysis for the generalized fractional Oldroyd-B fluid model. Firstly, a high-order compact finite difference scheme is derived with accuracy \(O\left(\tau^{\min{\{3-\gamma, 2-\alpha}\}}+h^4\right) \), where \(\gamma \in (1,2)\) and \(\alpha \in (0,1)\) are the orders of the time fractional derivatives. Then, by means of a new inner product, the unconditional stability and convergence in the maximum norm of the derived high-order numerical method have been discussed rigorously using the energy method. Finally, numerical experiments are presented to test the convergence order in the temporal and spatial direction, respectively. To precisely demonstrate the computational efficiency of the derived high-order numerical method, the maximum norm error and the CPU time are measured in contrast with the second-order finite difference scheme for the same temporal grid size. Additionally, the derived high-order numerical method has been applied to solve and analyze the flow problem of an incompressible Oldroyd-B fluid with fractional derivative model bounded by two infinite parallel rigid plates.A linearised three-point combined compact difference method with weighted approximation for nonlinear time fractional Klein-Gordon equationshttps://www.zbmath.org/1475.650882022-01-14T13:23:02.489162Z"Zhang, Chun-Hua"https://www.zbmath.org/authors/?q=ai:zhang.chunhua.1|zhang.chunhua"Sun, Hai-Wei"https://www.zbmath.org/authors/?q=ai:sun.haiweiSummary: A numerical method for nonlinear time fractional Klein-Gordon equations is studied. Discretising spatial and temporal variables by a combined compact difference and a weighted approximation, respectively, we develop a linearised method for the equations under consideration. It has at least sixth-order accuracy in space and second-order accuracy in time. Numerical examples demonstrate the effectiveness and accuracy of the method.Numerical solution of nonlinear advection diffusion reaction equation using high-order compact difference methodhttps://www.zbmath.org/1475.650892022-01-14T13:23:02.489162Z"Zhang, Lin"https://www.zbmath.org/authors/?q=ai:zhang.lin.2|zhang.lin.3|zhang.lin|zhang.lin.1"Ge, Yongbin"https://www.zbmath.org/authors/?q=ai:ge.yongbinSummary: In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order \textit{Padé} formula of the first-order derivative, and the time derivative term is discretized by using the fourth-order backward differencing formula. An unconditionally stable five-step fourth-order fully implicit compact difference scheme is developed. This scheme has fourth-order accuracy in both time and space. Secondly, for the calculations of the start-up time steps, the time derivative term is discretized by the Crank-Nicolson method, and Richardson extrapolation formula is used to improve the accuracy in time direction from the second-order to the fourth-order. Thirdly, convergence and stability of the difference scheme in \(H^1\) seminorm, \(L^\infty\) and \(L^2\) norms, existence and uniqueness of the numerical solutions are proved, respectively. Fourthly, the Thomas algorithm is used to solve the nonlinear algebraic equations at each time step, and a time advancement algorithm with linearized iteration strategy is established. Finally, the accuracy, stability and efficiency of the present approach are verified by some numerical experiments.A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamicshttps://www.zbmath.org/1475.650902022-01-14T13:23:02.489162Z"El Harrak, Anouar"https://www.zbmath.org/authors/?q=ai:el-harrak.anouar"Tayeq, Hatim"https://www.zbmath.org/authors/?q=ai:tayeq.hatim"Bergam, Amal"https://www.zbmath.org/authors/?q=ai:bergam.amalSummary: This work gives a posteriori error estimates for a finite volume implicit scheme, applied to a two-time nonlinear reaction-diffusion problem in population dynamics, whose evolution processes occur at two different time scales, represented by a parameter \( \varepsilon>0 \) small enough. This work consists of building error indicators concerning time and space approximations and using them as a tool of adaptive mesh refinement in order to find approximate solutions to such models, in population dynamics, that are often hard to be handled analytically and also to be approximated numerically using the classical approach.
An application of the theoretical results is provided to emphasize the efficiency of our approach compared to the classical one for a spatial inter-specific model with constant diffusivity and population growth given by a logistic law in population dynamics.A posteriori subcell finite volume limiter for general \(P_NP_M\) schemes: applications from gasdynamics to relativistic magnetohydrodynamicshttps://www.zbmath.org/1475.650912022-01-14T13:23:02.489162Z"Gaburro, Elena"https://www.zbmath.org/authors/?q=ai:gaburro.elena"Dumbser, Michael"https://www.zbmath.org/authors/?q=ai:dumbser.michaelThe authors propose a new simple, robust, accurate and computationally efficient limiting strategy for the general family of ADER \(P_NP_M\) schemes, allowing the use of hybrid reconstructed methods (\(N > 0\), \(M > N\) ) in the modeling of discontinuous phenomena. This new approach has been applied to many different systems of hyperbolic conservation laws, providing highly accurate numerical results in all cases. The performance of the class of intermediate \(P_N P_M\) schemes with \(M > N > 0\) is compared with pure the Discontinuous Galerkin (DG) schemes (\(M = N\)). It is remarked that in the most cases the intermediate \(P_N P_M\) schemes lead to reduced computational cost compared with the pure DG methods. A new efficient posteriori subcell finite volume limiting strategy that is valid for the entire class of \(P_NP_M\) schemes is presented.
Reviewer: Abdallah Bradji (Annaba)Finite volume scheme for AMSS modelhttps://www.zbmath.org/1475.650922022-01-14T13:23:02.489162Z"Handlovičová, Angela"https://www.zbmath.org/authors/?q=ai:handlovicova.angelaSummary: We propose a new finite volume numerical scheme for the approximation of the Affine Morphological Scale Space (AMSS) model. We derive the basic scheme and its iterative improvement. For both schemes, several numerical experiments using examples where the exact solution is known are presented. Then the numerical errors and experimental order of convergence of the proposed schemes is studied.High-order non-conservative simulation of hyperbolic moment models in partially-conservative formhttps://www.zbmath.org/1475.650932022-01-14T13:23:02.489162Z"Koellermeier, J."https://www.zbmath.org/authors/?q=ai:koellermeier.julian"Castro, M. J."https://www.zbmath.org/authors/?q=ai:castro.manuel-jSummary: In this paper the first dedicated study on high-order non-conservative numerical schemes for hyperbolic moment models is presented. The implementation uses a new formulation that allows for explicit evaluation of the model while satisfying conservation of mass, momentum, and energy. The high-order numerical schemes use a path-conservative treatment of the non-conservative terms and a new consistent evaluation of the eigenvalues. The numerical results of two initial value problems, one stationary test case and a boundary value problem, yield stable and accurate solutions with convergence towards the reference solution despite the presence of a non-conservative term. A large speedup or accuracy gain in comparison to existing first-order codes could be demonstrated.The accuracy of numerical simulation of the acoustic wave propagations in a liquid medium based on Navier-Stokes equationshttps://www.zbmath.org/1475.650942022-01-14T13:23:02.489162Z"Kozelkov, Andrey Sergeevich"https://www.zbmath.org/authors/?q=ai:kozelkov.andrey-sergeevich"Krutyakova, Olga Leonidovna"https://www.zbmath.org/authors/?q=ai:krutyakova.olga-leonidovna"Kurulin, Vadim Viktorovich"https://www.zbmath.org/authors/?q=ai:kurulin.vadim-viktorovich"Strelets, Dmitry Yurievich"https://www.zbmath.org/authors/?q=ai:strelets.dmitry-yurievich"Shishlenin, Maxim Aleksandrovich"https://www.zbmath.org/authors/?q=ai:shishlenin.maxim-aSummary: The space and time resolution needed to simulate the propagation of acoustic perturbations in a liquid medium is estimated. The dependence of the solution accuracy on the parameters of an iterative procedure and a numerical discretization of the equations is analyzed. As a numerical method, a widely used method called SIMPLE is used together with a finite-volume discretization of the equations. A problem of propagation of perturbations in a liquid medium from a harmonic source of oscillations is considered for the estimation. Estimates of the required space and time resolution are obtained to provide an acceptable accuracy of the solution. The estimates are tested using the problem of propagation of harmonic waves from a point source in a liquid medium.Macrophage image segmentation by thresholding and subjective surface methodhttps://www.zbmath.org/1475.650952022-01-14T13:23:02.489162Z"Park, Seol Ah"https://www.zbmath.org/authors/?q=ai:park.seol-ah"Sipka, Tamara"https://www.zbmath.org/authors/?q=ai:sipka.tamara"Krivá, Zuzana"https://www.zbmath.org/authors/?q=ai:kriva.zuzana"Ambroz, Martin"https://www.zbmath.org/authors/?q=ai:ambroz.martin"Kollár, Jozef"https://www.zbmath.org/authors/?q=ai:kollar.jozef"Kósa, Balázs"https://www.zbmath.org/authors/?q=ai:kosa.balazs"Chi, Mai Nguyen"https://www.zbmath.org/authors/?q=ai:chi.mai-nguyen"Lutfalla, Georges"https://www.zbmath.org/authors/?q=ai:lutfalla.georges"Mikula, Karol"https://www.zbmath.org/authors/?q=ai:mikula.karolSummary: We introduce two level-set method approaches to segmentation of 2D macrophage images. The first one is based on the Otsu thresholding and the second one on the information entropy thresholding, both followed by the classical subjective surface (SUBSURF) method. Results of both methods are compared with the semi-automatic Lagrangian method in which the segmentation curve evolves along the edge of the macrophage and it is controlled by an expert user. We present the comparison of all three methods with respect to the Hausdorff distance of resulting segmentation curves and we compare also their perimeter and enclosed area. We show that accuracy of the automatic SUBSURF method is comparable to the results of the semi-automatic Lagrangian segmentation.Convergence of projection difference method of approximate solution in parabolic equation with symmetric operator and integral condition for the solutionhttps://www.zbmath.org/1475.650962022-01-14T13:23:02.489162Z"Nguyen Thuong Huyen"https://www.zbmath.org/authors/?q=ai:nguyen-thuong-huyen.Summary: In the Hilbert space the abstract linear parabolic equation with symmetric operator and nonlocal integral condition for the solution is resolved approximate by the projection difference method with using implicit Euler scheme to time. Approximation to the spatial variables is oriented on the finite element method. The established error estimations of approximate solutions, the convergence of approximate solution to exact solution and the orders of speed of convergence.Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrödinger equationshttps://www.zbmath.org/1475.650972022-01-14T13:23:02.489162Z"Yang, He"https://www.zbmath.org/authors/?q=ai:yang.heSummary: The Klein-Gordon-Schrödinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank-Nicholson time discretization for the Schrödinger equation part, a Crank-Nicholson leap frog method in time for the Klein-Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein-Gordon or Schrödinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.A diagonalization-based parareal algorithm for dissipative and wave propagation problemshttps://www.zbmath.org/1475.650982022-01-14T13:23:02.489162Z"Gander, Martin J."https://www.zbmath.org/authors/?q=ai:gander.martin-j"Wu, Shu-Lin"https://www.zbmath.org/authors/?q=ai:wu.shulinThis paper proposes a new parareal algorithm for solving initial value problems of the form \(\partial_t u+f(t,u)=0\), where \(f:(0,T)\times \mathbb{R}^m\to \mathbb{R}^m\), \(m\ge 1\). The algorithm allows the use of a coarse propagator that discretizes the underlying problem on the same mesh as the fine propagator. The coarse propagator is approximated with a head-tail coupled condition such that it can be parallelized using diagonalization in time. It is shown that with an optimal choice of the parameter in the head-tail condition, the new parareal algorithm converges rapidly for both linear and nonlinear problems under certain conditions. Numerical experiments for solving PDEs with linear and nonlinear fractional Laplacian and the nonlinear wave equation are included.
Reviewer: Zhiming Chen (Beijing)The Newmark method and a space-time FEM for the second-order wave equationhttps://www.zbmath.org/1475.650992022-01-14T13:23:02.489162Z"Zank, Marco"https://www.zbmath.org/authors/?q=ai:zank.marcoSummary: For the second-order wave equation, we compare the Newmark Galerkin method with a stabilised space-time finite element method for tensor-product space-time discretisations with piecewise multilinear, continuous ansatz and test functions leading to an unconditionally stable Galerkin-Petrov scheme, which satisfies a space-time error estimate. We show that both methods require to solve a linear system with the same system matrix. In particular, the stabilised space-time finite element method can be solved sequentially in time as the Newmark Galerkin method. However, the treatment of the right-hand side of the wave equation is different, where the Newmark Galerkin method requires more regularity.
For the entire collection see [Zbl 1471.65009].On the Gauss Runge-Kutta and method of lines transpose for initial-boundary value parabolic PDEshttps://www.zbmath.org/1475.651002022-01-14T13:23:02.489162Z"Zhang, Bo"https://www.zbmath.org/authors/?q=ai:zhang.bo.5"Chen, Dangxing"https://www.zbmath.org/authors/?q=ai:chen.dangxing"Huang, Jingfang"https://www.zbmath.org/authors/?q=ai:huang.jingfangSummary: It has been shown in existing analysis that the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation is super-convergent when applied to the initial value problem of ordinary differential equations (ODEs) in that the discretization error is order 2s when s Gaussian nodes are used. Additionally, the discretized system can be solved accurately and efficiently using the spectral deferred correction (SDC) or Krylov deferred correction (KDC) method. In this paper, we combine the GRK formulation with the Method of Lines Transpose (\(\mathrm{MoL}^T\)) to solve time-dependent parabolic partial differential equations (PDEs). For the GRK-\(\mathrm{MoL}^T\) formulation, we show how the coupled spatial differential equations can be decoupled and efficiently solved using the SDC or KDC method. Preliminary analysis of the GRK-\(\mathrm{MoL}^T\) algorithm reveals that the super-convergent property of the GRK formulation no longer holds in the PDE case for general boundary conditions, and there exists a new type of ``stiffness'' in the semi-discrete spatial elliptic differential equations. We present numerical experiments to validate the theoretical findings.Numerical approximations and error analysis of the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditionshttps://www.zbmath.org/1475.651012022-01-14T13:23:02.489162Z"Bao, Xuelian"https://www.zbmath.org/authors/?q=ai:bao.xuelian"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.2|zhang.hui.9|zhang.hui.3|zhang.hui.8|zhang.hui.1|zhang.hui.7|zhang.hui.5|zhang.hui.11|zhang.hui.10|zhang.hui.6|zhang.hui|zhang.hui.4The authors present a semi-discretization (discretisation in time) for a Cahn-Hilliard equation, with reaction rate dependent dynamic boundary conditions, introduced in [\textit{P. Knopf} et al., ESAIM, Math. Model. Numer. Anal. 55, No. 1, 229--282 (2021; Zbl 1470.35173)]. A first-order in time, linear and energy stable scheme for solving this model is proposed. Error analysis is also provided. Some numerical experiments in two dimensions are presented to confirm the accuracy of the proposed scheme. The discretisation in space in these simulations is performed using the second-order central finite difference method on uniform meshes. The convergence results for the relaxation parameter \(K\rightarrow 0\) and \(K\rightarrow \infty\) are also illustrated, which are consistent with the former work.
Reviewer: Abdallah Bradji (Annaba)Computing ill-posed time-reversed 2D Navier-Stokes equations, using a stabilized explicit finite difference scheme marching backward in timehttps://www.zbmath.org/1475.651022022-01-14T13:23:02.489162Z"Carasso, Alfred S."https://www.zbmath.org/authors/?q=ai:carasso.alfred-sSummary: This paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier-Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on \((- \Delta)^p\), with real \(p>2\), can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Navier-Stokes initial value problems. Several reconstruction examples are included, based on the \textit{stream function-vorticity} formulation, and focusing on \(256 \times 256\) pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time \(T>0\). Successful backward recovery is shown to be possible at parameter values exceeding expectations.A numerical scheme based on discrete mollification method using Bernstein basis polynomials for solving the inverse one-dimensional Stefan problemhttps://www.zbmath.org/1475.651032022-01-14T13:23:02.489162Z"Bodaghi, Soheila"https://www.zbmath.org/authors/?q=ai:bodaghi.soheila"Zakeri, Ali"https://www.zbmath.org/authors/?q=ai:zakeri.ali"Amiraslani, Amir"https://www.zbmath.org/authors/?q=ai:amiraslani.amirSummary: This paper concerns a one-phase inverse Stefan problem in one-dimensional space. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We also consider noisy data for this problem. As such, we first regularize the proposed problem by the discrete mollification method. We apply the integration matrix using Bernstein basis polynomials for the discrete mollification method. Through this method, the execution time was gradually decreased. We then extend the space marching algorithm for solving our problem. Moreover, proofs of stability and convergence of the process are given. Finally, the results of this paper have been illustrated and examined by some numerical examples. Numerical examples confirm the efficiency of the proposed method.A meshless computational approach for solving two-dimensional inverse time-fractional diffusion problem with non-local boundary conditionhttps://www.zbmath.org/1475.651042022-01-14T13:23:02.489162Z"Ghehsareh, Hadi Roohani"https://www.zbmath.org/authors/?q=ai:ghehsareh.hadi-roohani"Zabetzadeh, Sayyed Mahmood"https://www.zbmath.org/authors/?q=ai:zabetzadeh.sayyed-mahmoodSummary: This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov-Galerkin method is implemented to solve the governing inverse problem. Firstly, an implicit time integration scheme is used to discretize the model in the temporal direction. To fully discretize the model, the primary spatial domain is represented by a set of distributed nodes and data-dependent basis functions are constructed by using the radial point interpolation method. Then, the local meshless Petrov-Galerkin method is used to discretize the problem in the spatial direction. Numerical examples are presented to verify the accuracy and efficiency of the proposed technique. The stability of the method is examined when the input data are contaminated with noise.A new method based on polynomials equipped with a parameter to solve two parabolic inverse problems with a nonlocal boundary conditionhttps://www.zbmath.org/1475.651052022-01-14T13:23:02.489162Z"Hajishafieiha, J."https://www.zbmath.org/authors/?q=ai:hajishafieiha.j"Abbasbandy, S."https://www.zbmath.org/authors/?q=ai:abbasbandy.saeidSummary: In this paper, a new method is used based on polynomials equipped with a parameter to solve two parabolic inverse problems. These inverse problems have nonlocal boundary conditions and over-determination of data that make it difficult to solve these problems. In this method, we use the combination of the finite difference method and the finite element method. In each point \( t_j\), a nonlinear equation system is solved via the least-squares method, and then, we obtain an approximate function for the solution of the problem by using the interpolation of these points.Pseudospectral method for a one-dimensional fractional inverse problemhttps://www.zbmath.org/1475.651062022-01-14T13:23:02.489162Z"Karimi, Maryam"https://www.zbmath.org/authors/?q=ai:karimi.maryam"Behroozifar, Mahmoud"https://www.zbmath.org/authors/?q=ai:behroozifar.mahmoudSummary: In this paper, a method is implemented to a one-dimensional inverse problem with a parabolic differential equation of fractional order in which the fractional derivative is in the Caputo sense. The considered inverse problem involves a time-dependent source control parameter \(p(t)\). In order to numerically solve the problem, first, the main problem is converted to a homogeneous problem by Lagrange interpolation. Consequently, a new problem is derived by a practical technique that verifies all the conditions of the main problem. Finally, a system of nonlinear algebraic equations is solved by Newton's method to obtain the unknown coefficients. It is notable that all the needed computations are done in Mathematica. In this work, operational matrices of Bernoulli polynomials are stated and applied to approximate functions. Illustrative examples are included to prove the efficiency and applicability of the proposed methods. In the numerical tests, a low amount of polynomials is needed to acquire a precise estimate solution. For demonstrating the low running time of this method, CPU time for all examples is exhibited.Analytic series solutions of 2D forward and backward heat conduction problems in rectangles and a new regularizationhttps://www.zbmath.org/1475.651072022-01-14T13:23:02.489162Z"Liu, Chein-Shan"https://www.zbmath.org/authors/?q=ai:liu.chein-shan"Chang, Chih-Wen"https://www.zbmath.org/authors/?q=ai:chang.chih-wenSummary: In the paper, we solve a non-homogeneous heat conduction equation with non-homogeneous boundary conditions in a 2D rectangle. First, we derive the domain/boundary integral equations for both the forward and backward heat conduction problems. Then, by using the technique of homogenization, inserting the adjoint Trefftz test functions into the derived integral equations and expanding the solutions in terms of eigenfunctions, we can obtain the expansion coefficients in closed form. Hence, the analytic series solutions of forward heat conduction problems (FHCPs) and backward heat conduction problems (BHCPs) are available. For the FHCPs, only a few terms in the series render very high-order accurate solutions at any time, with errors of the order \(10^{- 10}\). For the BHCPs, we require to modify the closed-form series solutions via a new spring-damping regularization technique. Numerical tests for the BHCPs in a large space-time domain reveal that the present analytic series solution is very accurate to recover the initial temperature with an error of the order \(10^{- 10}\), although the measured final time temperature is very small when \(t_f\) is large up to 100 and is even polluted by a large relative noise up to the level \(20\%\).A total variation regularization method for an inverse problem of recovering an unknown diffusion coefficient in a parabolic equationhttps://www.zbmath.org/1475.651082022-01-14T13:23:02.489162Z"Li, Zhaoxing"https://www.zbmath.org/authors/?q=ai:li.zhaoxing.1"Deng, Zhiliang"https://www.zbmath.org/authors/?q=ai:deng.zhiliangSummary: This paper studies an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation. We adopt a total variation regularization method to deal with the ill-posedness. This method has the advantage to solve problems that the solution is non-smooth or discontinuous. By transforming the problem into an optimal control problem, we derive a necessary condition of the control functional. Through some prior estimates of the direct problem, the uniqueness and stability of the minimizer are obtained. In the numerical part, a Gauss-Jacobi iteration scheme is used to deal with the non-linear term. Some numerical examples are presented to illustrate the performance of the proposed algorithm.On the choice of Lagrange multipliers in the iterated Tikhonov method for linear ill-posed equations in Banach spaceshttps://www.zbmath.org/1475.651092022-01-14T13:23:02.489162Z"Machado, M. P."https://www.zbmath.org/authors/?q=ai:machado.matilde-p|machado.m-penton"Margotti, F."https://www.zbmath.org/authors/?q=ai:margotti.fabio"Leitão, A."https://www.zbmath.org/authors/?q=ai:leitao.antonioSummary: This article is devoted to the study of \textit{nonstationary Iterated Tikhonov} (nIT) type methods [\textit{M. Hanke} and \textit{C. W. Groetsch}, J. Optim. Theory Appl. 98, No. 1, 37--53 (1998; Zbl 0910.47005); \textit{H. W. Engl} et al., Regularization of inverse problems. Dordrecht: Kluwer Academic Publishers (1996; Zbl 0859.65054)] for obtaining stable approximations to linear ill-posed problems modelled by operators mapping between Banach spaces. Here we propose and analyse an \textit{a posteriori} strategy for choosing the sequence of regularization parameters for the nIT method, aiming to obtain a pre-defined decay rate of the residual. Convergence analysis of the proposed nIT type method is provided (convergence, stability and semi-convergence results). Moreover, in order to test the method's efficiency, numerical experiments for three distinct applications are conducted: (i) a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space); (ii) a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space); (iii) a 2D elliptic inverse potential problem.Source strength identification problem for the three-dimensional inverse heat conduction equationshttps://www.zbmath.org/1475.651102022-01-14T13:23:02.489162Z"Min, Tao"https://www.zbmath.org/authors/?q=ai:min.tao"Zang, Shunquan"https://www.zbmath.org/authors/?q=ai:zang.shunquan"Chen, Shengnan"https://www.zbmath.org/authors/?q=ai:chen.shengnanSummary: In this paper, we consider the source strength identification problem for the three-dimensional inverse heat conduction equations. The problem is to determine an unknown heat source strength from the measurement data for a specified location. In this process, the direct problem is solved by applying the Green's function method. Then, this problem can be converted into a Volterra integral equation of the first kind. Further, the Tikhonov and truncated singular value decomposition regularization methods are developed to identify the unknown source strength based on the discrepancy principle for choosing the regularization parameter. Finally, numerical examples are presented to show the feasibility and efficiency of the proposed method.A fractional-order quasi-reversibility method to a backward problem for the time fractional diffusion equationhttps://www.zbmath.org/1475.651112022-01-14T13:23:02.489162Z"Shi, Wanxia"https://www.zbmath.org/authors/?q=ai:shi.wanxia"Xiong, Xiangtuan"https://www.zbmath.org/authors/?q=ai:xiong.xiangtuan"Xue, Xuemin"https://www.zbmath.org/authors/?q=ai:xue.xueminSummary: In this paper, we consider the regularization of the backward problem of diffusion process with time-fractional derivative. Since the equation under consideration involves the time-fractional derivative, we introduce a new perturbation which is related to the time-fractional derivative into the original equation. This leads to a fractional-order quasi-reversibility method. In theory, we give the regularity of the regularized solution and the corresponding convergence rate is also proved under the appropriate regularization parameter choice rule. In numerics, some numerical experiments are presented to illustrate the effectiveness of our method and some numerical comparison with the existing quasi-reversibility method is conducted. Both theoretical and numerical results show the advantage of the new method.Identifying an unknown source term in a time-space fractional parabolic equationhttps://www.zbmath.org/1475.651122022-01-14T13:23:02.489162Z"Van Thang, Nguyen"https://www.zbmath.org/authors/?q=ai:thang.nguyen-van"Van Duc, Nguyen"https://www.zbmath.org/authors/?q=ai:duc.nguyen-van"Minh, Luong Duy Nhat"https://www.zbmath.org/authors/?q=ai:minh.luong-duy-nhat"Thành, Nguyen Trung"https://www.zbmath.org/authors/?q=ai:thanh.nguyen-trungSummary: An inverse problem of identifying an unknown space-dependent source term in a time-space fractional parabolic equation is considered in this paper. Under reasonable boundedness assumptions about the source function, a Hölder-type stability estimate of optimal order is proved. To regularize the inverse source problem, a mollification regularization method is applied. Error estimates of the regularized solution are proved for both \textit{a priori} and \textit{a posteriori} rules for choosing the mollification parameter. A direct numerical method for solving the regularized problem is proposed and numerical examples are presented to illustrate its effectiveness.A potential-free field inverse Schrödinger problem: optimal error bound analysis and regularization methodhttps://www.zbmath.org/1475.651132022-01-14T13:23:02.489162Z"Yang, Fan"https://www.zbmath.org/authors/?q=ai:yang.fan.1"Fu, Jun-Liang"https://www.zbmath.org/authors/?q=ai:fu.jun-liang"Li, Xiao-Xiao"https://www.zbmath.org/authors/?q=ai:li.xiaoxiaoSummary: In this paper, an inverse Schrödinger problem of potential-free field is studied. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an \textit{a priori} assumption, the optimal errorbound analysis is given. Moreover, two different regularization methods are used to solve this problem, respectively. Under an \textit{a priori} and an \textit{a posteriori} regularization parameters choice rule, the convergent error estimates are all obtained. Compared with Landweber iterative regularization method, the convergent estimate between the exact solution and the regularization solution obtained by a modified kernel method is optimal for the \textit{priori} regularization parameter choice rule, and the \textit{posteriori} error estimate is order-optimal. Finally, some numerical examples are given to illustrate the effectiveness, stability and superiority of these methods.A class of multistep numerical difference schemes applied in inverse heat conduction problem with a control parameterhttps://www.zbmath.org/1475.651142022-01-14T13:23:02.489162Z"Zhang, Yong-Fu"https://www.zbmath.org/authors/?q=ai:zhang.yongfu"Li, Chong-Jun"https://www.zbmath.org/authors/?q=ai:li.chongjunFrom the abstract (with slight modifications for language): This paper is concerned with a class of multistep numerical difference techniques to solve one-dimensional parabolic inverse problems with source control parameter. We apply a linear multistep method combined with Lagrange interpolation to develop three different numerical difference schemes. As the problem of numerical differentiation with noisy scattered data is mildly ill-posed, a smoothing spline model based on Tikhonov regularization is developed to compute numerical derivatives contaminated by noise error. Simultaneously, the truncation error estimates and convergence results are derived for the above difference methods. The results of numerical tests with different noise levels are given to show that the presented algorithms are accurate and effective.
Reviewer: Christian Clason (Graz)Adaptive time stepping methods within a data assimilation framework applied to non-isothermal flow dynamicshttps://www.zbmath.org/1475.651152022-01-14T13:23:02.489162Z"Evert Uilhoorn, Ferdinand"https://www.zbmath.org/authors/?q=ai:uilhoorn.ferdinand-evertSummary: This contribution discusses the performance of time stepping schemes within a data assimilation framework, applied to the method of lines solutions of the non-isothermal compressible gas flow equations. We consider important classes of schemes, namely an embedded explicit Runge-Kutta (ERK) scheme, a diagonally implicit Runge-Kutta (DIRK) scheme, a fully implicit Runge-Kutta (IRK) scheme and a Rosenbrock-Krylov (ROK) scheme. For the numerical illustration, we estimated the flow transients in a subsea pipeline system. Errors from numerical discretization, missing and variability of physical parameters and inaccuracy of initial and boundary conditions are assumed non-Gaussian. Efficiency, robustness and estimation accuracy were evaluated. Results showed that the DIRK scheme is a good compromise between efficiency and robustness. Spurious oscillations were filtered out by the sequential Monte-Carlo algorithm.
For the entire collection see [Zbl 1471.65009].A restricted linearised augmented Lagrangian method for Euler's elastica modelhttps://www.zbmath.org/1475.651162022-01-14T13:23:02.489162Z"Zhang, Yinghui"https://www.zbmath.org/authors/?q=ai:zhang.yinghui"Deng, Xiaojuan"https://www.zbmath.org/authors/?q=ai:deng.xiaojuan"Zhao, Xing"https://www.zbmath.org/authors/?q=ai:zhao.xing"Li, Hongwei"https://www.zbmath.org/authors/?q=ai:li.hongweiSummary: A simple cutting-off strategy for the augmented Lagrangian formulation for minimising the Euler's elastica energy is introduced. It is connected to a discovered internal inconsistency of the model and helps to decouple the tricky dependence between auxiliary splitting variables, thus fixing the problem mentioned. Numerical experiments show that the method converges much faster than conventional algorithms, provides a better parameter-tuning and ensures the higher quality of image restorations.Finite element methods for non-Fourier thermal wave model of bio heat transfer with an interfacehttps://www.zbmath.org/1475.651172022-01-14T13:23:02.489162Z"Deka, Bhupen"https://www.zbmath.org/authors/?q=ai:deka.bhupen"Dutta, Jogen"https://www.zbmath.org/authors/?q=ai:dutta.jogenSummary: We propose a fitted finite element method for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell-Cattaneo equation on the physical media that have a heterogeneous conductivity. Well-posedness of the model interface problem is established. A continuous piecewise linear finite element space is employed for the spatially semidiscrete approximation and the temporal discretization is based on backward scheme. Optimal order error estimates for both semidiscrete and fully discrete schemes are proved in \(L^{\infty}(H^1)\) norm. Finally, we give numerical examples to verify our theoretical results. The new results and finite element schemes can be applied in the fields of engineering, medicine, and biotechnology.A second order finite element method with mass lumping for wave equations in \(H(\mathrm{div})\)https://www.zbmath.org/1475.651182022-01-14T13:23:02.489162Z"Egger, Herbert"https://www.zbmath.org/authors/?q=ai:egger.herbert"Radu, Bogdan"https://www.zbmath.org/authors/?q=ai:radu.bogdanSummary: We consider the efficient numerical approximation of acoustic wave propagation in time domain by a finite element method with mass lumping. In the presence of internal damping, the problem can be reduced to a second order formulation in time for the velocity field alone. For the spatial approximation we consider \(H(\mathrm{div})\)-conforming finite elements of second order. In order to allow for an efficient time integration, we propose a mass-lumping strategy based on approximation of the \(L^2\)-scalar product by inexact numerical integration which leads to a block-diagonal mass matrix. A careful error analysis allows to show that second order accuracy is not reduced by the quadrature errors which is illustrated also by numerical tests.
For the entire collection see [Zbl 1471.65009].A variational formulation for LTI-systems and model reductionhttps://www.zbmath.org/1475.651192022-01-14T13:23:02.489162Z"Feuerle, Moritz"https://www.zbmath.org/authors/?q=ai:feuerle.moritz"Urban, Karsten"https://www.zbmath.org/authors/?q=ai:urban.karstenSummary: We consider a variational formulation of Linear Time-Invariant (LTI)-systems and derive a model reduction in dimension and time inspired by space-time variational reduced basis (RB) methods for parabolic problems. A residual-type RB error estimator is derived whose effectivity is investigated numerically.
For the entire collection see [Zbl 1471.65009].Foundations of space-time finite element methods: polytopes, interpolation, and integrationhttps://www.zbmath.org/1475.651202022-01-14T13:23:02.489162Z"Frontin, Cory V."https://www.zbmath.org/authors/?q=ai:frontin.cory-v"Walters, Gage S."https://www.zbmath.org/authors/?q=ai:walters.gage-s"Witherden, Freddie D."https://www.zbmath.org/authors/?q=ai:witherden.freddie-d"Lee, Carl W."https://www.zbmath.org/authors/?q=ai:lee.carl-w"Williams, David M."https://www.zbmath.org/authors/?q=ai:williams.david-m"Darmofal, David L."https://www.zbmath.org/authors/?q=ai:darmofal.david-lSummary: The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures (usually orthonormal polynomial bases), and numerical integration procedures (usually quadrature rules). It is well known that each of these areas has yet to be fully explored, and in the present article, we attempt to directly address this issue. We begin by developing a concrete, sequential procedure for constructing generic four-dimensional elements (4-polytopes). Thereafter, we review the key numerical properties of several canonical elements: the tesseract, tetrahedral prism, and pentatope. Here, we provide explicit expressions for orthonormal polynomial bases on these elements. Next, we construct fully symmetric quadrature rules with positive weights that are capable of exactly integrating high-degree polynomials, e.g. up to degree 17 on the tesseract. Finally, the quadrature rules are successfully tested using a set of canonical numerical experiments on polynomial and transcendental functions.Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equationshttps://www.zbmath.org/1475.651212022-01-14T13:23:02.489162Z"Hao, Zhaopeng"https://www.zbmath.org/authors/?q=ai:hao.zhaopeng"Zhang, Zhongqiang"https://www.zbmath.org/authors/?q=ai:zhang.zhongqiangNonlocal models of advection-diffusion-reaction equations with fractional Laplacian are studied, which can be considered as a simplified model for the fractional Navier-Stokes equations. The convergence rate of a spectral Galerkin method is analyzed for the one-dimensional case. Sharp regularity estimates of the solution are proven in weighted Sobolev spaces. Hereby a factorization of the solution is employed. A matrix free iterative solver is likewise proposed which has almost optimal complexity. Different numerical experiments considering smooth right-hand sides, weakly singular ones, or a boundary singularity case, illustrate the theoretical results for different parameter settings.
Reviewer: Kai Schneider (Marseille)Enriched Galerkin method for the shallow-water equationshttps://www.zbmath.org/1475.651222022-01-14T13:23:02.489162Z"Hauck, Moritz"https://www.zbmath.org/authors/?q=ai:hauck.moritz"Aizinger, Vadym"https://www.zbmath.org/authors/?q=ai:aizinger.vadym"Frank, Florian"https://www.zbmath.org/authors/?q=ai:frank.florian"Hajduk, Hennes"https://www.zbmath.org/authors/?q=ai:hajduk.hennes"Rupp, Andreas"https://www.zbmath.org/authors/?q=ai:rupp.andreasIn this paper, the authors have considered the two-dimensional (2D) shallow-water equations. First, they have introduced the enriched Galerkin method for the system of 2D shallow-water equations. Then, they have shown the accuracy and robustness of the proposed method using an analytical convergence test. Finally, they have compared the enriched Galerkin method and discontinuous Galerkin method in terms of accuracy, stability, and robustness using artificial and realistic test problems.
Reviewer: J. Manimaran (Ponda)Matrix oriented reduction of space-time Petrov-Galerkin variational problemshttps://www.zbmath.org/1475.651232022-01-14T13:23:02.489162Z"Henning, Julian"https://www.zbmath.org/authors/?q=ai:henning.julian"Palitta, Davide"https://www.zbmath.org/authors/?q=ai:palitta.davide"Simoncini, Valeria"https://www.zbmath.org/authors/?q=ai:simoncini.valeria"Urban, Karsten"https://www.zbmath.org/authors/?q=ai:urban.karstenSummary: Variational formulations of time-dependent PDEs in space and time yield \((d + 1)\)-dimensional problems to be solved numerically. This increases the number of unknowns as well as the storage amount. On the other hand, this approach enables adaptivity in space and time as well as model reduction w.r.t. both type of variables. In this paper, we show that matrix oriented techniques can significantly reduce the computational timings for solving the arising linear systems outperforming both time-stepping schemes and other solvers.
For the entire collection see [Zbl 1471.65009].A \(\theta\)-\(L\) formulation-based finite element method for solving axisymmetric solid-state dewetting problemshttps://www.zbmath.org/1475.651242022-01-14T13:23:02.489162Z"Huang, Weijie"https://www.zbmath.org/authors/?q=ai:huang.weijie"Jiang, Wei"https://www.zbmath.org/authors/?q=ai:jiang.wei.4|jiang.wei.3|jiang.wei.2|jiang.wei|jiang.wei.1|jiang.wei.5"Zhao, Quan"https://www.zbmath.org/authors/?q=ai:zhao.quanSummary: We propose a \(\theta\)-\(L\) formulation-based finite element method for the sharp-interface model of solid-state dewetting with axisymmetric geometry. The model describes the film/vapor interface using the radial curve in cylindrical coordinates, and is governed by a fourth-order geometric partial differential equation with complex boundary conditions at the moving contact lines. By introducing an appropriate tangential velocity, we derive an equivalent system for the original sharp-interface model. This gives the kinetic equation for the tangential angle \(\theta\) and the total length \(L\) of the radial curve. The new formulation can alleviate the stiffness of the original model and help to maintain mesh equidistribution during the evolution. We present an efficient finite element method for solving the resulting \(\theta\)-\(L\) formulation based on its weak form. Numerical examples are reported to demonstrate the accuracy and efficiency of the numerical scheme.Weak Galerkin finite element method for solving one-dimensional coupled Burgers' equationshttps://www.zbmath.org/1475.651252022-01-14T13:23:02.489162Z"Hussein, Ahmed J."https://www.zbmath.org/authors/?q=ai:hussein.ahmed-jabbar"Kashkool, Hashim A."https://www.zbmath.org/authors/?q=ai:kashkool.hashim-aSummary: In this paper, we apply a weak Galerkin method for solving one dimensional coupled Burgers' equations. Based on a conservation form for nonlinear term and some of the technical derivational. Theorticly, we drive the optimal order error in \(L^2\) and \(H^1\) norm for both continuous and discrete time weak Galerkin finite element schemes, also the stability of continuous time weak Galerkin finite element method is proved. Numerically, the accuracy and effectiveness of the weak Galerkin finite element method are illustrated by using Numerical examples with the lower order Raviart-Thomas element \(RT_k\) for discrete weak derivative space.Approximate deconvolution with correction: a member of a new class of models for high Reynolds number flowshttps://www.zbmath.org/1475.651262022-01-14T13:23:02.489162Z"Labovsky, Alexander E."https://www.zbmath.org/authors/?q=ai:labovsky.alexander-eThis paper introduces a two-step defect correction method to the modeling of the high-Reynolds number flows, which combines the defect correction approach and the turbulence modeling. The method is investigated numerically and theoretically based on the approximate deconvolution models. The competitive performance of the method is shown on several benchmark problems, including a benchmark problem of finding maximal drag and lift coefficients, flow past the step, and the discussion on Taylor-Green vortex solutions.
Reviewer: Zhiming Chen (Beijing)Discretization of Euler's equations using optimal transport: Cauchy and boundary value problemshttps://www.zbmath.org/1475.651272022-01-14T13:23:02.489162Z"Mérigot, Quentin"https://www.zbmath.org/authors/?q=ai:merigot.quentinSummary: This note presents a numerical method based on optimal transport to construct minimal geodesics along the group of volume preserving maps, equipped with the \(\mathrm{L}^2\) metric. As observed by Arnold, such geodesics solve the Euler equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete optimal transport and it is robust enough to extract non-classical, multi-valued solutions of Euler's equations predicted by Brenier and Schnirelman [\textit{Q. Mérigot} and \textit{J.-M. Mirebeau}, SIAM J. Numer. Anal. 54, No. 6, 3465--3492 (2016; Zbl 1354.65136)]. In a second part, we explain how this approach also leads to a numerical scheme able to approximate regular solutions to the Cauchy problem for Euler's equations [\textit{T. O. Gallouët} and \textit{Q. Mérigot}, Found. Comput. Math. 18, No. 4, 835--865 (2018; Zbl 1410.35103)].Convergens of Galerkin's method of approximate solution in parabolic equation with integral condition for the solutionhttps://www.zbmath.org/1475.651282022-01-14T13:23:02.489162Z"Nguyen Thuong Huyen"https://www.zbmath.org/authors/?q=ai:nguyen-thuong-huyen."Smagin, Victor Vasilievich"https://www.zbmath.org/authors/?q=ai:smagin.viktor-vSummary: In the Hilbert space the abstract linear parabolic equation with nonlocal integral condition for the solution is resolved by approximate by the Galerkin's method. The established error estimations of approximate solutions, the convergence of approximate solution to exact solution and the orders of speed of convergence.Finite element method for two-dimensional linear advection equations based on spline methodhttps://www.zbmath.org/1475.651292022-01-14T13:23:02.489162Z"Qu, Kai"https://www.zbmath.org/authors/?q=ai:qu.kai"Dong, Qi"https://www.zbmath.org/authors/?q=ai:dong.qi"Li, Chanjie"https://www.zbmath.org/authors/?q=ai:li.chanjie"Zhang, Feiyu"https://www.zbmath.org/authors/?q=ai:zhang.feiyuSummary: A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.Solving a non-linear fractional convection-diffusion equation using local discontinuous Galerkin methodhttps://www.zbmath.org/1475.651302022-01-14T13:23:02.489162Z"Safdari, Hamid"https://www.zbmath.org/authors/?q=ai:safdari.hamid"Rajabzadeh, Majid"https://www.zbmath.org/authors/?q=ai:rajabzadeh.majid"Khalighi, Moein"https://www.zbmath.org/authors/?q=ai:khalighi.moeinSummary: We propose a local discontinuous Galerkin method for solving a nonlinear convection-diffusion equation consisting of a fractional diffusion described by a fractional Laplacian operator of order \(0<p<2\), a nonlinear diffusion, and a nonlinear convection term. The algorithm is developed by the local discontinuous Galerkin method using Spline interpolations to achieve higher accuracy. In this method, we convert the main problem to a first-order system and approximate the outcome by the Galerkin method. In this study, in contrast to the direct Galerkin method using Legender polynomials, we demonstrate that the proposed method can be suitable for the general fractional convection-diffusion problem, remarkably improve stability and provide convergence order \(O(h^{k+1})\), when \(k\) indicates the degree of polynomials. Numerical results have illustrated the accuracy of this scheme and compare it for different conditions.The weak solvability of the Cauchy problem for a parabolic equation and the mean square convergence of the semi-discrete Galerkin methodhttps://www.zbmath.org/1475.651312022-01-14T13:23:02.489162Z"Smagin, V. V."https://www.zbmath.org/authors/?q=ai:smagin.viktor-vSummary: Sufficient conditions for the weak solvability of the Cauchy problem for a parabolic equation in a Hilbert space. By these conditions, the mean square convergence of approximate solutions which was found using the semi-discrete Galerkin method is established. For the projection subspaces of finite elements type speeds estimates which ave exact in the approximation order ave found.Convergence of the Galerkin's method for an approximate solution of a parabolic equation with periodic conditions for solutionshttps://www.zbmath.org/1475.651322022-01-14T13:23:02.489162Z"Smagin, V. V."https://www.zbmath.org/authors/?q=ai:smagin.viktor-vSummary: In the Hilbert space the abstract linear parabolic equation with periodic conditions for the solution is approximative \(y\) solved by means of semi-discrete Galerkin's method. We establish error estimates for approximate solutions, the convergence of approximate solution to the exact solution and the orders of the rate of convergence.Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equationhttps://www.zbmath.org/1475.651332022-01-14T13:23:02.489162Z"Streitbürger, Florian"https://www.zbmath.org/authors/?q=ai:streitburger.florian"Engwer, Christian"https://www.zbmath.org/authors/?q=ai:engwer.christian"May, Sandra"https://www.zbmath.org/authors/?q=ai:may.sandra"Nüßing, Andreas"https://www.zbmath.org/authors/?q=ai:nussing.andreasSummary: For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called \textit{small cell problem} is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the potentially arbitrarily small cut cells as well. For explicit time stepping schemes this leads to instabilities. In a recent paper [SIAM J. Sci. Comput. 42, No. 6, A3677--A3703 (2020; Zbl 1469.65147)], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation. The usage of the proposed stabilization terms results in stable schemes of first and second order in one and two space dimensions. In one dimension, for piecewise constant data in space and explicit Euler in time, the stabilized scheme can even be shown to be monotone. In this contribution, we will examine the conditions for monotonicity in more detail.
For the entire collection see [Zbl 1471.65009].Finite element approximation of a system coupling curve evolution with prescribed normal contact to a fixed boundary to reaction-diffusion on the curvehttps://www.zbmath.org/1475.651342022-01-14T13:23:02.489162Z"Styles, Vanessa"https://www.zbmath.org/authors/?q=ai:styles.vanessa"Van Yperen, James"https://www.zbmath.org/authors/?q=ai:van-yperen.jamesSummary: We consider a finite element approximation for a system consisting of the evolution of a curve evolving by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The curve evolves inside a given domain \(\Omega \subset \mathbb{R}^2\) and meets \(\partial \Omega\) orthogonally. We present numerical experiments and show the experimental order of convergence of the approximation.
For the entire collection see [Zbl 1471.65009].Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equationshttps://www.zbmath.org/1475.651352022-01-14T13:23:02.489162Z"Yang, Yun-Bo"https://www.zbmath.org/authors/?q=ai:yang.yunbo"Jiang, Yao-Lin"https://www.zbmath.org/authors/?q=ai:jiang.yaolinThe paper deals with the numerical solution of the Schrödinger-Helmholtz equations by the linearized backward Euler Galerkin finite element methods. The authors derive unconditionally optimal error estimates using the temporal-spatial error splitting techniques, where the error between the exact solution and the numerical solution is split into two parts which are called the temporal error and the spatial error. First, they prove the uniform boundedness for the solution of this time-discrete system in some strong norms and derive error estimates in temporal direction. Second, by the above achievements, they obtain the boundedness of the numerical solution in the \(L^\infty\)-norm. Then, the optimal \(L^2\) error estimates for \(r\)-order FEMs are derived without any restriction on the time step size. Numerical results in both two- and three-dimensional spaces justify the theoretical predictions and demonstrate the efficiency of the methods.
Reviewer: Vit Dolejsi (Praha)A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equationshttps://www.zbmath.org/1475.651362022-01-14T13:23:02.489162Z"Yin, Baoli"https://www.zbmath.org/authors/?q=ai:yin.baoli"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.11"Li, Hong"https://www.zbmath.org/authors/?q=ai:li.hong"Zeng, Fanhai"https://www.zbmath.org/authors/?q=ai:zeng.fanhaiSummary: A family of novel time-stepping methods for the fractional calculus operators is presented with a shifted parameter. The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature. To improve the efficiency, two aspects are considered, that i) a fast algorithm is developed to reduce the computation complexity from \(O(N_t^2)\) to \(O( N_t\log N_t)\) and the memory requirement from \(O(N_t)\) to \(O(\log N_t)\), where \(N_t\) denotes the number of successive time steps, and ii) correction terms are added to deal with the initial singularity of the solution. The stability analysis and error estimates are provided in detail where in temporal direction the novel time-stepping methods are applied and the spatial variable is discretized by the finite element method. Numerical results for \(d\)-dimensional examples \((d=1,2,3)\) confirm our theoretical conclusions and the efficiency of the fast algorithm.An exact realization of a modified Hilbert transformation for space-time methods for parabolic evolution equationshttps://www.zbmath.org/1475.651372022-01-14T13:23:02.489162Z"Zank, Marco"https://www.zbmath.org/authors/?q=ai:zank.marcoSummary: We present different possibilities of realizing a modified Hilbert type transformation as it is used for Galerkin-Bubnov discretizations of space-time variational formulations for parabolic evolution equations in anisotropic Sobolev spaces of spatial order 1 and temporal order \(\frac{1}{2}\). First, we investigate the series expansion of the definition of the modified Hilbert transformation, where the truncation parameter has to be adapted to the mesh size. Second, we introduce a new series expansion based on the Legendre chi function to calculate the corresponding matrices for piecewise polynomial functions. With this new procedure, the matrix entries for a space-time finite element method for parabolic evolution equations are computable to machine precision independently of the mesh size. Numerical results conclude this work.Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equationshttps://www.zbmath.org/1475.651382022-01-14T13:23:02.489162Z"Zhao, Jingjun"https://www.zbmath.org/authors/?q=ai:zhao.jingjun"Zhao, Wenjiao"https://www.zbmath.org/authors/?q=ai:zhao.wenjiao"Xu, Yang"https://www.zbmath.org/authors/?q=ai:xu.yang.1Summary: This paper provides a Lagrange nodal discontinuous Galerkin method for solving the time-dependent incompressible space fractional Navier-Stokes equations numerically. The existence and uniqueness of weak solutions are obtained. By combining the Lagrange method in temporal discretization and the hybridized discontinuous Galerkin method in spatial direction, the fully discrete scheme is presented and the stability is proved rigorously. Furthermore, the error estimates for the \(L^2\)-norm are derived in both the velocity and the pressure. Finally, some numerical experiments are given to illustrate the performance of the proposed method and validate the theoretical result.Numerical simulations for the quasi-3D fluid streamer propagation model: methods and applicationshttps://www.zbmath.org/1475.651392022-01-14T13:23:02.489162Z"Zhuang, Chijie"https://www.zbmath.org/authors/?q=ai:zhuang.chijie"Huang, Mengmin"https://www.zbmath.org/authors/?q=ai:huang.mengmin"Zeng, Rong"https://www.zbmath.org/authors/?q=ai:zeng.rongSummary: In this work, we propose and compare four different strategies for simulating the fluid model of quasi-three-dimensional streamer propagation, consisting of Poisson's equation for the particle velocity and two continuity equations for particle transport in the cylindrical coordinate system with angular symmetry. Each strategy involves one method for solving Poisson's equation, a discontinuous Galerkin method for solving the continuity equations, and a total variation-diminishing Runge-Kutta method in temporal discretization. The numerical methods for Poisson's equation include discontinuous Galerkin methods, the mixed finite element method, and the least-squares finite element method. The numerical method for continuity equations is the Oden-Babuška-Baumann discontinuous Galerkin method. A slope limiter for the DG methods in the cylindrical coordinate system is proposed to conserve the physical property. Tests and comparisons show that all four strategies are compatible in the sense that solutions to particle densities converge. Finally, different types of streamer propagation phenomena were simulated using the proposed method, including double-headed streamer in nitrogen and \(\mathrm{SF}_6\) between parallel plates, a streamer discharge in a point-to-plane gap.High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problemshttps://www.zbmath.org/1475.651402022-01-14T13:23:02.489162Z"Zhu, Jun"https://www.zbmath.org/authors/?q=ai:zhu.jun"Shu, Chi-Wang"https://www.zbmath.org/authors/?q=ai:shu.chi-wang"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianSummary: Since the classical WENO schemes [\textit{G.-S. Jiang} and \textit{C.-W. Shu}, J. Comput. Phys. 126, No. 1, 202--228 (1996; Zbl 0877.65065)] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [\textit{J. Zhu} and \textit{C.-W. Shu}, J. Comput. Phys. 375, 659--683 (2018; Zbl 1416.65286)] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical \(L^2\) projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.A local radial basis function method for the Laplace-Beltrami operatorhttps://www.zbmath.org/1475.651412022-01-14T13:23:02.489162Z"Álvarez, Diego"https://www.zbmath.org/authors/?q=ai:alvarez.diego.1"González-Rodríguez, Pedro"https://www.zbmath.org/authors/?q=ai:gonzalez-rodriguez.pedro"Kindelan, Manuel"https://www.zbmath.org/authors/?q=ai:kindelan.manuel-seguraThis article discusses a local meshfree method for the approximation of the Laplace-Beltrami operator on a smooth surface in 3D. The approach relies on radial basis functions augmented with multivariate polynomials. Also, the approach does not require an explicit expression of the surface, which can be simply defined by a set of scattered nodes. The convergence, accuracy and other computational characteristics of the proposed method are discussed. Numerical experiments related to the Turing model for pattern formation and the Schaeffer's model for electrical cardiac tissue behavior are included.
Reviewer: Marius Ghergu (Dublin)A preconditioned conjugated gradient method for computing ground states of rotating dipolar Bose-Einstein condensates via kernel truncation method for dipole-dipole interaction evaluationhttps://www.zbmath.org/1475.651422022-01-14T13:23:02.489162Z"Antoine, Xavier"https://www.zbmath.org/authors/?q=ai:antoine.xavier"Tang, Qinglin"https://www.zbmath.org/authors/?q=ai:tang.qinglin"Zhang, Yong"https://www.zbmath.org/authors/?q=ai:zhang.yong.2Summary: In this paper, we propose an efficient and accurate method to compute the ground state of 2D/3D rotating dipolar BEC by incorporating the Kernel Truncation Method (KTM) for Dipole-Dipole Interaction (DDI) evaluation into the newly-developed Preconditioned Conjugate Gradient (PCG) method [\textit{X. Antoine} et al., J. Comput. Phys. 343, 92--109 (2017; Zbl 1380.81496)]. Adaptation details of KTM and PCG, including multidimensional discrete convolution acceleration for KTM, choice of the preconditioners in PCG, are provided. The performance of our method is confirmed with extensive numerical tests, with emphasis on spectral accuracy of KTM and efficiency of ground state computation with PCG. Application of our method shows some interesting vortex lattice patterns in 2D and 3D respectively.A hybrid method for computing the Schrödinger equations with periodic potential with band-crossings in the momentum spacehttps://www.zbmath.org/1475.651432022-01-14T13:23:02.489162Z"Chai, Lihui"https://www.zbmath.org/authors/?q=ai:chai.lihui"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi"Markowich, Peter A."https://www.zbmath.org/authors/?q=ai:markowich.peter-alexanderSummary: We propose a hybrid method which combines the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number \(\varepsilon \rightarrow 0\), the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an \(\mathcal{O}(1)\) difference between the solutions of the Schrödinger equation and its Dirac approximation.On entropy-stable discretizations and the entropy adjointhttps://www.zbmath.org/1475.651442022-01-14T13:23:02.489162Z"Hicken, Jason E."https://www.zbmath.org/authors/?q=ai:hicken.jason-eFor symmetrizable conservation laws, the entropy variables are adjoints for a functional that balances entropy flux into the domain with sources of entropy inside the domain. In this paper, it was shown that entropy-stable summation by parts (SBP) discretizations mimic an additional property of the continuous equations: the entropy adjoint. The entropy variables satisfy the discrete adjoint equation for a discretized entropy-balance functional if they are evaluated from the discrete solution. They are identical to the discrete adjoint variables up to machine accuracy. The theoretical results are verified for steady inviscid (Euler equation) and viscous flows (Navier-Stokes equation) over an airfoil.
Reviewer: Bülent Karasözen (Ankara)An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equationhttps://www.zbmath.org/1475.651452022-01-14T13:23:02.489162Z"Jia, Junqing"https://www.zbmath.org/authors/?q=ai:jia.junqing"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.1|zhang.hui.8|zhang.hui.3|zhang.hui|zhang.hui.7|zhang.hui.11|zhang.hui.2|zhang.hui.10|zhang.hui.4|zhang.hui.5|zhang.hui.6|zhang.hui.9"Xu, Huanying"https://www.zbmath.org/authors/?q=ai:xu.huanying"Jiang, Xiaoyun"https://www.zbmath.org/authors/?q=ai:jiang.xiaoyunSummary: In this paper, we give a stabilized second order scheme for the time fractional Allen-Cahn equation. The scheme uses the fractional backward difference formula (FBDF) for the time fractional derivative and the Legendre spectral method for the space approximation. The nonlinear terms are treated implicitly with a second order stabilized term. Based on the fractional Grönwall inequality, we strictly prove that the proposed scheme converges to second order accuracy in time and spectral accuracy in space. To save computation time and storage, a fast evaluation is developed. Finally, we give some numerical examples to show the configurations of phase field evolution and verify the effectiveness of the proposed methods.A parallel, non-spatial iterative, and rotational pressure projection method for the nonlinear fluid-fluid interactionhttps://www.zbmath.org/1475.651462022-01-14T13:23:02.489162Z"Li, Jian"https://www.zbmath.org/authors/?q=ai:li.jian.1"Gao, Jiawei"https://www.zbmath.org/authors/?q=ai:gao.jiawei"Shu, Yu"https://www.zbmath.org/authors/?q=ai:shu.yuSummary: In this paper, a parallel, non-spatial iterative, and rotational pressure projection method for the coupled Navier-Stokes equations is proposed and developed. As for each Navier-Stokes equation, one elliptic equation and one Possion equation at each time step determine the values of the velocity and pressure, respectively. Moreover, we only need to solve four simple linear equations, and the computational time of the whole system is greatly reduced. The rotational pressure projection method achieves the same accuracy as the traditional decoupled fractional time-stepping method. Furthermore, we apply the presented method, the rotational pressure projection method and the linear decoupled fractional time-stepping method to the submarine mountain problem. Compared to the streamline diagrams of these three methods from two aspects of accuracy and CPU times, we obtain that the presented method has good parallelism and is the most efficient method, the rotational pressure projection method is more efficient than the traditional decoupled fractional time-stepping method. Finally, numerical results verify that our proposed method has the high efficiency.Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equationhttps://www.zbmath.org/1475.651472022-01-14T13:23:02.489162Z"Li, Xiao"https://www.zbmath.org/authors/?q=ai:li.xiao"Qiao, Zhonghua"https://www.zbmath.org/authors/?q=ai:qiao.zhonghua"Wang, Cheng"https://www.zbmath.org/authors/?q=ai:wang.cheng.1This paper provides a convergence analysis for a first order stabilized semi-implicit numerical scheme for the non-local Cahn-Hilliard equation which is proposed in [\textit{Q. Du} et al., J. Comput. Phys. 363, 39--54 (2018; Zbl 1395.65099)]. The error estimate is established in the discrete \(L^\infty(0,T; H^{-1})\) norm and the \(L^2(0,T;L^2)\) norm. The energy stability of the scheme is proved under the condition that the stabilizing constant is sufficiently large.
Reviewer: Zhiming Chen (Beijing)Hermite spectral collocation methods for fractional PDEs in unbounded domainshttps://www.zbmath.org/1475.651482022-01-14T13:23:02.489162Z"Tang, Tao"https://www.zbmath.org/authors/?q=ai:tang.tao"Yuan, Huifang"https://www.zbmath.org/authors/?q=ai:yuan.huifang"Zhou, Tao"https://www.zbmath.org/authors/?q=ai:zhou.taoSummary: This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite points. In this work, two Hermite-type functions are employed to serve as basis functions. Our main task is to find corresponding differentiation matrices which are computed recursively. Two important issues relevant to condition numbers and scaling factors will be discussed. Applications of the spectral collocation methods to multi-term fractional PDEs are also presented. Several numerical examples are carried out to demonstrate the effectiveness of the proposed methods.Nonstandard Fourier pseudospectral time domain (PSTD) schemes for partial differential equationshttps://www.zbmath.org/1475.651492022-01-14T13:23:02.489162Z"Treeby, Bradley E."https://www.zbmath.org/authors/?q=ai:treeby.bradley-e"Wise, Elliott S."https://www.zbmath.org/authors/?q=ai:wise.elliott-s"Cox, B. T."https://www.zbmath.org/authors/?q=ai:cox.ben-tSummary: A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or \(k\)-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.Conservative numerical schemes for the nonlinear fractional Schrödinger equationhttps://www.zbmath.org/1475.651502022-01-14T13:23:02.489162Z"Wu, Longbin"https://www.zbmath.org/authors/?q=ai:wu.longbin"Ma, Qiang"https://www.zbmath.org/authors/?q=ai:ma.qiang"Ding, Xiaohua"https://www.zbmath.org/authors/?q=ai:ding.xiaohuaSummary: This paper deals with the Crank-Nicolson Fourier collocation method for the nonlinear fractional Schrödinger equation containing a fractional derivative. We prove that at each discrete time the method preserves the discrete mass and energy conservation laws. The existence, uniqueness and convergence of the numerical solution are also investigated. In particular, we show that the method has the second-order accuracy in time and the spectral accuracy in space. Since the proposed schemes are implicit, they are solved by an iteration algorithm with FFT. Two examples illustrate the efficiency and accuracy of the numerical schemes.Spectral computation of low probability tails for the homogeneous Boltzmann equationhttps://www.zbmath.org/1475.651512022-01-14T13:23:02.489162Z"Zweck, John"https://www.zbmath.org/authors/?q=ai:zweck.john-w"Chen, Yanping"https://www.zbmath.org/authors/?q=ai:chen.yanping.2|chen.yanping.1|chen.yanping.3"Goeckner, Matthew J."https://www.zbmath.org/authors/?q=ai:goeckner.matthew-j"Shen, Yannan"https://www.zbmath.org/authors/?q=ai:shen.yannanThe authors focus on low-probability tails for the velocity distribution of the spatially homogeneous Boltzmann equation. They invest the spectral-Lagrangian method of \textit{I. M. Gamba} and \textit{S. H. Tharkabhushanam} [J. Comput. Phys. 228, No. 6, 2012--2036 (2009; Zbl 1159.82320); J. Comput. Math. 28, No. 4, 430--460 (2010; Zbl 1228.76138)] in this context and demonstrate numerically the importance of the choice of the truncation parameter. They present an error estimate, guiding the choice of the truncation parameter.
Reviewer: Hendrik Ranocha (Münster)The positive numerical solution for stochastic age-dependent capital system based on explicit-implicit algorithmhttps://www.zbmath.org/1475.651522022-01-14T13:23:02.489162Z"Du, Yanyan"https://www.zbmath.org/authors/?q=ai:du.yanyan"Zhang, Qimin"https://www.zbmath.org/authors/?q=ai:zhang.qimin"Meyer-Baese, Anke"https://www.zbmath.org/authors/?q=ai:meyer-base.ankeSummary: We know that the exact solutions for most of stochastic age-structured capital systems are difficult to find. The numerical approximation method becomes an important tool to study properties for stochastic age-structured capital models. In the paper, we study the numerical solution for stochastic age-dependent capital system based on explicit-implicit algorithm and discuss the convergence of numerical solution. For the practical significance of capital, we need to consider the positivity of the numerical solution. Therefore, we introduce a penalty factor in the stochastic age-dependent capital system to maintain the positivity, and analyze the convergence of the positive numerical solution. Finally, an example is given to verify our theoretical results.Analysis of a splitting method for stochastic balance lawshttps://www.zbmath.org/1475.651532022-01-14T13:23:02.489162Z"Karlsen, K. H."https://www.zbmath.org/authors/?q=ai:karlsen.kenneth-hvistendahl"Storrøsten, E. B."https://www.zbmath.org/authors/?q=ai:storrosten.erlend-briseidSummary: We analyse a semidiscrete splitting method for conservation laws driven by a semilinear noise term. Making use of fractional bounded variation (BV) estimates, we show that the splitting method generates approximate solutions converging to the exact solution, as the time step \(\Delta t \rightarrow 0\). Under the assumption of a homogenous noise function, and thus the availability of BV estimates, we provide an \(L^1\)-error estimate. Bringing into play a generalization of Kružkov's entropy condition, permitting the `Kružkov constants' to be Malliavin differentiable random variables, we establish an \(L^1\)-convergence rate of order \(\frac13\) in \(\Delta t\).Meshless singular boundary method for two-dimensional pseudo-parabolic equation: analysis of stability and convergencehttps://www.zbmath.org/1475.651542022-01-14T13:23:02.489162Z"Aslefallah, Mohammad"https://www.zbmath.org/authors/?q=ai:aslefallah.mohammad"Abbasbandy, Saeid"https://www.zbmath.org/authors/?q=ai:abbasbandy.saeid"Shivanian, Elyas"https://www.zbmath.org/authors/?q=ai:shivanian.elyasSummary: In this study, the singular boundary method is applied to solve time-dependent pseudo-parabolic equations in two space dimensions with initial and Dirichlet-type boundary conditions. A splitting procedure is used to split the solution of the inhomogeneous governing equation into a homogeneous solution and a particular solution. This work presents the numerical operation for calculating the particular solution and homogeneous solution. Several numerical examples are provided to show the accuracy and efficiency of the method. Furthermore, the analysis of stability and convergence is presented.An efficient approach for the numerical solution of fifth-order KdV equationshttps://www.zbmath.org/1475.651552022-01-14T13:23:02.489162Z"Ahmad, Hijaz"https://www.zbmath.org/authors/?q=ai:ahmad.hijaz"Khan, Tufail A."https://www.zbmath.org/authors/?q=ai:khan.tufail-a"Yao, Shao-Wen"https://www.zbmath.org/authors/?q=ai:yao.shaowenSummary: The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.Nonlocal wave propagation in unbounded multi-scale mediahttps://www.zbmath.org/1475.651562022-01-14T13:23:02.489162Z"Du, Qiang"https://www.zbmath.org/authors/?q=ai:du.qiang"Zhang, Jiwei"https://www.zbmath.org/authors/?q=ai:zhang.jiwei"Zheng, Chunxiong"https://www.zbmath.org/authors/?q=ai:zheng.chunxiongSummary: This paper focuses on the simulation of nonlocal wave propagations in unbounded multi-scale mediums. To this end, we consider two issues: (a) the design of artificial/absorbing boundary conditions; and (b) the construction of an asymptotically compatible (AC) scheme for the nonlocal operator with general kernels. The design of ABCs facilitates us to reformulate unbounded domain problems into bounded domain problems. The construction of AC scheme facilitates us to simulate nonlocal wave propagations in multi-scale mediums. By applying the proposed ABCs and the proposed AC scheme, we investigate different wave propagation behaviors in the ``local'' and nonlocal mediums through numerical examples.Proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientshttps://www.zbmath.org/1475.651572022-01-14T13:23:02.489162Z"Jentzen, Arnulf"https://www.zbmath.org/authors/?q=ai:jentzen.arnulf"Salimova, Diyora"https://www.zbmath.org/authors/?q=ai:salimova.diyora-f"Welti, Timo"https://www.zbmath.org/authors/?q=ai:welti.timoSummary: In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). These numerical simulations indicate that DNNs seem to have the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy \(\varepsilon>0\) and the dimension \(d\in\mathbb{N}\) of the function which the DNN aims to approximate in such computational problems. There is also a large number of rigorous mathematical approximation results for artificial neural networks in the scientific literature but there are only a few special situations where results in the literature can rigorously justify the success of DNNs to approximate high-dimensional functions. The key contribution of this article is to reveal that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. We prove that the number of parameters used to describe the employed DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy \(\varepsilon>0\) and the PDE dimension \(d\in\mathbb{N}\). A crucial ingredient in our proof is the fact that the artificial neural network used to approximate the solution of the PDE is indeed a deep artificial neural network with a large number of hidden layers.A new efficient method for time-fractional sine-Gordon equation with the Caputo and Caputo-Fabrizio operatorshttps://www.zbmath.org/1475.651582022-01-14T13:23:02.489162Z"Khalouta, Ali"https://www.zbmath.org/authors/?q=ai:khalouta.ali"Kadem, Abdelouahab"https://www.zbmath.org/authors/?q=ai:kadem.abdelouahabSummary: In this work, a new efficient method called, Elzaki's fractional decomposition method (EFDM) has been used to give an approximate series solutions to time-fractional Sine-Gordon equation. The time-fractional derivatives are described in the Caputo and Caputo-Fabrizio sense. The EFDM is based on the combination of two different methods which are: the Elzaki transform method and the Adomian decomposition method. To demonstrate the accuracy and efficiency of the proposed method, a numerical example is provided. The obtained results indicate that the EFDM is simple and practical for solving the fractional partial differential equations which appear in various fields of applied sciences.The introduction of the surfing scheme for shock capturing with high-stability and high-speed convergencehttps://www.zbmath.org/1475.651592022-01-14T13:23:02.489162Z"Mollaei, Mehdi"https://www.zbmath.org/authors/?q=ai:mollaei.mehdi"Malek Jafarian, Seyyed Majid"https://www.zbmath.org/authors/?q=ai:malek-jafarian.seyyed-majidSummary: In this paper, a novel shock capturing scheme with high stability and speed of solution is presented. The scheme divides the dependent variables into two parts, i.e. Surf and Surfer, like surfing sport. The Surf part is calculated by applying the concept of Sobolev gradient on the dependent variables. Furthermore, it has the least difference with these variables, along with the lowest discretization error and the same equation. Hence, it is expected that this part can be solved at high stability conditions and large time steps. In addition, the Surfer part is obtained from the difference between the Surf part and the dependent variables. Therefore, only the error-maker operators (for example, a discontinuity like a shock) are in this part and accordingly it has a local nature. Due to this feature, it can be solved with fewer points compared to the Surf part. Therefore, its equation is solved as quickly as Surf equation, despite its limited stability conditions. Finally, the summation of the Surf and Surfer solutions makes the main solution. The scheme has been applied to the inviscid Burgers' equation with an initial value of the step function, and one-dimensional inviscid Euler flow through a nozzle with a shock wave. The stability condition for the Burgers' equation has been increased to a Courant number of \(10^{10}\) and the solution time for the Euler flow case has been decreased by one-fifth. Comparison of the results with traditional methods indicates the ultra-high stability and ultra-speed convergence of this scheme.An error of the iteration method for a Timoshenko nonhomogeneous equationhttps://www.zbmath.org/1475.651602022-01-14T13:23:02.489162Z"Peradze, J."https://www.zbmath.org/authors/?q=ai:peradze.jemal"Tsiklauri, Z."https://www.zbmath.org/authors/?q=ai:tsiklauri.zviad-iSummary: We consider the initial boundary value problem for an integro-differential equation describing the vibration of a beam. Using the Galerkin method and a symmetric difference scheme, the solution is approximates with respect to a spatial and a time variable. Thus the problem is reduced to a system of nonlinear discrete equations which is solved by the iteration method. The convergence of the method is proved.A novel iterative solution for time-fractional Boussinesq equation by reproducing kernel methodhttps://www.zbmath.org/1475.651612022-01-14T13:23:02.489162Z"Sakar, Mehmet Giyas"https://www.zbmath.org/authors/?q=ai:sakar.mehmet-giyas"Saldır, Onur"https://www.zbmath.org/authors/?q=ai:saldir.onurSummary: In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, ``good'' Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics.Data-driven modeling for wave-propagationhttps://www.zbmath.org/1475.651622022-01-14T13:23:02.489162Z"van Leeuwen, Tristan"https://www.zbmath.org/authors/?q=ai:van-leeuwen.tristan"van Leeuwen, Peter Jan"https://www.zbmath.org/authors/?q=ai:van-leeuwen.peter-jan"Zhuk, Sergiy"https://www.zbmath.org/authors/?q=ai:zhuk.sergiy-mSummary: Many imaging modalities, such as ultrasound and radar, rely heavily on the ability to accurately model wave propagation. In most applications, the response of an object to an incident wave is recorded and the goal is to characterize the object in terms of its physical parameters (e.g., density or soundspeed). We can cast this as a joint parameter and state estimation problem. In particular, we consider the case where the inner problem of estimating the state is a weakly constrained data-assimilation problem. In this paper, we discuss a numerical method for solving this variational problem.
For the entire collection see [Zbl 1471.65009].Error estimate for discrete approximation of the radiative-conductive heat transfer problem in a system of absolutely black rodshttps://www.zbmath.org/1475.651632022-01-14T13:23:02.489162Z"Amosov, A. A."https://www.zbmath.org/authors/?q=ai:amosov.andrei"Krymov, N. E."https://www.zbmath.org/authors/?q=ai:krymov.nikita-eThe authors develop discrete and semidiscrete approximations of the radiative-conductive heat transfer problem. The problem domain is assumed to consist of \(n^2\) absolutely black rods with circular sections, separated by vacuum. Firstly, some properties of the continuous problem are presented, such as existence and uniqueness, and maximum principle-type results. Then finite difference approximation of the problem is developed. The main result of the paper is error estimate, which establishes the convergence of the solution of the discrete to the exact one at a rate of \(\mathcal{O}\left( \frac{\sqrt{\varepsilon}}{\lambda} \right)\), where \(\varepsilon\) is the radius of rods and \(\lambda\) is the heat-conductivity coefficient.
Reviewer: Aziz Takhirov (Sharja)Monotone and second order consistent scheme for the two dimensional Pucci equationhttps://www.zbmath.org/1475.651642022-01-14T13:23:02.489162Z"Bonnans, Joseph Frédéric"https://www.zbmath.org/authors/?q=ai:bonnans.joseph-frederic"Bonnet, Guillaume"https://www.zbmath.org/authors/?q=ai:bonnet.guillaume"Mirebeau, Jean-Marie"https://www.zbmath.org/authors/?q=ai:mirebeau.jean-marieSummary: We introduce a new strategy for the design of second-order accurate discretizations of non-linear second order operators of Bellman type, which preserves degenerate ellipticity. The approach relies on Selling's formula, a tool from lattice geometry, and is applied to the Pucci equation, discretized on a two dimensional Cartesian grid. Numerical experiments illustrate the robustness and the accuracy of the method.
For the entire collection see [Zbl 1471.65009].A multi-level ADMM algorithm for elliptic PDE-constrained optimization problemshttps://www.zbmath.org/1475.651652022-01-14T13:23:02.489162Z"Chen, Xiaotong"https://www.zbmath.org/authors/?q=ai:chen.xiaotong"Song, Xiaoliang"https://www.zbmath.org/authors/?q=ai:song.xiaoliang"Chen, Zixuan"https://www.zbmath.org/authors/?q=ai:chen.zixuan"Yu, Bo"https://www.zbmath.org/authors/?q=ai:yu.bo.1In this paper, a multi-level alternating direction method of multipliers ADMM (mADMM) algorithm is employed for solving optimization problems with PDE constraints applying the `optimize-discretize-optimize' strategy. In this way, the subproblems of the inexact ADMM algorithm are discretized by different discretization schemes. Following the multi-level strategy, the grid is refined gradually and the subproblems are solved inexactly with Krylov-based methods. Overall a fast convergent mADMM algorithm is designed, which significantly reduces the computational cost. The authors show that the iteration complexity is of order \({\mathcal O} (1/k) \), where \(k\) is the number of iterations. Numerical results demonstrate the efficiency of the proposed mADMM algorithm.
Reviewer: Bülent Karasözen (Ankara)A generalized optimal fourth-order finite difference scheme for a 2D Helmholtz equation with the perfectly matched layer boundary conditionhttps://www.zbmath.org/1475.651662022-01-14T13:23:02.489162Z"Dastour, Hatef"https://www.zbmath.org/authors/?q=ai:dastour.hatef"Liao, Wenyuan"https://www.zbmath.org/authors/?q=ai:liao.wenyuanSummary: A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference schemes, optimal point-weighting 25p and optimal point-weighting 17p. It is shown that the two schemes are consistent with the Helmholtz equation with PML. Moreover, an error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, we implement the refined choice strategy for selecting optimal parameters and present refined point-weighting 25p and refined point-weighting 17p finite difference schemes. Furthermore, three numerical examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion.A new development of sixth order accurate compact scheme for the Helmholtz equationhttps://www.zbmath.org/1475.651672022-01-14T13:23:02.489162Z"Kumar, Neelesh"https://www.zbmath.org/authors/?q=ai:kumar.neelesh"Dubey, Ritesh Kumar"https://www.zbmath.org/authors/?q=ai:dubey.ritesh-kumarSummary: A standard sixth order compact finite difference scheme for two dimensional Helmholtz equation is further improved and a more accurate scheme is presented for the two dimensional Helmholtz equation which is also compact. The novel feature of the present scheme is that it is less sensitive to the associated wave number when compared with that for available sixth order schemes. Theoretical analysis is presented for the newly constructed scheme. The high accuracy of the proposed scheme is illustrated by comparing numerical solutions for solving the two dimensional Helmholtz equations using available sixth-order schemes and the present scheme.Crank-Nicolson method of a two-grid finite volume element algorithm for nonlinear parabolic equationshttps://www.zbmath.org/1475.651682022-01-14T13:23:02.489162Z"Gong, Yunjie"https://www.zbmath.org/authors/?q=ai:gong.yunjie"Chen, Chuanjun"https://www.zbmath.org/authors/?q=ai:chen.chuanjun"Lou, Yuzhi"https://www.zbmath.org/authors/?q=ai:lou.yuzhi"Xue, Guanyu"https://www.zbmath.org/authors/?q=ai:xue.guanyuSummary: A two-grid finite volume element algorithm based on Crank-Nicolson scheme for nonlinear parabolic equations is proposed. In this method, the nonlinear problem is solved on a coarse grid of size \(H\) and a linear problem is considered on a fine grid of size \(h\) by using the coarse-grid solution and one Newton iteration. This helps to improve the computing efficiency while keeping the accuracy. It is proved that the two-grid method can achieve asymptotically optimal error estimates in spaces and second order accuracy in time. Numerical results are consistent with the theoretical findings.Multistage preconditioning for adaptive discretization of porous media two-phase flowhttps://www.zbmath.org/1475.651692022-01-14T13:23:02.489162Z"Kane, Birane"https://www.zbmath.org/authors/?q=ai:kane.biraneSummary: We present a constrained pressure residual (CPR) two-stage preconditioner applied to a discontinuous Galerkin discretization of a two-phase flow in strongly heterogeneous porous media. We consider a fully implicit, locally conservative, higher order discretization on adaptively generated meshes. The implementation is based on the open-source PDE software framework Dune and its PETSc binding.
For the entire collection see [Zbl 1471.65009].A monotone finite volume scheme with second order accuracy for convection-diffusion equations on deformed meshes.https://www.zbmath.org/1475.651702022-01-14T13:23:02.489162Z"Lan, Bin"https://www.zbmath.org/authors/?q=ai:lan.bin"Sheng, Zhiqiang"https://www.zbmath.org/authors/?q=ai:sheng.zhiqiang"Yuan, Guangwei"https://www.zbmath.org/authors/?q=ai:yuan.guangweiSummary: In this paper, we present a new monotone finite volume scheme for the steady state convection-diffusion equation. The discretization of diffusive flux [\textit{Z. Sheng} and \textit{G. Yuan}, J. Comput. Phys. 315, 182--193 (2016; Zbl 1349.65582)] is utilised and a new corrected upwind scheme with second order accuracy for the discretization of convective flux is proposed based on some available informations of diffusive flux. The scheme is locally conservative and monotone on deformed meshes, and has only cell-centered unknowns. Numerical results are presented to show that the scheme obtains second-order accuracy for the solution and first-order accuracy for the flux.A quadratic finite volume method for nonlinear elliptic problemshttps://www.zbmath.org/1475.651712022-01-14T13:23:02.489162Z"Zhang, Yuanyuan"https://www.zbmath.org/authors/?q=ai:zhang.yuanyuan"Chen, Chuanjun"https://www.zbmath.org/authors/?q=ai:chen.chuanjun"Bi, Chunjia"https://www.zbmath.org/authors/?q=ai:bi.chunjiaThe authors consider a high order finite volume finite element method to approximate a large class of noninear elliptic equations in two dimensions. The numerical method uses the quadratic finite element spaces on the primal mesh. A nonlinear scheme is established. The existence and uniqueness of the discrete solution is proved. An error estimate of order \(O(h|\ln h|)\) in the \(W^{1,\infty}\)-norm is derived. It is also shown that the discrete solution has an optimal error estimate of order \(O(h^2)\) in the \(H^{1}\)-norm. Some numerical tests are presented to confirm the theoretical results. These error estimates are proved under some suitable assumptions on the exact solution.
Reviewer: Abdallah Bradji (Annaba)A stabilizer free weak Galerkin finite element method on polytopal mesh. IIIhttps://www.zbmath.org/1475.651722022-01-14T13:23:02.489162Z"Ye, Xiu"https://www.zbmath.org/authors/?q=ai:ye.xiu"Zhang, Shangyou"https://www.zbmath.org/authors/?q=ai:zhang.shangyouSummary: A weak Galerkin (WG) finite element method without stabilizers was introduced in [the authors, ibid. 371, Article ID 112699, 9 p. (2020; Zbl 1434.65285)] on polytopal mesh. Then it was improved in [the authors, ibid. 394, Article ID 113525, 11 p. (2021; Zbl 1475.65205)] with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG method on polytopal mesh. This method has convergence rates two orders higher than the optimal convergence rates for the corresponding WG solution in both an energy norm and the \(L^2\) norm. The numerical examples are tested for low and high order elements in two and three dimensional spaces.Wells' identification and transmissivity estimation in porous mediahttps://www.zbmath.org/1475.651732022-01-14T13:23:02.489162Z"Ameur, Hend Ben"https://www.zbmath.org/authors/?q=ai:ameur.hend-ben"Hariga-Tlatli, Nejla"https://www.zbmath.org/authors/?q=ai:tlatli.nejla-hariga"Mansouri, Wafa"https://www.zbmath.org/authors/?q=ai:mansouri.wafaSummary: This paper deals with the inverse problems of wells' location and transmissivity estimation in a saturated porous media. Wells are considered as circular holes and the heterogeneous domain is divided into zones with constant transmissivity in each one. The main used tool for wells' location is the topological gradient method applied to a design function defined with respect to available data. Moreover, this technique is incorporated in an adaptive parameterization algorithm leading, in a progressive way, to recover interfaces between hydrogeological zones and transmissivity values. The obtained algorithm allows to recover jointly the transmissivities and the wells' locations. Then the proposed method is tested on a simplified model inspired from the Rocky Mountain aquifer.Solution of Cauchy problems by the multiple scale method of particular solutions using polynomial basis functionshttps://www.zbmath.org/1475.651742022-01-14T13:23:02.489162Z"Lin, Ji"https://www.zbmath.org/authors/?q=ai:lin.ji"Zhang, Yuhui"https://www.zbmath.org/authors/?q=ai:zhang.yuhui"Dangal, Thir"https://www.zbmath.org/authors/?q=ai:dangal.thir"Chen, C. S."https://www.zbmath.org/authors/?q=ai:chen.ching-shyangSummary: We have recently proposed a new meshless method for solving second order partial differential equations where the polynomial particular solutions are obtained analytically [\textit{T. Dangal} et al., Comput. Math. Appl. 73, No. 1, 60--70 (2017; Zbl 1368.65255)]. In this paper, we further extend this new method for the solution of general two- and three-dimensional Cauchy problems. The resulting system of linear equations is ill-conditioned, and therefore, the solution will be regularized by using a multiple scale technique in conjunction with the Tikhonov regularization method, while the \(L\)-curve approach is used for the determination of a suitable regularization parameter. Numerical examples including 2D and 3D problems in both smooth and piecewise smooth geometries are given to demonstrate the validity and applicability of the new approach.Identification of obstacles immersed in a stationary Oseen fluid via boundary measurementshttps://www.zbmath.org/1475.651752022-01-14T13:23:02.489162Z"Karageorghis, Andreas"https://www.zbmath.org/authors/?q=ai:karageorghis.andreas"Lesnic, Daniel"https://www.zbmath.org/authors/?q=ai:lesnic.danielSummary: In this paper we consider the interior inverse problem of identifying a rigid boundary of an annular infinitely long cylinder within which there is a stationary Oseen viscous fluid, by measuring various quantities such as the fluid velocity, fluid traction (stress force) and/or the pressure gradient on portions of the outer accessible boundary of the annular geometry. The inverse problems are nonlinear with respect to the variable polar radius parameterizing the unknown star-shaped obstacle. Although for the type of boundary data that we are considering the obstacle can be uniquely identified based on the principle of analytic continuation, its reconstruction is still unstable with respect to small errors in the measured data. In order to deal with this instability, the nonlinear Tikhonov regularization is employed. Obstacles of various shapes are numerically reconstructed using the method of fundamental solutions for approximating the fluid velocity and pressure combined with the \(\mathrm{MATLAB}^{\copyright}\) toolbox routine lsqnonlin for minimizing the nonlinear Tikhonov's regularization functional subject to simple bounds on the variables.A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equationhttps://www.zbmath.org/1475.651762022-01-14T13:23:02.489162Z"Zhao, Zhenyu"https://www.zbmath.org/authors/?q=ai:zhao.zhenyu"You, Lei"https://www.zbmath.org/authors/?q=ai:you.lei"Meng, Zehong"https://www.zbmath.org/authors/?q=ai:meng.zehongSummary: In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysishttps://www.zbmath.org/1475.651772022-01-14T13:23:02.489162Z"Tani, Mattia"https://www.zbmath.org/authors/?q=ai:tani.mattiaSummary: In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the recently-developed weighted quadrature for the Galerkin discretization. In this paper, we are interested in the solution of the linear systems arising from the discretization of the Poisson problem using one of these approaches. In [\textit{G. Sangalli} and \textit{M. Tani} [SIAM J. Sci. Comput. 38, No. 6, A3644--A3671 (2016; Zbl 1353.65035)], a well-established direct solver for linear systems with tensor structure was used as a preconditioner in the context of Galerkin isogeometric analysis, yielding promising results. In particular, this preconditioner is robust with respect to the mesh size \(h\) and the spline degree \(p\). In the present work, we discuss how a similar approach can be applied to the considered nonsymmetric linear systems. The efficiency of the proposed preconditioning strategy is assessed with numerical experiments on two-dimensional and three-dimensional problems.A skeletal finite element method can compute lower eigenvalue boundshttps://www.zbmath.org/1475.651782022-01-14T13:23:02.489162Z"Carstensen, Carsten"https://www.zbmath.org/authors/?q=ai:carstensen.carsten"Zhai, Qilong"https://www.zbmath.org/authors/?q=ai:zhai.qilong"Zhang, Ran"https://www.zbmath.org/authors/?q=ai:zhang.ran.1The computation of eigenvalues of the Laplacian with homogeneous Dirichlet boundary conditions on regular triangulations is analyzed. The numerical method is a hybridized discontinuous Galerkin finite element method with Lehrenfeld-Schöberl stabilization. Guaranteed error control of the eigenvalues, in particular for the lower bound, is derived using a lower bound for the stabilization parameter. Numerical experiments in 2D \(L\)-shaped domains illustrate the behavior of the eigenvalues for different maximal mesh size and confirm the theoretical results.
Reviewer: Kai Schneider (Marseille)Smoothed-adaptive perturbed inverse iteration for elliptic eigenvalue problemshttps://www.zbmath.org/1475.651792022-01-14T13:23:02.489162Z"Giani, Stefano"https://www.zbmath.org/authors/?q=ai:giani.stefano"Grubišić, Luka"https://www.zbmath.org/authors/?q=ai:grubisic.luka"Heltai, Luca"https://www.zbmath.org/authors/?q=ai:heltai.luca"Mulita, Ornela"https://www.zbmath.org/authors/?q=ai:mulita.ornelaSummary: We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We couple the perturbed inverse iteration approach with mesh refinement strategy based on residual estimators. We demonstrate our approach on model problems in two and three dimensions.On effects of perforated domains on parameter-dependent free vibrationhttps://www.zbmath.org/1475.651802022-01-14T13:23:02.489162Z"Giani, Stefano"https://www.zbmath.org/authors/?q=ai:giani.stefano"Hakula, Harri"https://www.zbmath.org/authors/?q=ai:hakula.harriSummary: Free vibration characteristics of thin perforated shells of revolution vary depending not only on the dimensionless thickness of the shell but also on the perforation structure. All holes are assumed to be free, that is, without any kinematical constraints. For a given configuration there exists a critical value of the dimensionless thickness below which homogenisation fails, since the modes do not have corresponding counterparts in the non-perforated reference shell. For a regular \(g\times g\)-perforation pattern, the critical thickness is reached when the lowest mode has an angular wave number of \(g/2\). This observation is supported both by geometric arguments and numerical experiments. The numerical experiments have been computed in 2D with high-order finite element method supporting Pitkäranta's mathematical shell model [\textit{L. Beirão da Veiga} et al., Math. Models Methods Appl. Sci. 18, No. 11, 1983--2002 (2008; Zbl 1155.74025); \textit{J. Pitkäranta} et al., Comput. Methods Appl. Mech. Eng. 190, No. 22--23, 2943--2975 (2001; Zbl 0983.74039)].Solving elliptic eigenproblems with adaptive multimesh \(hp\)-FEMhttps://www.zbmath.org/1475.651812022-01-14T13:23:02.489162Z"Giani, Stefano"https://www.zbmath.org/authors/?q=ai:giani.stefano"Solin, Pavel"https://www.zbmath.org/authors/?q=ai:solin.pavelSummary: This paper proposes a novel adaptive higher-order finite element \((hp\)-FEM) method for solving elliptic eigenvalue problems, where \(n\) eigenpairs are calculated simultaneously, but on individual higher-order finite element meshes. The meshes are automatically \(hp\)-refined independently of each other, with the goal to use an optimal mesh sequence for each eigenfunction. The method and the adaptive algorithm are described in detail. Numerical examples clearly demonstrate the superiority of the novel method over the standard approach where all eigenfunctions are approximated on the same finite element mesh.A mortar spectral element method for full-potential electronic structure calculationshttps://www.zbmath.org/1475.651822022-01-14T13:23:02.489162Z"Guo, Yichen"https://www.zbmath.org/authors/?q=ai:guo.yichen"Jia, Lueling"https://www.zbmath.org/authors/?q=ai:jia.lueling"Chen, Huajie"https://www.zbmath.org/authors/?q=ai:chen.huajie"Li, Huiyuan"https://www.zbmath.org/authors/?q=ai:li.huiyuan"Zhang, Zhimin"https://www.zbmath.org/authors/?q=ai:zhang.zhiminSummary: In this paper, we propose an efficient mortar spectral element approximation scheme for full-potential electronic structure calculations. As a subsequent work of
\textit{H. Li} and \textit{Z. Zhang} [SIAM J. Sci. Comput. 39, No. 1, A114--A140 (2017; Zbl 1355.65150)], the paper adopts a similar domain decomposition that the computational domain is first decomposed into a number of cuboid subdomains satisfying each nucleus is located in the center of one cube, in which a small ball element centered at the site of the nucleus is attached, and the remainder of the cube is further partitioned into six curvilinear hexahedrons. Specially designed Sobolev-orthogonal basis is adopted in each ball. Classic conforming spectral element approximations using mapped Jacobi polynomials are implemented on the curvilinear hexahedrons and the cuboid elements without nuclei. A mortar technique is applied to patch the different discretizations. Numerical experiments are carried out to demonstrate the efficiency of our scheme, especially the spectral convergence rates of the ground state approximations. Essentially the algorithm can be extended to general eigenvalue problems with the Coulomb singularities.Computing eigenvalues and eigenfunctions of Schrödinger equations using a model reduction approachhttps://www.zbmath.org/1475.651832022-01-14T13:23:02.489162Z"Li, Shuangping"https://www.zbmath.org/authors/?q=ai:li.shuangping"Zhang, Zhiwen"https://www.zbmath.org/authors/?q=ai:zhang.zhiwenSummary: We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a low-dimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schrödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problemshttps://www.zbmath.org/1475.651842022-01-14T13:23:02.489162Z"Xu, Fei"https://www.zbmath.org/authors/?q=ai:xu.fei.3|xu.fei|xu.fei.1|xu.fei.2"Yue, Meiling"https://www.zbmath.org/authors/?q=ai:yue.meiling"Zheng, Bin"https://www.zbmath.org/authors/?q=ai:zheng.binSummary: Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.Overlapping multipatch isogeometric method with minimal stabilizationhttps://www.zbmath.org/1475.651852022-01-14T13:23:02.489162Z"Antolin, Pablo"https://www.zbmath.org/authors/?q=ai:antolin.pablo"Buffa, Annalisa"https://www.zbmath.org/authors/?q=ai:buffa.annalisa"Puppi, Riccardo"https://www.zbmath.org/authors/?q=ai:puppi.riccardo"Wei, Xiaodong"https://www.zbmath.org/authors/?q=ai:wei.xiaodongThe paper present a novel method for isogeometric analysis (IGA) to directly work on geometries constructed by Boolean operations including difference (i.e., trimming), union, and intersection. This work focuses on the union operation, which involves multiple independent, generally nonconforming and trimmed, spline patches. The Nitsche's method is employed to weakly couple independent patches through visible interfaces. The so-called minimal stabilization method is proposed in the context of union to address the stability issue that arises from bad cut elements. The proposed method recovers stability and guarantees well-posedness of the problem as well as optimal error estimates. Finally, the authors verify numerically the theory by solving the Poisson's equation on various geometries that are obtained by the union operation.
Reviewer: Vit Dolejsi (Praha)An unfitted hybrid high-order method with cell agglomeration for elliptic interface problemshttps://www.zbmath.org/1475.651862022-01-14T13:23:02.489162Z"Burman, Erik"https://www.zbmath.org/authors/?q=ai:burman.erik"Cicuttin, Matteo"https://www.zbmath.org/authors/?q=ai:cicuttin.matteo"Delay, Guillaume"https://www.zbmath.org/authors/?q=ai:delay.guillaume"Ern, Alexandre"https://www.zbmath.org/authors/?q=ai:ern.alexandreThe authors propose and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems by means of a consistent penalty Nitsche method. The possible curved interface can cut through the mesh cells in a rather general fashion. Robustness with respect to the cuts is achieved by using a cell agglomeration technique, and robustness with respect to the contrast in the diffusion coefficients is achieved by using a different gradient reconstruction on each side of the interface. A key novel feature of the gradient reconstruction is to incorporate a jump term across the interface, thereby releasing the Nitsche penalty parameter from the constraint of being large enough. Error estimates with optimal convergence rates are established. A robust cell agglomeration procedure limiting the agglomerations to the nearest neighbors is devised. Several numerical examples with various interface shapes justify the theoretical results.
Reviewer: Vit Dolejsi (Praha)\(p\)- and \(hp\)-virtual elements for the Stokes problemhttps://www.zbmath.org/1475.651872022-01-14T13:23:02.489162Z"Chernov, A."https://www.zbmath.org/authors/?q=ai:chernov.alexey"Marcati, C."https://www.zbmath.org/authors/?q=ai:marcati.carlo"Mascotto, L."https://www.zbmath.org/authors/?q=ai:mascotto.lorenzoThis article discusses the \(p\)- and \(hp\)-versions of the virtual element method for the Stokes problem on polygonal domains. The approach relies on the use of the one-to-one mapping between the Poisson-like and Stokes-like virtual element spaces for the velocities. The authors obtain an exponential rate of convergence for the \(hp\)-virtual element method and algebraic and exponential convergence rate of the \(p\)-version of the method.
Reviewer: Marius Ghergu (Dublin)A posteriori analysis of the Newton method applied to the Navier-Stokes problemhttps://www.zbmath.org/1475.651882022-01-14T13:23:02.489162Z"Dakroub, Jad"https://www.zbmath.org/authors/?q=ai:dakroub.jad"Faddoul, Joanna"https://www.zbmath.org/authors/?q=ai:faddoul.joanna"Sayah, Toni"https://www.zbmath.org/authors/?q=ai:sayah.toniSummary: In this paper we study the a posteriori error estimates for the Navier-Stokes equations. The problem is discretized using the finite element method and solved using the Newton iterative algorithm. A posteriori error estimate has been established based on two types of error indicators. Finally, numerical experiments and comparisons with previous works validate the proposed scheme and show the effectiveness of the studied algorithm.Development of finite element field solver in gyrokinetic toroidal codehttps://www.zbmath.org/1475.651892022-01-14T13:23:02.489162Z"Feng, Hongying"https://www.zbmath.org/authors/?q=ai:feng.hongying"Zhang, Wenlu"https://www.zbmath.org/authors/?q=ai:zhang.wenlu"Lin, Zhihong"https://www.zbmath.org/authors/?q=ai:lin.zhihong"Zhufu, Xiaohe"https://www.zbmath.org/authors/?q=ai:zhufu.xiaohe"Xu, Jin"https://www.zbmath.org/authors/?q=ai:xu.jin"Cao, Jintao"https://www.zbmath.org/authors/?q=ai:cao.jintao"Li, Ding"https://www.zbmath.org/authors/?q=ai:li.dingSummary: A new finite element (FE) field solver has been implemented in the gyrokinetic toroidal code (GTC) in attempt to extend the simulation domain to magnetic axis and beyond the last closed flux surface, which will enhance the capability the GTC code since the original finite difference (FD) solver will lose its capability in such circumstances. A method of manufactured solution is employed in the unit fidelity test for the new FE field solver, which is then further verified through integrated tests with three typical physical cases for the comparison between the new FE field solver and the original finite difference field solver. The results by the newly implemented FE field solver are in great accord with the original solver.Stochastic Galerkin approximation of the Reynolds equation with irregular film thicknesshttps://www.zbmath.org/1475.651902022-01-14T13:23:02.489162Z"Gustafsson, Tom"https://www.zbmath.org/authors/?q=ai:gustafsson.tom"Hakula, Harri"https://www.zbmath.org/authors/?q=ai:hakula.harri"Leinonen, Matti"https://www.zbmath.org/authors/?q=ai:leinonen.mattiSummary: We consider the approximation of the Reynolds equation with an uncertain film thickness. The resulting stochastic partial differential equation is solved numerically by the stochastic Galerkin finite element method with high-order discretizations both in spatial and stochastic domains. We compute the pressure field of a journal bearing in various numerical examples that demonstrate the effectiveness and versatility of the approach. The results suggest that the stochastic Galerkin method is capable of supporting design when manufacturing imperfections are the main sources of uncertainty.Nitsche's master-slave method for elastic contact problemshttps://www.zbmath.org/1475.651912022-01-14T13:23:02.489162Z"Gustafsson, Tom"https://www.zbmath.org/authors/?q=ai:gustafsson.tom"Stenberg, Rolf"https://www.zbmath.org/authors/?q=ai:stenberg.rolf"Videman, Juha"https://www.zbmath.org/authors/?q=ai:videman.juha-hansSummary: We survey the Nitsche's master-slave finite element method for elastic contact problems analysed in our paper [SIAM J. Sci. Comput. 42, No. 2, B425--B446 (2020; Zbl 1447.65143)]. The main steps of the error analysis are recalled and numerical benchmark computations are presented.
For the entire collection see [Zbl 1471.65009].On mesh regularity conditions for simplicial finite elementshttps://www.zbmath.org/1475.651922022-01-14T13:23:02.489162Z"Khademi, Ali"https://www.zbmath.org/authors/?q=ai:khademi.ali"Korotov, Sergey"https://www.zbmath.org/authors/?q=ai:korotov.sergey"Vatne, Jon Eivind"https://www.zbmath.org/authors/?q=ai:vatne.jon-eivindSummary: We review here various results (including own very recent ones) on mesh regularity conditions commonly imposed on simplicial finite element meshes in the interpolation theory and finite element analysis. Several open problems are listed as well.
For the entire collection see [Zbl 1471.65009].A new algebraically stabilized method for convection-diffusion-reaction equationshttps://www.zbmath.org/1475.651932022-01-14T13:23:02.489162Z"Knobloch, Petr"https://www.zbmath.org/authors/?q=ai:knobloch.petrSummary: This paper is devoted to algebraically stabilized finite element methods for the numerical solution of convection-diffusion-reaction equations. First, the algebraic flux correction scheme with the popular Kuzmin limiter is presented. This limiter has several favourable properties but does not guarantee the validity of the discrete maximum principle for non-Delaunay meshes. Therefore, a generalization of the algebraic flux correction scheme and a modification of the limiter are proposed which lead to the discrete maximum principle for arbitrary meshes. Numerical results demonstrate the advantages of the new method.
For the entire collection see [Zbl 1471.65009].Importance of parameter optimization in a nonlinear stabilized method adding a crosswind diffusionhttps://www.zbmath.org/1475.651942022-01-14T13:23:02.489162Z"Knobloch, Petr"https://www.zbmath.org/authors/?q=ai:knobloch.petr"Lukáš, Petr"https://www.zbmath.org/authors/?q=ai:lukas.petr"Solin, Pavel"https://www.zbmath.org/authors/?q=ai:solin.pavelThe problem under consideration is the following
\[
-\varepsilon\Delta u+\mathbf{b}\cdot u+cu=f \text{ in }\Omega,\quad u=u_b \text{ on } \Gamma^D,\quad \varepsilon\frac{\partial u}{\partial n}=g\text{ on } \Gamma^N,
\]
where \(\Omega\subset\mathbb R^d\), \(d=2,3\) is a bounded domain with Lipschitz polyhedral boundary \(\partial\Omega\), \(\Gamma^D\) and \(\Gamma^N\) are relatively open subsets of \(\partial\Omega\), \(\Gamma^D\cap\Gamma^N=\emptyset\), and \(\Gamma^D\cup\Gamma^N=\partial\Omega\). Furthermore, \(\varepsilon>0\) is a constant diffusivity, \(\mathbf{b}\in W^{1,\infty}(\Omega)^d\) is a convective field, \(c\in L^{\infty}(\Omega)\) is a reaction coefficient, \(f\in L^2(\Omega)\) is an outer source of \(u\), \(u_b\in H^{1/2}(\Gamma^D)\) is the Dirichlet boundary condition, and \(g\in L^2(\Gamma^N)\) is a function prescribing the Neumann boundary condition and some additional assumptions.
The authors introduce a finite element space and use so called streamline upwind/Petrov-Galerkin (SUPG) method [\textit{A. N. Brooks} and \textit{T. J. R. Hughes}, Comput. Methods Appl. Mech. Eng. 32, 199--259 (1982; Zbl 0497.76041)] for approximation. They describe a certain iteration procedure which is called SOLD method and assert that `` this approach can lead to more physically meaningful solutions than techniques considered before''. Numerical experiments are given.
Reviewer: Vladimir Vasilyev (Belgorod)Quasi-optimal and pressure robust discretizations of the Stokes equations by moment- and divergence-preserving operatorshttps://www.zbmath.org/1475.651952022-01-14T13:23:02.489162Z"Kreuzer, Christian"https://www.zbmath.org/authors/?q=ai:kreuzer.christian"Verfürth, Rüdiger"https://www.zbmath.org/authors/?q=ai:verfurth.rudiger"Zanotti, Pietro"https://www.zbmath.org/authors/?q=ai:zanotti.pietroSummary: We approximate the solution of the Stokes equations by a new quasi-optimal and pressure robust discontinuous Galerkin discretization of arbitrary order. This means quasi-optimality of the velocity error independent of the pressure. Moreover, the discretization is well-defined for any load which is admissible for the continuous problem and it also provides classical quasi-optimal estimates for the sum of velocity and pressure errors. The key design principle is a careful discretization of the load involving a linear operator, which maps discontinuous Galerkin test functions onto conforming ones thereby preserving the discrete divergence and certain moment conditions on faces and elements.Assembly of multiscale linear PDE operatorshttps://www.zbmath.org/1475.651962022-01-14T13:23:02.489162Z"Kuchta, Miroslav"https://www.zbmath.org/authors/?q=ai:kuchta.miroslavSummary: In numerous applications the mathematical model consists of different processes coupled across a lower dimensional manifold. Due to the multiscale coupling, finite element discretization of such models presents a challenge. Assuming that only \textit{singlescale} finite element forms can be assembled we present here a simple algorithm for representing multiscale models as linear operators suitable for Krylov methods. Flexibility of the approach is demonstrated by numerical examples with coupling across dimensionality gap 1 and 2. Preconditioners for several of the problems are discussed.
For the entire collection see [Zbl 1471.65009].A least-squares Galerkin gradient recovery method for fully nonlinear elliptic equationshttps://www.zbmath.org/1475.651972022-01-14T13:23:02.489162Z"Lakkis, Omar"https://www.zbmath.org/authors/?q=ai:lakkis.omar"Mousavi, Amireh"https://www.zbmath.org/authors/?q=ai:mousavi.amirehSummary: We propose a least squares Galerkin based gradient recovery to approximate Dirichlet problems for strong solutions of linear elliptic problems in nondivergence form and corresponding a priori and a posteriori error bounds. This approach is used to tackle fully nonlinear elliptic problems, e.g., Monge-Ampère, Hamilton-Jacobi-Bellman, using the smooth (vanilla) and the semismooth Newton linearization. We discuss numerical results, including adaptive methods based on the a posteriori error indicators.
For the entire collection see [Zbl 1471.65009].The finite element method as a tool to solve the oblique derivative boundary value problem in geodesyhttps://www.zbmath.org/1475.651982022-01-14T13:23:02.489162Z"Macák, Marek"https://www.zbmath.org/authors/?q=ai:macak.marek"Minarechová, Zuzana"https://www.zbmath.org/authors/?q=ai:minarechova.zuzana"Čunderlík, Róbert"https://www.zbmath.org/authors/?q=ai:cunderlik.robert"Mikula, Karol"https://www.zbmath.org/authors/?q=ai:mikula.karolSummary: In this paper, we propose a novel approach to approximate the solution of the Laplace equation with an oblique derivative boundary condition by the finite element method. We present and analyse diverse testing experiments to study its behaviour and convergence. Finally, the usefulness of this approach is demonstrated by using it to gravity field modelling, namely, to approximate the solution of a geodetic boundary value problem in Himalayas.Fast finite element method for the three-dimensional Poisson equation in infinite domainshttps://www.zbmath.org/1475.651992022-01-14T13:23:02.489162Z"Ma, Xiang"https://www.zbmath.org/authors/?q=ai:ma.xiang"Zheng, Chunxiong"https://www.zbmath.org/authors/?q=ai:zheng.chunxiongSummary: We aim at a fast finite element method for the Poisson equation in three-dimensional infinite domains. Both the exterior and strip-tail problems are considered. By introducing a suitable artificial boundary and imposing the exact boundary condition of Dirichlet-to-Neumann (DtN) type, we reduce the original infinite domain problem into a truncated finite domain problem. The point is how to efficiently implement this exact artificial boundary condition. The traditional modal expansion method is hard to apply for the strip-tail problem with a general cross section. We develop a fast algorithm based on the Padé approximation for the square root function involved in the exact artificial boundary condition. The most remarkable advantage of our method is that it is unnecessary to compute the full eigen system associated with the Laplace-Beltrami operator on the artificial boundary. Besides, compared with the modal expansion method, the computational cost of the DtN mapping is significantly reduced. We perform a complete numerical analysis on the fast algorithm. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.Parameter robust preconditioning for multi-compartmental Darcy equationshttps://www.zbmath.org/1475.652002022-01-14T13:23:02.489162Z"Piersanti, Eleonora"https://www.zbmath.org/authors/?q=ai:piersanti.eleonora"Rognes, Marie E."https://www.zbmath.org/authors/?q=ai:rognes.marie-e"Mardal, Kent-Andre"https://www.zbmath.org/authors/?q=ai:mardal.kent-andreSummary: In this paper, we propose a new finite element solution approach to the multi-compartmental Darcy equations describing flow and interactions in a porous medium with multiple fluid compartments. We introduce a new numerical formulation and a block-diagonal preconditioner. The robustness with respect to variations in material parameters is demonstrated by theoretical considerations and numerical examples.
For the entire collection see [Zbl 1471.65009].The concept of prehandling as direct preconditioning for Poisson-like problemshttps://www.zbmath.org/1475.652012022-01-14T13:23:02.489162Z"Ruda, Dustin"https://www.zbmath.org/authors/?q=ai:ruda.dustin"Turek, Stefan"https://www.zbmath.org/authors/?q=ai:turek.stefan"Zajac, Peter"https://www.zbmath.org/authors/?q=ai:zajac.peter"Ribbrock, Dirk"https://www.zbmath.org/authors/?q=ai:ribbrock.dirkSummary: To benefit from current trends in HPC hardware, such as increasing availability of low precision hardware, we present the concept of prehandling as a direct way of preconditioning and the hierarchical finite element method which is exceptionally well-suited to apply prehandling to Poisson-like problems, at least in 1D and 2D. Such problems are known to cause ill-conditioned stiffness matrices and therefore high computational errors due to round-off. We show by means of numerical results that by prehandling via the hierarchical finite element method the condition number can be significantly reduced (while advantageous properties are preserved) which enables us to obtain sufficiently accurate solutions to Poisson-like problems even if lower computing precision (i.e. single or half precision format) is used.
For the entire collection see [Zbl 1471.65009].A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problemhttps://www.zbmath.org/1475.652022022-01-14T13:23:02.489162Z"Song, Shicang"https://www.zbmath.org/authors/?q=ai:song.shicang"Lu, Lijuan"https://www.zbmath.org/authors/?q=ai:lu.lijuanSummary: A new \(C^0\) nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space \(P_3\) and some bubble functions. Different from non-\(C^0\) nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution \(u\in H_0^2(\Omega)\cap H^4(\Omega)\), the scheme is of linear order convergence by the measurement of discrete norm \(\Vert \cdot \Vert_h\). Some numerical results are included to further illustrate the convergence analysis.A posteriori error estimates of discontinuous streamline diffusion methods for transport equationshttps://www.zbmath.org/1475.652032022-01-14T13:23:02.489162Z"Sun, Juan"https://www.zbmath.org/authors/?q=ai:sun.juan"Zhou, Zhaojie"https://www.zbmath.org/authors/?q=ai:zhou.zhaojie"Liu, Huipo"https://www.zbmath.org/authors/?q=ai:liu.huipoSummary: Residual-based posteriori error estimates for discontinuous streamline diffusion methods for transport equations are studied in this paper. Computable upper bounds of the errors are measured based on mesh-dependent energy norm and negative norm. The estimates obtained are locally efficient, and thus suitable for adaptive mesh refinement applications. Numerical experiments are provided to illustrate underlying features of the estimators.A direct projection to low-order level for \(p\)-multigrid methods in isogeometric analysishttps://www.zbmath.org/1475.652042022-01-14T13:23:02.489162Z"Tielen, Roel"https://www.zbmath.org/authors/?q=ai:tielen.roel"Möller, Matthias"https://www.zbmath.org/authors/?q=ai:moller.matthias"Vuik, Kees"https://www.zbmath.org/authors/?q=ai:vuik.cSummary: Isogeometric Analysis (IgA) can be considered as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. The development of efficient solvers for discretizations arising in IgA is a challenging task, as most (standard) iterative solvers have a detoriating performance for increasing values of the approximation order \(p\) of the basis functions. Recently, \(p\)-multigrid methods have been developed as an alternative solution strategy. With \(p\)-multigrid methods, a multigrid hierarchy is constructed based on the approximation order pinstead of the mesh width \(h\) (i.e. \(h\)-multigrid). The coarse grid correction is then obtained at level \(p = 1\), where B-spline basis functions coincide with standard Lagrangian \(P_1\) basis functions, enabling the use of well known solution strategies developed for the Finite Element Method to solve the residual equation. Different projection schemes can be adopted to go from the high-order level to level \(p = 1\). In this paper, we compare a direct projection to level \(p = 1\) with a projection between each level \(1 \leq k \leq p\) in terms of iteration numbers and CPU times. Numerical results, including a spectral analysis, show that a direct projection leads to the most efficient method for both single patch and multipatch geometries.
For the entire collection see [Zbl 1471.65009].A stabilizer free weak Galerkin finite element method on polytopal mesh. IIhttps://www.zbmath.org/1475.652052022-01-14T13:23:02.489162Z"Ye, Xiu"https://www.zbmath.org/authors/?q=ai:ye.xiu"Zhang, Shangyou"https://www.zbmath.org/authors/?q=ai:zhang.shangyouSummary: A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper [the authors, ibid. 371, Article ID 112699, 9 p. (2020; Zbl 1434.65285)]. Removing stabilizers from discontinuous finite element methods simplifies formulations and reduces programming complexity. The purpose of this paper is to introduce a new WG method without stabilizers on polytopal mesh that has convergence rates one order higher than optimal convergence rates. This method is the first WG method that achieves superconvergence on polytopal mesh. Numerical examples in 2D and 3D are presented verifying the theorem.A weak Galerkin finite element method for \(p\)-Laplacian problemhttps://www.zbmath.org/1475.652062022-01-14T13:23:02.489162Z"Ye, Xiu"https://www.zbmath.org/authors/?q=ai:ye.xiu"Zhang, Shangyou"https://www.zbmath.org/authors/?q=ai:zhang.shangyouSummary: In this paper, we introduce a weak Galerkin (WG) finite element method for \(p\)-Laplacian problem on general polytopal mesh. The quasi-optimal error estimates of the weak Galerkin finite element approximation are obtained. The numerical examples confirm the theory.Poly-Sinc solution of stochastic elliptic differential equationshttps://www.zbmath.org/1475.652072022-01-14T13:23:02.489162Z"Youssef, Maha"https://www.zbmath.org/authors/?q=ai:youssef.maha"Pulch, Roland"https://www.zbmath.org/authors/?q=ai:pulch.rolandThis article discusses a numerical approach to stochastic partial differential equations of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. The authors combine a stochastic Galerkin method with polynomial chaos, which yields a system of deterministic partial differential equations to be solved by using a small number of collocation points in space. Two applications using Legendre polynomials for stochastic variables are dicussed.
Reviewer: Marius Ghergu (Dublin)Convergence analysis of the scaled boundary finite element method for the Laplace equationhttps://www.zbmath.org/1475.652082022-01-14T13:23:02.489162Z"Bertrand, Fleurianne"https://www.zbmath.org/authors/?q=ai:bertrand.fleurianne"Boffi, Daniele"https://www.zbmath.org/authors/?q=ai:boffi.daniele"G. de Diego, Gonzalo"https://www.zbmath.org/authors/?q=ai:g-de-diego.gonzaloThis paper dicsusses the convergence analysis of a scaled boundary finite element method for approximation of solutions to partial differential equations without the need of a fundamental solution. The approach relies on constructing a space of semi-discrete functions and an interpolation operator onto this space. Error estimates for the interpolation operator and optimal convergence is achieved.
Reviewer: Marius Ghergu (Dublin)Boundary element methods for acoustic scattering by fractal screenshttps://www.zbmath.org/1475.652092022-01-14T13:23:02.489162Z"Chandler-Wilde, Simon N."https://www.zbmath.org/authors/?q=ai:chandler-wilde.simon-n"Hewett, David P."https://www.zbmath.org/authors/?q=ai:hewett.david-p"Moiola, Andrea"https://www.zbmath.org/authors/?q=ai:moiola.andrea"Besson, Jeanne"https://www.zbmath.org/authors/?q=ai:besson.jeanneBoundary element methods (BEM) for scattering problems by fractal screens are studied in this article.
The main idea is to approximate the fractal screen by a sequence of smoother screens (prefractals) where well-posedness of the corresponding boundary integral equations is known and, consequently, classical methods like Galerkin BEM can be used for the discretization. Two types of fractal screens are considered: (i) sets with fractal boundaries and (ii) compact fractal sets with empty interior.
The (numerical) analysis of boundary integral equations on fractal sets requires various non-standard results (e.g., Sobolev spaces defined on screens) which are given in Sections 2--3.
Convergence results for sound-soft screen problems are found in Sections 4--5. The case where the fractal screen is a fixed point of an iterative function system is studied in detail. The main results are Theorem 5.2 and Theorem 5.3 where convergence of the Galerkin BEM is proven for the two types (i), (ii).
Section 6 contains various examples of screens, namely, Cantor sets in 2D and 3D, the Sierpinski triangle and snowflakes. Numerical experiments for these examples are found in Section 7.
Reviewer: Thomas Führer (Santiago)A Cartesian grid-based boundary integral method for an elliptic interface problem on closely packed cellshttps://www.zbmath.org/1475.652102022-01-14T13:23:02.489162Z"Ying, Wenjun"https://www.zbmath.org/authors/?q=ai:ying.wenjunSummary: In this work, we propose a second-order version and a fourth-order version of a Cartesian grid-based boundary integral method for an interface problem of the Laplace equation on closely packed cells. When the cells are closely packed, the boundary integrals involved in the boundary integral formulation for the interface problem become nearly singular. Direct evaluation of the boundary integrals has accuracy issues. The grid-based method evaluates a boundary integral by first solving an equivalent, simple interface problem on a Cartesian grid with a fast Fourier transform based Poisson solver, then interpolating the grid solution to get values of the boundary integral at discretization points of the interface. The grid-based method presents itself as an alternative but accurate numerical method for evaluating nearly singular, singular and hyper-singular boundary integrals. This work can be regarded as a further development of the kernel-free boundary integral method [\textit{W. Ying} and \textit{C. S. Henriquez}, J. Comput. Phys. 227, No. 2, 1046--1074 (2007; Zbl 1128.65102)] for problems in unbounded domains. Numerical examples with both second-order and fourth-order versions of the grid-based method are presented to demonstrate accuracy of the method.Single and multiple springback technique for construction and control of thick prismatic mesh layershttps://www.zbmath.org/1475.652112022-01-14T13:23:02.489162Z"Garanzha, Vladimir A."https://www.zbmath.org/authors/?q=ai:garanzha.vladimir-a"Kudryavtseva, Lyudmila N."https://www.zbmath.org/authors/?q=ai:kudryavtseva.lyudmila-n"Belokrys-Fedotov, Aleksei I."https://www.zbmath.org/authors/?q=ai:belokrys-fedotov.aleksei-iSummary: We suggest an algorithm for construction of semi-structured thick prismatic mesh layers which guarantees an absence of inverted prismatic cells in resulting layer and allows one to control near-surface mesh orthogonality. Initial mesh is modelled as a thin layer of highly compressed prisms made of hyperelastic material glued to the triangulated surface. In order to compute robust normals at the vertices of the surface mesh we use quadratic programming algorithm based on the nearest ball concept. This pre-stressed material expands, possibly with self-penetration and extrusion to exterior of computational domain until target layer thickness is attained. Special preconditioned relaxation procedure is proposed based on the solution of stationary springback problem. It is shown that preconditioner can handle very stiff problems. Once an offset prismatic mesh is constructed, self-intersections are eliminated using iterative prism cutting procedure. Next, variational advancing front procedure is applied for refinement and precise orthogonalization of prismatic layer near boundaries. We demonstrate that resulting mesh layer is `almost mesh-independent' in a sense that the dependence of thickness and shape of the layer on mesh resolution and triangle quality is weak. It is possible to apply elastic springback technique sequentially layer by layer. We compare single springback technique with multiple springback technique in terms of mesh quality, stiffness of local variational problems and mesh orthogonality or/and layer thickness balance.Multilevel optimized Schwarz methodshttps://www.zbmath.org/1475.652122022-01-14T13:23:02.489162Z"Gander, Martin J."https://www.zbmath.org/authors/?q=ai:gander.martin-j"Vanzan, Tommaso"https://www.zbmath.org/authors/?q=ai:vanzan.tommasoThis paper defines a two-level optimized Schwarz method (OSM) for elliptic partial differential equations, which solves the coarse space problem on a coarse mesh. A local mode convergence analysis is given both for overlapping and nonoverlapping decompositions, which also suggests how to choose the optimized transmission conditions. The two-level method is extended to a multilevel OSM which uses OSMs as smoothers on each level. The paper provides extensive numerical experiments for solving the advection diffusion equation, Helmholtz equation with a dispersion correction, and Stokes-Darcy coupling to illustrate the robustness of the proposed multilevel OSMs.
Reviewer: Zhiming Chen (Beijing)Machine learning in adaptive FETI-DP: reducing the effort in samplinghttps://www.zbmath.org/1475.652132022-01-14T13:23:02.489162Z"Heinlein, Alexander"https://www.zbmath.org/authors/?q=ai:heinlein.alexander"Klawonn, Axel"https://www.zbmath.org/authors/?q=ai:klawonn.axel"Lanser, Martin"https://www.zbmath.org/authors/?q=ai:lanser.martin"Weber, Janine"https://www.zbmath.org/authors/?q=ai:weber.janineSummary: The convergence rate of classic domain decomposition methods in general deteriorates severely for large discontinuities in the coefficient functions of the considered partial differential equation. To retain the robustness for such highly heterogeneous problems, the coarse space can be enriched by additional coarse basis functions. These can be obtained by solving local generalized eigenvalue problems on subdomain edges. In order to reduce the number of eigenvalue problems and thus the computational cost, we use a neural network to predict the geometric location of critical edges, i.e., edges where the eigenvalue problem is indispensable. As input data for the neural network, we use function evaluations of the coefficient function within the two subdomains adjacent to an edge. In the present article, we examine the effect of computing the input data only in a neighborhood of the edge, i.e., on slabs next to the edge. We show numerical results for both the training data as well as for a concrete test problem in form of a microsection subsection for linear elasticity problems. We observe that computing the sampling points only in one half or one quarter of each subdomain still provides robust algorithms.
For the entire collection see [Zbl 1471.65009].Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problemshttps://www.zbmath.org/1475.652142022-01-14T13:23:02.489162Z"Li, Yukun"https://www.zbmath.org/authors/?q=ai:li.yukun"Zhang, Yi"https://www.zbmath.org/authors/?q=ai:zhang.yi.6This paper proposes some efficient and accurate adaptive two-grid finite element algorithms for linear and nonlinear PDEs. The main idea of these algorithms is to utilize the solutions on the \(k\)-th level adaptive meshes to find the solutions on the \((k +1)\)-th level adaptive meshes which are constructed by performing adaptive element bisections on the \(k\)-th level adaptive meshes. The proposed algorithms are both accurate and efficient, they do not need to solve the nonsymmetric or nonlinear systems; the number of degrees of freedom is low, they are easily implemented, and the interpolation errors are small. Next, this paper constructs residual-type a posteriori error estimators which are shown to be reliable and efficient. The key ingredient in proving the efficiency is to establish an upper bound of the oscillation terms, which may not be higher-order terms due to the low regularity of the numerical solution. Furthermore, the convergence of the algorithms is proved when bisection is used for the mesh refinements. Finally, numerical experiments are provided to verify the accuracy and efficiency of the adaptive finite element algorithms compared to regular adaptive finite element algorithms and two-grid finite element algorithms.
Reviewer: Vit Dolejsi (Praha)Sparse compression of expected solution operatorshttps://www.zbmath.org/1475.652152022-01-14T13:23:02.489162Z"Feischl, Michael"https://www.zbmath.org/authors/?q=ai:feischl.michael"Peterseim, Daniel"https://www.zbmath.org/authors/?q=ai:peterseim.danielThis very interesting paper considers the elliptic partial differential equation \(-\operatorname{div}(A(\omega)\nabla u)=f\) in a bounded Lipschitz polytope \(D\subset\mathbb{R}^d\), \(d=1,2,3\), whose coefficient \(A(\omega)\) is a random field in some probability space with positive lower and upper bounds. The source \(f\) is assumed to be deterministic. The paper introduces an algorithm to approximate the inverse of the expected solution operator \(\mathcal{A}^{-1}=\mathbb{E}[A(\omega)^{-1}]\) in any given accuracy \(\delta>0\) in the \(L^2\) norm with the computational costs scale like \(\delta^{-d}\) up to logarithmic-in-\(\delta\) terms. The result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as the stochastic averaging. The results do not rely on the probabilistic structure of the random coefficient \(A(\omega)\), such as stationarity, ergodicity, or any characteristic length of correlation. Numerical experiments are included.
Reviewer: Zhiming Chen (Beijing)Numerical solution to the interface problem in a general domain using Moser's deformation methodhttps://www.zbmath.org/1475.652162022-01-14T13:23:02.489162Z"Hong, Eunhye"https://www.zbmath.org/authors/?q=ai:hong.eunhye"Lee, Eunjung"https://www.zbmath.org/authors/?q=ai:lee.eunjung"Jung, Younghoon"https://www.zbmath.org/authors/?q=ai:jung.younghoon"Lim, Mikyoung"https://www.zbmath.org/authors/?q=ai:lim.mikyoungSummary: When seeking a solution to the interface problem, the potential theory to represent solution has been widely used. As the solution representation of the interface problem is only well-known for domains with simple inclusion, such as a disk, conformal mapping is often used to transform the arbitrary inclusion to a manageable inclusion to utilize the known solution representation. This paper proposes the construction of a conformal mapping based on Moser's grid deformation method that generates a transformation with positive Jacobian determinant. By establishing an optimal control problem that minimizes the dissimilarity between transformed and reference images with desired constraints, we constructed an appropriate conformal mapping. Several numerical results prove the validity of our approach.A posteriori model error analysis of 3D-1D coupled PDEshttps://www.zbmath.org/1475.652172022-01-14T13:23:02.489162Z"Laurino, Federica"https://www.zbmath.org/authors/?q=ai:laurino.federica"Brambilla, Stefano"https://www.zbmath.org/authors/?q=ai:brambilla.stefano"Zunino, Paolo"https://www.zbmath.org/authors/?q=ai:zunino.paoloSummary: The objective of this work is to extend the model reduction technique for coupled 3D-1D elliptic PDEs, previously proposed by the authors, with an a posteriori analysis of the model error, defined as the difference between the solutions of the reference and reduced problem. More precisely, we introduce an estimator for a user-defined functional of the error, computed using a duality approach. This result is particularly useful since it allows to localize the model error on the computational mesh and to investigate the reliability of the model reduction approach.
For the entire collection see [Zbl 1471.65009].Approximate solution of singular IVPs of Lane-Emden type and error estimation via advanced Adomian decomposition methodhttps://www.zbmath.org/1475.652182022-01-14T13:23:02.489162Z"Umesh"https://www.zbmath.org/authors/?q=ai:umesh.v|umesh.k-n"Kumar, Manoj"https://www.zbmath.org/authors/?q=ai:kumar.manoj.3|yadav.manoj-kumar|kumar.manoj.1|kumar.manoj.2|kumar.manojSummary: This article aims to present a simple and effective method, named as advanced Adomian decomposition method, to attain the approximate solution of singular initial value problems of Lane-Emden type. Also, convergence analysis and error analysis with an upper bound of the absolute error for the proposed method are discussed. The proposed method is capable to remove the singular behaviour of the problems and provides an approximate solution up to the desired order. To illustrate the reliability and validity of the proposed method with error estimate several examples that arise in applications are considered and the attained outcomes are compared with some existing numerical methods.A uniformly convergent scheme for radiative transfer equation in the diffusion limit up to the boundary and interface layershttps://www.zbmath.org/1475.652192022-01-14T13:23:02.489162Z"Chen, Hongfei"https://www.zbmath.org/authors/?q=ai:chen.hongfei"Chen, Gaoyu"https://www.zbmath.org/authors/?q=ai:chen.gaoyu"Hong, Xiang"https://www.zbmath.org/authors/?q=ai:hong.xiang"Gao, Hao"https://www.zbmath.org/authors/?q=ai:gao.hao"Tang, Min"https://www.zbmath.org/authors/?q=ai:tang.min|tang.min.1Summary: In this paper, we present a numerical scheme for the steady-state radiative transfer equation (RTE) with Henyey-Greenstein scattering kernel. The scattering kernel is anisotropic but not highly forward peaked. On the one hand, for the velocity discretization, we approximate the anisotropic scattering kernel by a discrete matrix that can preserve the diffusion limit. On the other hand, for the space discretization, a uniformly convergent scheme up to the boundary or interface layer is proposed. The idea is that we first approximate the scattering coefficients as well as source by piecewise constant functions, then, in each cell, the true solution is approximated by the summation of a particular solution and a linear combinations of general solutions to the homogeneous RTE. Second-order accuracy can be observed, uniformly with respect to the mean free path up to the boundary and interface layers. The scheme works well for heterogenous medium, anisotropic sources as well as for the strong source regime.A method of solving a nonlinear boundary value problem for the Fredholm integro-differential equationhttps://www.zbmath.org/1475.652202022-01-14T13:23:02.489162Z"Dzhumabaev, Dulat S."https://www.zbmath.org/authors/?q=ai:dzhumabaev.dulat-syzdykbekovich"Mynbayeva, Sandugash"https://www.zbmath.org/authors/?q=ai:mynbayeva.sandugash-tabyldyevnaThe parametrization method is a common technique used to study the solvability of linear, linear non-homogenous Fredholm integro-differential equations (IDEs) and boundary value problems (BVPs). The authors consider nonlinear BVPs for two Fredholm integro-differential equations (Equations (1) and (2) in the paper). Employing some additional parameters and initial conditions at the left-end points of the subintervals, the problems are transformed into equivalent BVPs for a system of nonlinear IDE with parameters on certain subintervals. Special Cauchy problems are obtained for these systems for the unbounded closed operator which yields two nonlinear equations (Equations (6) and (7) in the paper). Further, these equations are solved using iterative methods and the convergence conditions for iterations is also proposed. The necessary and sufficient conditions for the existence of a unique solution to the special Cauchy problems are then obtained.
The numerical algorithm is presented for solving two auxiliary problems: a Cauchy problem for ODEs on subintervals and an evaluation of definite integrals on subintervals. The accuracy of the numerical solution is defined by means of a fourth-order Runge-Kutta method and Simpson's formula.
Reviewer: Deshna Loonker (Jodhpur)About methods of approximate solutions for composite bodies weakened by cracks in the case of antiplane problems of elasticity theoryhttps://www.zbmath.org/1475.652212022-01-14T13:23:02.489162Z"Papukashvili, A."https://www.zbmath.org/authors/?q=ai:papukashvili.a-r|papukashvili.archil"Gordeziani, D."https://www.zbmath.org/authors/?q=ai:gordeziani.david"Davitashvili, T."https://www.zbmath.org/authors/?q=ai:davitashvili.teimurazi|davitashvili.t-d"Sharikadze, M."https://www.zbmath.org/authors/?q=ai:sharikadze.meri"Manelidze, G."https://www.zbmath.org/authors/?q=ai:manelidze.gela"Kurdghelashvili, G."https://www.zbmath.org/authors/?q=ai:kurdghelashvili.gSummary: These problems lead to a system of singular integral equations with immovable singularity with respect to leap of the tangent stress. The problems of behavior of solutions at the boundary are studied. In the present work questions of the approached decision of one system (pair) of the singular integral equations are investigated. The study of boundary value problems for the composite bodies weakened by cracks has a great practical significance. The system of singular integral equations is solved by a collocation method, in particular, a discrete singular method in cases both uniform, and non-uniformly located knots.Numeric solution and testing methods of rigid integral-differential problem of isoelectric focusinghttps://www.zbmath.org/1475.652222022-01-14T13:23:02.489162Z"Sakharova, L. V."https://www.zbmath.org/authors/?q=ai:sakharova.l-vSummary: The article is devoted to solution development of rigid integral-differential problem connected with the model of Isoelectric Focusing (IEF) in so-called ``anomalous'' regimes. Besides, testing methods of the problem are built. It is found the analytic transformation of the initial integral-differential problem to the ordinary boundary-value problem which is suitable to evaluation by Runge-Kutta's method. Further, it is overcame the uncontrollable accumulation of calculation errors which are connected with the problem rigidity. It is done by means of solution representation in the special exponential form. For model testing two methods have been developed. They are the asymptotic method and the tangent one. Both of them show the high degree correspondence of numeric and asymptotic solutions.Exponential fitting collocation methods for a class of Volterra integral equationshttps://www.zbmath.org/1475.652232022-01-14T13:23:02.489162Z"Zhao, Longbin"https://www.zbmath.org/authors/?q=ai:zhao.longbin"Huang, Chengming"https://www.zbmath.org/authors/?q=ai:huang.chengmingSummary: In this paper, we propose a collocation method for a class of Volterra integral equations whose solutions contain periodic functions. Since the exponential fitting interpolation has an advantage in approximating periodic functions, we consider employing it with collocation method to construct our scheme. The global convergence analysis of the scheme is also presented based on the interpolation error. The theoretical results, as well as the superiority of the method, are verified in the numerical part.Biorthogonal boundary multiwaveletshttps://www.zbmath.org/1475.652242022-01-14T13:23:02.489162Z"Keinert, Fritz"https://www.zbmath.org/authors/?q=ai:keinert.fritzSummary: The discrete wavelet transform is defined for functions on the entire real line. One way to implement the transform on a finite interval is by using special boundary functions. For orthogonal multiwavelets, this has been studied in previous papers. We describe the generalization of some of these results to biorthogonal multiwavelets.
For the entire collection see [Zbl 1471.65009].Machine learning control by symbolic regressionhttps://www.zbmath.org/1475.680052022-01-14T13:23:02.489162Z"Diveev, Askhat"https://www.zbmath.org/authors/?q=ai:diveev.askhat"Shmalko, Elizaveta"https://www.zbmath.org/authors/?q=ai:shmalko.elizavetaPublisher's description: This book provides comprehensive coverage on a new direction in computational mathematics research: automatic search for formulas. Formulas must be sought in all areas of science and life: these are the laws of the universe, the macro and micro world, fundamental physics, engineering, weather and natural disasters forecasting; the search for new laws in economics, politics, sociology. Accumulating many years of experience in the development and application of numerical methods of symbolic regression to solving control problems, the authors offer new possibilities not only in the field of control automation, but also in the design of completely different optimal structures in many fields.
For specialists in the field of control, \textit{Machine Learning Control by Symbolic Regression} opens up a new promising direction of research and acquaints scientists with the methods of automatic construction of control systems.
For specialists in the field of machine learning, the book opens up a new, much broader direction than neural networks: methods of symbolic regression. This book makes it easy to master this new area in machine learning and apply this approach everywhere neural networks are used. For mathematicians, the book opens up a new approach to the construction of numerical methods for obtaining analytical solutions to unsolvable problems; for example, numerical analytical solutions of algebraic equations, differential equations, non-trivial integrals, etc.
For specialists in the field of artificial intelligence, the book offers a machine way to solve problems, framed in the form of analytical relationships.Computing a minimum-width square annulus in arbitrary orientation (extended abstract)https://www.zbmath.org/1475.684022022-01-14T13:23:02.489162Z"Bae, Sang Won"https://www.zbmath.org/authors/?q=ai:bae.sang-wonSummary: In this paper, we address the problem of computing a minimum-width square annulus in arbitrary orientation that encloses a given set of \(n\) points in the plane. A square annulus is the region between two concentric squares. We present an \(O(n^3 \log n)\)-time algorithm that finds such a square annulus over all orientations.
For the entire collection see [Zbl 1331.68015].Two and three dimensional image registration based on B-spline composition and level setshttps://www.zbmath.org/1475.684222022-01-14T13:23:02.489162Z"Chan, Chiu Ling"https://www.zbmath.org/authors/?q=ai:chan.chiu-ling"Anitescu, Cosmin"https://www.zbmath.org/authors/?q=ai:anitescu.cosmin"Zhang, Yongjie"https://www.zbmath.org/authors/?q=ai:zhang.yongjie"Rabczuk, Timon"https://www.zbmath.org/authors/?q=ai:rabczuk.timonSummary: A method for non-rigid image registration that is suitable for large deformations is presented. Conventional registration methods embed the image in a B-spline object, and the image is evolved by deforming the B-spline object. In this work, we represent the image using B-spline and deform the image using a composition approach. We also derive a computationally efficient algorithm for calculating the B-spline coefficients and gradients of the image by adopting ideas from signal processing using image filters. We demonstrate the application of our method on several different types of 2D and 3D images and compare it with existing methods.Numerical solution of some plane boundary value problems of the theory of binary mixtures by the boundary element methodhttps://www.zbmath.org/1475.740102022-01-14T13:23:02.489162Z"Zirakashvili, N."https://www.zbmath.org/authors/?q=ai:zirakashvili.n-g|zirakashvili.natela|zirakashvili.nathela"Janjgava, R."https://www.zbmath.org/authors/?q=ai:janjgava.romanSummary: The paper deals with the application of the method of boundary elements to the numerical solution of plane boundary problems in the case of the linear theory of elastic mixtures. First the Kelvin problem is solved analytically when concentrated force is applied to a point in an infinite domain filled with a binary mixture of two isotropic elastic materials. By integrating the solution of this problem we obtain a solution of the problem when constant forces are distributed over an interval segment. The obtained singular solutions are used for applying one of the boundary element methods called the fictitious load method to the solution of various boundary value problems for both finite and infinite domains.Dynamic and weighted stabilizations of the \(L\)-scheme applied to a phase-field model for fracture propagationhttps://www.zbmath.org/1475.741062022-01-14T13:23:02.489162Z"Engwer, Christian"https://www.zbmath.org/authors/?q=ai:engwer.christian"Pop, Iuliu Sorin"https://www.zbmath.org/authors/?q=ai:pop.iuliu-sorin"Wick, Thomas"https://www.zbmath.org/authors/?q=ai:wick.thomasSummary: We consider a phase-field fracture propagation model, which consists of two (nonlinear) coupled partial differential equations. The first equation describes the displacement evolution, and the second is a smoothed indicator variable, describing the crack position. We propose an iterative scheme, the so-called \(L\)-scheme, with a dynamic update of the stabilization parameters during the iterations. Our algorithmic improvements are substantiated with two numerical tests. The dynamic adjustments of the stabilization parameters lead to a significant reduction of iteration numbers in comparison to constant stabilization values.
For the entire collection see [Zbl 1471.65009].Adaptive numerical simulation of a phase-field fracture model in mixed form tested on an L-shaped specimen with high Poisson ratioshttps://www.zbmath.org/1475.741112022-01-14T13:23:02.489162Z"Mang, Katrin"https://www.zbmath.org/authors/?q=ai:mang.katrin"Walloth, Mirjam"https://www.zbmath.org/authors/?q=ai:walloth.mirjam"Wick, Thomas"https://www.zbmath.org/authors/?q=ai:wick.thomas"Wollner, Winnifried"https://www.zbmath.org/authors/?q=ai:wollner.winnifriedSummary: This work presents a new adaptive approach for the numerical simulation of a phase-field model for fractures in nearly incompressible solids. In order to cope with locking effects, we use a recently proposed mixed form where we have a hydro-static pressure as additional unknown besides the displacement field and the phase-field variable. To fulfill the fracture irreversibility constraint, we consider a formulation as a variational inequality in the phase-field variable. For adaptive mesh refinement, we use a recently developed residual-type a posteriori error estimator for the phase-field variational inequality which is efficient and reliable, and robust with respect to the phase-field regularization parameter. The proposed model and the adaptive error-based refinement strategy are demonstrated by means of numerical tests derived from the L-shaped panel test, originally developed for concrete. Here, the Poisson's ratio is changed from the standard setting towards the incompressible limit \(\nu \rightarrow 0.5\).
For the entire collection see [Zbl 1471.65009].Modal analysis of elastic vibrations of incompressible materials based on a variational multiscale finite element methodhttps://www.zbmath.org/1475.741192022-01-14T13:23:02.489162Z"Codina, Ramon"https://www.zbmath.org/authors/?q=ai:codina.ramon"Türk, Önder"https://www.zbmath.org/authors/?q=ai:turk.onderSummary: In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. The second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. The finite element formulation that is based on the variational multiscale concept preserves the linearity of the eigenproblem, and accommodates arbitrary interpolations. Several eigenvalues and eigenfunctions are computed, and then the time approximation to the continuous solution is obtained taking a few modes of the whole set, those with higher energy. We present an example of the vibration of a linear incompressible elastic material showing how our approach is able to approximate the problem. It is shown how the energy of the modes associated to higher frequencies rapidly decreases, allowing one to get good approximate solution with only a few modes.
For the entire collection see [Zbl 1471.65009].An improved formulation for hybridizable discontinuous Galerkin fluid-structure interaction modeling with reduced computational expensehttps://www.zbmath.org/1475.741222022-01-14T13:23:02.489162Z"Sheldon, Jason P."https://www.zbmath.org/authors/?q=ai:sheldon.jason-p"Miller, Scott T."https://www.zbmath.org/authors/?q=ai:miller.scott-t"Pitt, Jonathan S."https://www.zbmath.org/authors/?q=ai:pitt.jonathan-sSummary: This work presents two computational efficiency improvements for the hybridizable discontinuous Galerkin (HDG) fluid-structure interaction (FSI) model presented by Sheldon et al. A new formulation for the solid is presented that eliminates the global displacement, resulting in the velocity being the only global solid variable. This necessitates a change to the solid-mesh displacement coupling, which is accounted for by coupling the local solid displacement to the global mesh displacement. Additionally, the mesh basis and test functions are restricted to linear polynomials, rather than being equal-order with the fluid and solid. This change increases the computational efficiency dynamically, with greater benefit the higher order the computation, when compared to an equal-order formulation. These two improvements result in a 50\% reduction in the number of global degrees of freedom for high-order simulations for both the fluid and solid domains, as well as an approximately 50\% reduction in the number of local fluid domain degrees of freedom for high-order simulations. The new, more efficient formulation is compared against that from Sheldon et al. and negligible change of accuracy is found.Recursive POD expansion for the advection-diffusion-reaction equationhttps://www.zbmath.org/1475.741252022-01-14T13:23:02.489162Z"Azaïez, M."https://www.zbmath.org/authors/?q=ai:azaiez.mejdi"Chacón Rebollo, T."https://www.zbmath.org/authors/?q=ai:chacon-rebollo.tomas"Perracchione, E."https://www.zbmath.org/authors/?q=ai:perracchione.emma"Vega, J. M."https://www.zbmath.org/authors/?q=ai:vega.jose-manuel|vega.j-m-morenoSummary: This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [\textit{M. Azaïez} et al., Commun. Comput. Phys. 24, No. 5, 1556--1578 (2018; Zbl 07416706)] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Quasi-conforming analysis method for trimmed CAD surfaceshttps://www.zbmath.org/1475.741262022-01-14T13:23:02.489162Z"Wang, Changsheng"https://www.zbmath.org/authors/?q=ai:wang.changsheng"Zhu, Xuefeng"https://www.zbmath.org/authors/?q=ai:zhu.xuefeng"Zhang, Xiangkui"https://www.zbmath.org/authors/?q=ai:zhang.xiangkuiSummary: In this paper, the quasi-conforming method is introduced for the analysis of trimmed computer aided design (CAD) surfaces. The main benefit of the proposed method is that the boundary curves of elements are adopted for the numerical integration directly. In the quasi-conforming technique, the strains are approximated by using polynomials, and the weighted test functions are used to weaken the strain-displacement equations. The interpolation functions are introduced for strain integration. An appropriate choice of initial strain approximation and weighted test function ensures that inner-field functions are not required for strain integration, and this is used for the analysis of trimmed CAD surfaces. For example, the assumed stress quasi-conforming method is applied for the two-dimensional linear elastic problem. All the element edges are approximated by using quadratic Bézier curves for the conciseness, and this is easily incorporated into existing finite element codes and applied to Dirichlet boundary conditions. Numerical examples indicate the effectiveness and accuracy of the method.Stokes flow around a hypersphere in \(n\)-dimensional space and its visualizationhttps://www.zbmath.org/1475.760242022-01-14T13:23:02.489162Z"Yoshino, Takashi"https://www.zbmath.org/authors/?q=ai:yoshino.takashiSummary: We derived the Stokes equations and velocity potential around a hyperspherical obstacle in \(n\)-dimensional space. The objectives of this study were to understand the hyperspace through the physics in the space and to bring the analytical solution of fluid flow in hyperspace for numerical simulation. The equations were obtained from the \(n\)-dimensional Navier-Stokes equation assuming the low Reynolds number flow. These were generalized formulae from a 3-dimensional system to an \(n\)-dimensional one. Our results show that the effect of the hyperspherical obstacle on the uniform flow is localized in higher dimensional spaces. We visualized the flow using the collections of hypersections.Scalable domain decomposition algorithms for simulation of flows passing full size wind turbinehttps://www.zbmath.org/1475.760582022-01-14T13:23:02.489162Z"Chen, Rongliang"https://www.zbmath.org/authors/?q=ai:chen.rongliang"Yan, Zhengzheng"https://www.zbmath.org/authors/?q=ai:yan.zhengzheng"Zhao, Yubo"https://www.zbmath.org/authors/?q=ai:zhao.yubo"Cai, Xiao-Chuan"https://www.zbmath.org/authors/?q=ai:cai.xiao-chuanSummary: Accurate numerical simulation of fluid flows around wind turbine plays an important role in understanding the performance and also the design of the wind turbine. The computation is challenging because of the large size of the blades, the large computational mesh, the moving geometry, and the high Reynolds number. In this paper, we develop a highly parallel numerical algorithm for the simulation of fluid flows passing three-dimensional full size wind turbine including the rotor, nacelle, and tower with realistic geometry and Reynolds number. The flow in the moving domain is modeled by unsteady incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian form and a non-overlapping sliding-interface method is used to handle the relative motion of the rotor and the tower. A stabilized moving mesh finite element method is introduced to discretize the problem in space, and a fully implicit scheme is used to discretize the temporal variable. A parallel Newton-Krylov method with a new domain decomposition type preconditioner, which combines a non-overlapping method across the rotating interface and an overlapping Schwarz method in the remaining subdomains, is applied to solve the fully coupled nonlinear algebraic system at each time step. To understand the efficiency of the algorithm, we test the algorithm on a supercomputer for the simulation of a realistic 5MW wind turbine. The numerical results show that the newly developed algorithm is scalable with over 8000 processor cores for problems with tens of millions of unknowns.A three-field smoothed formulation for prediction of large-displacement fluid-structure interaction via the explicit relaxed interface coupling (ERIC) schemehttps://www.zbmath.org/1475.760602022-01-14T13:23:02.489162Z"He, Tao"https://www.zbmath.org/authors/?q=ai:he.taoSummary: A three-field smoothed formulation is proposed in this paper for the resolution of fluid-structure interaction (FSI) from the arbitrary Lagrangian-Eulerian perspective. The idea behind the proposed approach lies in different smoothing concepts. Both fluid and solid stress tensors are smoothly treated by the cell-based smoothed finite element method (CS-FEM) using four-node quadrilateral elements. In particular, the smoothed characteristic-based split technique is developed for the incompressible flows whereas the geometrically nonlinear solid is settled through CS-FEM as usual. The deformable mesh, often represented by a pseudo-structural system, is further tuned with the aid of a hybrid smoothing algorithm. The Explicit Relaxed Interface Coupling (ERIC) scheme is presented to interpret the nonlinear FSI effect, where all interacting fields are explicitly coupled in alliance with interface relaxation method for numerical stability. The promising ERIC solver is in detail validated against the previously published data for a large-displacement FSI benchmark. The good agreement is revealed in computed results and well-known flow-induced phenomena are accurately captured.Discontinuous Galerkin model order reduction of geometrically parametrized Stokes equationhttps://www.zbmath.org/1475.760612022-01-14T13:23:02.489162Z"Shah, Nirav Vasant"https://www.zbmath.org/authors/?q=ai:shah.nirav-vasant"Hess, Martin Wilfried"https://www.zbmath.org/authors/?q=ai:hess.martin-wilfried"Rozza, Gianluigi"https://www.zbmath.org/authors/?q=ai:rozza.gianluigiSummary: The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem. The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.
For the entire collection see [Zbl 1471.65009].An improved immersed boundary method with new forcing point searching scheme for simulation of bodies in free surface flowshttps://www.zbmath.org/1475.760632022-01-14T13:23:02.489162Z"Yan, Bin"https://www.zbmath.org/authors/?q=ai:yan.bin"Bai, Wei"https://www.zbmath.org/authors/?q=ai:bai.wei"Quek, Ser Tong"https://www.zbmath.org/authors/?q=ai:quek.ser-tongSummary: An improved immersed boundary method is proposed and applied to simulate fluid-structure interactions by combining a level set method for free water surface capturing. An efficient Navier-Stokes equation solver adopting the fractional step method at a staggered Cartesian grid system is used to solve the incompressible fluid motion. A new efficient algorithm to search forcing points near the immersed body boundary is developed. The searching schemes for forcing points located both inside and outside the solid phase with the linear interpolation schemes for the determination of velocities at forcing points are presented and compared via the case of dam break over obstacles. The accuracy and effectiveness of the proposed forcing point searching schemes are further demonstrated by the study of wave propagation over a submerged bar and more challenging cases of wedge with prescribed velocity or falling freely into the water. By the extensive comparison of present numerical results with other experimental and numerical data, it suggests that the present improved immersed boundary method with the new forcing point searching scheme has a better performance and is very promising due to its accuracy, efficiency and ease of implementation. Furthermore, the present numerical results show that the outside forcing scheme is superior over the inside forcing scheme.Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flowshttps://www.zbmath.org/1475.760642022-01-14T13:23:02.489162Z"Zheng, Bo"https://www.zbmath.org/authors/?q=ai:zheng.bo|zheng.bo.1"Shang, Yueqiang"https://www.zbmath.org/authors/?q=ai:shang.yueqiangAuthors' abstract: Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier-Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor independently computes an approximate solution in its own subdomain using a global composite mesh that is fine around its own subdomain and coarse elsewhere, making the methods be easy to implement based on existing codes and have low communication complexity. Under some (strong) uniqueness conditions, stability and convergence theory of the parallel iterative stabilized methods are derived. Numerical tests are also performed to demonstrate the stability, convergence orders and high efficiency of the proposed methods.
Reviewer: Dimitra Antonopoulou (Chester)Application of the GRP scheme for cylindrical compressible fluid flowshttps://www.zbmath.org/1475.760652022-01-14T13:23:02.489162Z"Chen, Rui"https://www.zbmath.org/authors/?q=ai:chen.rui"Li, Jiequan"https://www.zbmath.org/authors/?q=ai:li.jiequan"Tian, Baolin"https://www.zbmath.org/authors/?q=ai:tian.baolinSummary: This paper contributes to apply both the direct Eulerian and Lagrangian generalized Riemann problem (GRP) schemes for the simulation of compressible fluid flows in two-dimensional cylindrical geometry. Particular attention is paid to the treatment of numerical boundary conditions at the symmetric center besides the zero velocity (momentum) enforced by the symmetry. The new treatment precisely describes how the thermodynamical variables are discretized near the center using the conservation property. Moreover, the Lagrangian GRP scheme is verified rigorously to satisfy the properties of symmetry and conservation. Numerical results demonstrate the performance of such treatments and the symmetry preserving property of the scheme with second order accuracy both in space and time.A direct ALE multi-moment finite volume scheme for the compressible Euler equationshttps://www.zbmath.org/1475.760662022-01-14T13:23:02.489162Z"Jin, Peng"https://www.zbmath.org/authors/?q=ai:jin.peng"Deng, Xi"https://www.zbmath.org/authors/?q=ai:deng.xi"Xiao, Feng"https://www.zbmath.org/authors/?q=ai:xiao.fengSummary: A direct Arbitrary Lagrangian Eulerian (ALE) method based on multi-moment finite volume scheme is developed for the Euler equations of compressible gas in 1D and 2D space. Both the volume integrated average (VIA) and the point values (PV) at cell vertices, which are used for high-order reconstructions, are treated as the computational variables and updated simultaneously by numerical formulations in integral and differential forms respectively. The VIAs of the conservative variables are solved by a finite volume method in the integral form of the governing equations to ensure the numerical conservativeness; whereas, the governing equations of differential form are solved for the PVs of the primitive variables to avoid the additional source terms generated from moving mesh, which largely simplifies the solution procedure. Numerical tests in both 1D and 2D are presented to demonstrate the performance of the proposed ALE scheme. The present multi-moment finite volume formulation consistent with moving meshes provides a high-order and efficient ALE computational model for compressible flows.A high-resolution cell-centered Lagrangian method with a vorticity-based adaptive nodal solver for two-dimensional compressible Euler equationshttps://www.zbmath.org/1475.760672022-01-14T13:23:02.489162Z"Qi, Jin"https://www.zbmath.org/authors/?q=ai:qi.jin"Tian, Baolin"https://www.zbmath.org/authors/?q=ai:tian.baolin"Li, Jiequan"https://www.zbmath.org/authors/?q=ai:li.jiequanSummary: In this work, a second-order high-resolution LAgrangian method with a Vorticity-based Adaptive Nodal Solver (LAVANS) is proposed to overcome the numerical difficulty of traditional Lagrangian methods for the simulation of multidimensional flows. The work mainly include three aspects to improve the performance of the traditional CAVEAT-type cell-centered Lagrangian method. First, a vorticity-based adaptive least-squares method for vertex velocity computation is proposed to suppress nonphysical mesh distortion caused by the traditional five-point-stencil least-squares method. Second, a simple interface flux modification is proposed such that the geometry conservation law is satisfied. Third, a generalized Riemann problem solver is employed in the LAVANS scheme to achieve one-step time-space second-order accuracy. Some typical benchmark numerical tests validate the performance of the LAVANS scheme.Incorporation of NURBS boundary representation with an unstructured finite volume approximationhttps://www.zbmath.org/1475.760682022-01-14T13:23:02.489162Z"Xia, Yifan"https://www.zbmath.org/authors/?q=ai:xia.yifan"Wang, Gaofeng"https://www.zbmath.org/authors/?q=ai:wang.gaofeng"Zheng, Yao"https://www.zbmath.org/authors/?q=ai:zheng.yao"Ji, Tingwei"https://www.zbmath.org/authors/?q=ai:ji.tingwei"Loh, Ching Y."https://www.zbmath.org/authors/?q=ai:loh.ching-yuenSummary: For compressible flow computations, the present paper extends the ETAU (enhanced time-accurate upwind) unstructured finite volume (FV) scheme to handle curved domain boundary with better accuracy. For the interior cells in the computational domain or the boundary cells with straight line boundary, the original ETAU scheme with second order accuracy in space and time is applied. For those boundary cells with the curved geometry, a more accurate Non-Uniform Rational B-Spline (NURBS) representation of the boundary is considered. The NURBS is commonly employed in computer aided design (CAD) to construct complex geometries. Here, it yields an exact geometry expression of complex boundary geometry. By combining ETAU with NURBS, the NURBS incorporated ETAU scheme (NETAU) is proposed for more accurate geometrical representation and fluxes evaluation. Details of the computing procedure of the geometry and surface fluxes for cells on the curved boundary, such as special transformation strategies and merging of ETAU and NURBS, are introduced and implemented. With NURBS, the NETAU scheme are geometrically versatile and more flexible. Several two-dimensional (2D) numerical cases are investigated to demonstrate the performance, computing efficiency and benefits of the NETAU scheme. The numerical results show that, for flows with low speed and high Reynolds number, the NETAU scheme provides more accurate pressure distribution on curved boundary than the original ETAU scheme. Meanwhile, the high-speed flow case shows that the NETAU scheme is still stable for high Mach number problem with shocks. Thus, the NETAU scheme potentially provides an accurate tool to describe complex geometry in computational fluid dynamics (CFD) simulations. It will help to reduce computational costs and enhances accuracy for flow domain dominated by complex geometries, with features such as high curvature and sharp edges.For the stability of homogeneous explicit finite-difference scheme of two-dimensional non-steady flow of compressible viscid gas with microstructurehttps://www.zbmath.org/1475.760702022-01-14T13:23:02.489162Z"Verveiko, N. D."https://www.zbmath.org/authors/?q=ai:verveiko.nikolai-dmitrievich"Prosvetov, V. I."https://www.zbmath.org/authors/?q=ai:prosvetov.v-iSummary: This paper considers mathematical model of two-dimensional non-steady flow of compressible viscid gas with microstructure, its homogeneous explicit finite-difference scheme. Especially it was studied stability of finite-difference scheme subject to microstructure. It is obtained stability rating of geometric interval and time step under the condition, that dissipative term is added, when kinematic viscosity misses.Stationary flow predictions using convolutional neural networkshttps://www.zbmath.org/1475.760722022-01-14T13:23:02.489162Z"Eichinger, Matthias"https://www.zbmath.org/authors/?q=ai:eichinger.matthias"Heinlein, Alexander"https://www.zbmath.org/authors/?q=ai:heinlein.alexander"Klawonn, Axel"https://www.zbmath.org/authors/?q=ai:klawonn.axelSummary: Computational Fluid Dynamics (CFD) simulations are a numerical tool to model and analyze the behavior of fluid flow. However, accurate simulations are generally very costly because they require high grid resolutions. In this paper, an alternative approach for computing flow predictions using Convolutional Neural Networks (CNNs) is described; in particular, a classical CNN as well as the U-Net architecture are used. First, the networks are trained in an expensive offline phase using flow fields computed by CFD simulations. Afterwards, the evaluation of the trained neural networks is very cheap. Here, the focus is on the dependence of the stationary flow in a channel on variations of the shape and the location of an obstacle. CNNs perform very well on validation data, where the averaged error for the best networks is below 3\%. In addition to that, they also generalize very well to new data, with an averaged error below 10\%.
For the entire collection see [Zbl 1471.65009].A fully space-time least-squares method for the unsteady Navier-Stokes systemhttps://www.zbmath.org/1475.760732022-01-14T13:23:02.489162Z"Lemoine, Jérôme"https://www.zbmath.org/authors/?q=ai:lemoine.jerome"Münch, Arnaud"https://www.zbmath.org/authors/?q=ai:munch.arnaudSummary: We introduce and analyze a space-time least-squares method associated with the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we construct a minimizing sequence for the least-squares functional which converges strongly to a solution of the Navier-Stokes system. After a finite number of iterations related to the value of the viscosity coefficient, the convergence is quadratic. Numerical experiments within the two dimensional case support our analysis. This globally convergent least-squares approach is related to the damped Newton method.An efficient numerical scheme for fully coupled flow and reactive transport in variably saturated porous media including dynamic capillary effectshttps://www.zbmath.org/1475.760932022-01-14T13:23:02.489162Z"Illiano, Davide"https://www.zbmath.org/authors/?q=ai:illiano.davide"Pop, Iuliu Sorin"https://www.zbmath.org/authors/?q=ai:pop.iuliu-sorin"Radu, Florin Adrian"https://www.zbmath.org/authors/?q=ai:radu.florin-adrianSummary: In this paper we study a model for the transport of an external component, e.g., a surfactant, in variably saturated porous media. We discretize the model in time and space by combining a backward Euler method with the linear Galerkin finite elements. The Newton method and the L-Scheme are employed for the linearization and the performance of these schemes is studied numerically. A special focus is set on the effects of dynamic capillarity on the transport equation.
For the entire collection see [Zbl 1471.65009].Global random walk solutions for flow and transport in porous mediahttps://www.zbmath.org/1475.760972022-01-14T13:23:02.489162Z"Suciu, Nicolae"https://www.zbmath.org/authors/?q=ai:suciu.nicolae-petruSummary: This article presents a new approach to solve the equations of flow in heterogeneous porous media by using random walks on regular lattices. The hydraulic head is represented by computational particles which are spread globally from the lattice sites according to random walk rules, with jump probabilities determined by the hydraulic conductivity. The latter is modeled as a realization of a random function generated as a superposition of periodic random modes. One- and two-dimensional numerical solutions are validated by comparisons with analytical manufactured solutions. Further, an ensemble of divergence-free velocity fields computed with the new approach is used to conduct Monte Carlo simulations of diffusion in random fields. The transport equation is solved by a global random walk algorithm which moves computational particles representing the concentration of the solute on the same lattice as that used to solve the flow equations. The integrated flow and transport solution is validated by a good agreement between the statistical estimations of the first two spatial moments of the solute plume and the predictions of the stochastic theory of transport in groundwater.
For the entire collection see [Zbl 1471.65009].Mathematical optics. Classical, quantum, and computational methodshttps://www.zbmath.org/1475.780022022-01-14T13:23:02.489162ZPublisher's description: Going beyond standard introductory texts, Mathematical Optics: Classical, Quantum, and Computational Methods brings together many new mathematical techniques from optical science and engineering research. Profusely illustrated, the book makes the material accessible to students and newcomers to the field.
Divided into six parts, the text presents state-of-the-art mathematical methods and applications in classical optics, quantum optics, and image processing.
\begin{itemize}
\item Part I describes the use of phase space concepts to characterize optical beams and the application of dynamic programming in optical waveguides.
\item Part II explores solutions to paraxial, linear, and nonlinear wave equations.
\item Part III discusses cutting-edge areas in transformation optics (such as invisibility cloaks) and computational plasmonics.
\item Part IV uses Lorentz groups, dihedral group symmetry, Lie algebras, and Liouville space to analyze problems in polarization, ray optics, visual optics, and quantum optics.
\item Part V examines the role of coherence functions in modern laser physics and explains how to apply quantum memory channel models in quantum computers.
\item Part VI introduces super-resolution imaging and differential geometric methods in image processing.
\end{itemize}
As numerical/symbolic computation is an important tool for solving numerous real-life problems in optical science, many chapters include Mathematica code in their appendices. The software codes and notebooks as well as color versions of the book's figures are available at \url {www.crcpress.com}.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Padgett, Miles}, Orbital angular momentum: a ray optical interpretation, 3-12 [Zbl 1288.78028]
\textit{Alieva, Tatiana; Cámara, Alejandro; Bastiaans, Martin J.}, Wigner distribution moments for beam characterization, 13-51 [Zbl 1288.78026]
\textit{Calvo, Maria L.; Pérez-Ríos, Jesús; Lakshminarayanan, Vasudevan}, Dynamic programming applications in optics, 53-94 [Zbl 1288.78039]
\textit{Alonso, Miguel A.; Moore, Nicole J.}, Basis expansions for monochromatic field propagation in free space, 97-141 [Zbl 1288.78010]
\textit{Abramochkin, Eugeny; Alieva, Tatiana; Rodrigo, José A.}, Solutions of paraxial equations and families of Gaussian beams, 143-192 [Zbl 1288.78009]
\textit{Lakshminarayanan, Vasudevan; Nandy, Sudipta; Sridhar, Raghavendra}, The decomposition method to solve differential equations: optical applications, 193-232 [Zbl 1288.78012]
\textit{Kadic, Muamer; Guenneau, Sébastien; Enoch, Stefan}, An introduction to mathematics of transformational plasmonics, 235-277 [Zbl 1288.78019]
\textit{Sukharev, Maxim}, Plasmonics: computational approach, 279-299 [Zbl 1288.78029]
\textit{Başkai, Sibel; Kim, Y. S.}, Lorentz group in ray and polarization optics, 303-340 [Zbl 1288.78027]
\textit{Torre, Amalia}, Paraxial wave equation: Lie-algebra-based approach, 341-417 [Zbl 1288.78014]
\textit{Viana, Marlos}, Dihedral polynomials, 419-437 [Zbl 1288.78004]
\textit{Ban, Masashi}, Lie algebra and Liouville-space methods in quantum optics, 439-479 [Zbl 1290.81048]
\textit{Luis, Alfredo}, From classical to quantum light and vice versa: quantum phase-space methods, 483-506 [Zbl 1290.81227]
\textit{Zahid, Imrana Ashraf; Lakshminarayanan, Vasudevan}, Coherence functions in classical and quantum optics, 507-531 [Zbl 1288.78006]
\textit{Rybár, Tomáš; Ziman, Mário; Bužek, Vladimír}, Quantum memory channels in quantum optics, 533-552 [Zbl 1290.81020]
\textit{Simpkins, Jonathan D.; Stevenson, Robert L.}, An introduction to super-resolution imaging, 555-580 [Zbl 1298.94015]
\textit{ter Haar Romeny, Bart M.}, The differential structure of images, 581-597 [Zbl 1294.94007]The electromagnetic scattering problem by a cylindrical doubly connected domain at oblique incidence: the direct problemhttps://www.zbmath.org/1475.780052022-01-14T13:23:02.489162Z"Mindrinos, Leonidas"https://www.zbmath.org/authors/?q=ai:mindrinos.leonidasThe author analyses the direct electromagnetic scattering problem of time-harmonic oblique incident waves by an infinitely long, penetrable and doubly connected cylinder. The well-posedness is obtained, when considering transmission conditions on the outer boundary and impedance boundary condition on the inner boundary, by the hybrid method of using Green's formulas and the integral representation of the solutions. The numerical solution is exposed by considering a collocation method using trigonometric polynomial approximations.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)SPARCOM: sparsity based super-resolution correlation microscopyhttps://www.zbmath.org/1475.780062022-01-14T13:23:02.489162Z"Solomon, Oren"https://www.zbmath.org/authors/?q=ai:solomon.oren"Eldar, Yonina C."https://www.zbmath.org/authors/?q=ai:eldar.yonina-c"Mutzafi, Maor"https://www.zbmath.org/authors/?q=ai:mutzafi.maor"Segev, Mordechai"https://www.zbmath.org/authors/?q=ai:segev.mordechaiNumerical simulation of coupled electromagnetic and thermal problems in permanent magnet synchronous machineshttps://www.zbmath.org/1475.780082022-01-14T13:23:02.489162Z"Lotfi, A."https://www.zbmath.org/authors/?q=ai:lotfi.abdelhakim|lotfi.ali"Marcsa, D."https://www.zbmath.org/authors/?q=ai:marcsa.daniel"Horváth, Z."https://www.zbmath.org/authors/?q=ai:horvath.zoltan|horvath.zsolt|horvath.zsuzsanna"Prudhomme, C."https://www.zbmath.org/authors/?q=ai:prudhomme.christophe"Chabannes, V."https://www.zbmath.org/authors/?q=ai:chabannes.vincentSummary: The main objective of our task is to develop mathematical models, numerical techniques to analyse the thermal effects in electric machines, to implement the developed algorithm in multiprocessor or multi-core environments and to apply them to industrial use cases. In this study, we take into account coupled character of the electromagnetic and thermal features of the physical process. Both thermal and electromagnetic processes are considered transient, solved by means of the FEM method on independent meshes and the time-discretization is realized using time operator splitting. Two examples are presented to assess the accuracy of the developed coupled solvers and the numerical results are compared with the experimental ones, which are obtained from a prototype machine.
For the entire collection see [Zbl 1471.65009].Enhancing mesh-based photoacoustic tomography with parallel computing on multiprocessor schemehttps://www.zbmath.org/1475.780092022-01-14T13:23:02.489162Z"Sun, Yao"https://www.zbmath.org/authors/?q=ai:sun.yao"Yuan, Zhen"https://www.zbmath.org/authors/?q=ai:yuan.zhen"Jiang, Huaibei"https://www.zbmath.org/authors/?q=ai:jiang.huaibeiSummary: Photoacoustic tomography is an emerging technique in biomedicine that is capable of visualizing high resolution structural and functional information of tissue up to several centimeters deep. Mesh based numerical reconstruction algorithms have an unrivaled advantage over other reconstruction algorithms in photoacoustic imaging, due to its accurate mathematical modeling and the capability to recover multiple optical/acoustic parameters in the reconstruction. However, the slow reconstruction speed and huge memory cost hindered this advanced reconstruction algorithm from application areas where large scale reconstruction or real/near-real time reconstruction is required, for example, sub-millimeter or micrometer resolution imaging, photoacoustic guided cancer treatment, etc. In this study, we reported a high performance photoacoustic tomography method based on parallel computing strategy with multiprocessor scheme. Our simulation result has shown that the parallelized photoacoustic tomography method using multiprocessor scheme is capable of providing fine reconstructed images of blood vessel structures up to 0.14mm in diameter. Further phantom experiment demonstrated that cross hairs can be clearly reconstructed, when a mesh comprised of 28512 triangle finite elements is used. Therefore, our multiprocessor based parallelized photoacoustic tomography might be promising for large scale reconstruction or real/near-real time reconstruction in biomedical application of mesh-based photoacoustic tomography algorithm.A fast method for evaluating Green's function in irregular domains with application to charge interaction in a nanoporehttps://www.zbmath.org/1475.780122022-01-14T13:23:02.489162Z"Zhao, Qiyuan"https://www.zbmath.org/authors/?q=ai:zhao.qiyuan"Liu, Pei"https://www.zbmath.org/authors/?q=ai:liu.pei"Xu, Zhenli"https://www.zbmath.org/authors/?q=ai:xu.zhenliSummary: We develop a fast meshless algorithm for electrostatic interaction in an irregular domain with given potential boundary conditions, which is of importance in many applications such as electrochemical energy and electric structure calculations. The algorithm is based on an approximation of the Green's function using two-level image charges, in which the inner-layer charges are located nearby the boundary to eliminate the singularity of the induced polarization potential, and the outer-layer charges with fixed positions approximate the long-range tail of the potential. We find the number of inner-layer image charges can be very small and thus the total complexity of the algorithm is less expensive and potentially suitable for use in particle simulations. The numerical results show the performance of the algorithm is attractive. We also use the algorithm to investigate the electrostatic interaction for particles in a cylindrical nanopore and show that the electrostatic interaction within the pore has an exponential decay.Methodology of determining material parameters based on optimization techniqueshttps://www.zbmath.org/1475.800022022-01-14T13:23:02.489162Z"Petrasova, Iveta"https://www.zbmath.org/authors/?q=ai:petrasova.iveta"Kotlan, Vaclav"https://www.zbmath.org/authors/?q=ai:kotlan.vaclav"Panek, David"https://www.zbmath.org/authors/?q=ai:panek.david"Dolezel, Ivo"https://www.zbmath.org/authors/?q=ai:dolezel.ivoThe goal of numerical modelling while producing new devices is to avoid creating expensive prototypes in order to cheapen the production. The problem is that the characteristics of the materials involved in the modelling may be unknown in the temperature range of interest and their experimental determination is often impossible. For example, the heat conduction and the specific heat of some metal or alloy may be unknown above its melting point.
To find such temperature-dependent material properties the authors suggest the following method.
The temperature dependencies of the desired material characteristics are calibrated, i.e., compared with analogous properties of similar material. Then the desired function of temperature begins to depend on several adjustable parameters which must be found by solving an optimization problem. In turn, during optimization some process is modeled which involves the materials under study. The temperature field at this process is calculated in some moments of time and compared with experimental data. The adjustable parameters are selected to minimize some quality functional describing the difference between the experimental and calculated temperatures.
The method described is illustrated by two processes: laser welding and laser cladding. Several numerical optimization methods are discussed and compared.
Reviewer: Aleksey Syromyasov (Saransk)Bivariate rational approximations of the general temperature integralhttps://www.zbmath.org/1475.800082022-01-14T13:23:02.489162Z"Aghili, Alireza"https://www.zbmath.org/authors/?q=ai:aghili.alireza"Sukhorukova, Nadezda"https://www.zbmath.org/authors/?q=ai:sukhorukova.nadezda"Ugon, Julien"https://www.zbmath.org/authors/?q=ai:ugon.julienSummary: The non-isothermal analysis of materials with the application of the Arrhenius equation involves temperature integration. If the frequency factor in the Arrhenius equation depends on temperature with a power-law relationship, the integral is known as the general temperature integral. This integral which has no analytical solution is estimated by the approximation functions with different accuracies. In this article, the rational approximations of the integral were obtained based on the minimization of the maximal deviation of bivariate functions. Mathematically, these problems belong to the class of quasiconvex optimization and can be solved using the bisection method. The approximations obtained in this study are more accurate than all approximates available in the literature.A positivity-preserving finite volume scheme for heat conduction equation on generalized polyhedral mesheshttps://www.zbmath.org/1475.800092022-01-14T13:23:02.489162Z"Xie, Hui"https://www.zbmath.org/authors/?q=ai:xie.hui"Xu, Xuejun"https://www.zbmath.org/authors/?q=ai:xu.xuejun"Zhai, Chuanlei"https://www.zbmath.org/authors/?q=ai:zhai.chuanlei"Yong, Heng"https://www.zbmath.org/authors/?q=ai:yong.hengSummary: In this paper we present a nonlinear finite volume scheme preserving positivity for heat conduction equations. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are eliminated by our newly developed second-order explicit interpolation formula on generalized polyhedral meshes. With the help of the additional parameters, it is not necessary to choose the stencil adaptively to obtain the convex decomposition of the co-normal vector and also is not required to replace the interpolation formula with positivity-preserving but usually low-order accurate ones whenever negative interpolated auxiliary unknowns appear. Moreover, the new flux approximation has a fixed stencil. These features make our scheme more efficient compared with other existing methods based on Le Potier's nonlinear two-point approximation, especially in 3D. Numerical experiments show that the scheme maintains the positivity of the continuous solution and has nearly second-order accuracy for the solution on the distorted meshes where the diffusion tensor may be anisotropic and discontinuous.Computational software: polymer chain generation for coarse-grained models using radical-like polymerizationhttps://www.zbmath.org/1475.820242022-01-14T13:23:02.489162Z"Mahaud, Morgane"https://www.zbmath.org/authors/?q=ai:mahaud.morgane"Zhai, Zengqiang"https://www.zbmath.org/authors/?q=ai:zhai.zengqiang"Perez, Michel"https://www.zbmath.org/authors/?q=ai:perez.michel"Lame, Olivier"https://www.zbmath.org/authors/?q=ai:lame.olivier"Fusco, Claudio"https://www.zbmath.org/authors/?q=ai:fusco.claudio"Chazeau, Laurent"https://www.zbmath.org/authors/?q=ai:chazeau.laurent"Makke, Ali"https://www.zbmath.org/authors/?q=ai:makke.ali"Marque, Grégory"https://www.zbmath.org/authors/?q=ai:marque.gregory"Morthomas, Julien"https://www.zbmath.org/authors/?q=ai:morthomas.julienSummary: This paper presents major improvements in the efficiency of the so-called Radical-Like Polymerization (RLP) algorithm proposed in [\textit{M. Perez} et al., ``Polymer chain generation for coarse-grained models using radical-like polymerization'', J. Chem. Phys. 128, No. 23, Article ID 234904 (2008; \url{doi:10.1063/1.2936839})]. Three enhancements are detailed in this paper: (1) the capture radius of a radical is enlarged to increase the probability of finding a neighboring monomer; (2) between each growth step, equilibration is now performed with increasing the relaxation time depending on the actual chain size; (3) the RLP algorithm is now fully parallelized and proposed as a ``fix'' within the ``Lammps'' molecular dynamics simulation suite.An admissible asymptotic-preserving numerical scheme for the electronic \(M_1\) model in the diffusive limithttps://www.zbmath.org/1475.820252022-01-14T13:23:02.489162Z"Guisset, Sébastien"https://www.zbmath.org/authors/?q=ai:guisset.sebastien"Brull, Stéphane"https://www.zbmath.org/authors/?q=ai:brull.stephane"Dubroca, Bruno"https://www.zbmath.org/authors/?q=ai:dubroca.bruno"Turpault, Rodolphe"https://www.zbmath.org/authors/?q=ai:turpault.rodolpheSummary: This work is devoted to the derivation of an admissible asymptotic-preserving scheme for the electronic \(M_1\) model in the diffusive regime. A numerical scheme is proposed in order to deal with the mixed derivatives which arise in the diffusive limit leading to an anisotropic diffusion. The derived numerical scheme preserves the realisability domain and enjoys asymptotic-preserving properties correctly handling the diffusive limit recovering the relevant limit equation. In addition, the cases of non constants electric field and collisional parameter are naturally taken into account with the present approach. Numerical test cases validate the considered scheme in the non-collisional and diffusive limits.Comparison of preconditioning strategies in energy conserving implicit particle in cell methodshttps://www.zbmath.org/1475.820262022-01-14T13:23:02.489162Z"Siddi, Lorenzo"https://www.zbmath.org/authors/?q=ai:siddi.lorenzo"Cazzola, Emanuele"https://www.zbmath.org/authors/?q=ai:cazzola.emanuele"Lapenta, Giovanni"https://www.zbmath.org/authors/?q=ai:lapenta.giovanniSummary: This work presents a set of preconditioning strategies able to significantly accelerate the performance of fully implicit energy-conserving Particle-in-Cell methods to a level that becomes competitive with semi-implicit methods. We compare three different preconditioners. We consider three methods and compare them with a straight unpreconditioned Jacobian Free Newton Krylov (JFNK) implementation. The first two focus, respectively, on improving the handling of particles (particle hiding) or fields (field hiding) within the JFNK iteration. The third uses the field hiding preconditioner within a direct Newton iteration where a Schwarz-decomposed Jacobian is computed analytically. Clearly, field hiding used with JFNK or with the direct Newton-Schwarz (DNS) method outperforms all method. We compare these implementations with a recent semi-implicit energy conserving scheme. Fully implicit methods are still lag behind in cost per cycle but not by a large margin when proper preconditioning is used. However, for exact energy conservation, preconditioned fully implicit methods are significantly easier to implement compared with semi-implicit methods and can be extended to fully relativistic physics.Stabilized predictor-corrector schemes for gradient flows with strong anisotropic free energyhttps://www.zbmath.org/1475.820272022-01-14T13:23:02.489162Z"Shen, Jie"https://www.zbmath.org/authors/?q=ai:shen.jie"Xu, Jie"https://www.zbmath.org/authors/?q=ai:xu.jieSummary: Gradient flows with strong anisotropic free energy are difficult to deal with numerically with existing approaches. We propose a stabilized predictor-corrector approach to construct schemes which are second-order accurate, easy to implement, and maintain the stability of first-order stabilized schemes. We apply the new approach to three different types of gradient flows with strong anisotropic free energy: anisotropic diffusion equation, anisotropic Cahn-Hilliard equation, and Cahn-Hilliard equation with degenerate diffusion mobility. Numerical results are presented to show that the stabilized predictor-corrector schemes are second-order accurate, unconditionally stable for the first two equations, and allow larger time step than the first-order stabilized scheme for the last equation. We also prove rigorously that, for the isotropic Cahn-Hilliard equation, the stabilized predictor-corrector scheme is of second-order.GenEx: a simple generator structure for exclusive processes in high energy collisionshttps://www.zbmath.org/1475.820282022-01-14T13:23:02.489162Z"Kycia, R. A."https://www.zbmath.org/authors/?q=ai:kycia.radoslaw-antoni"Chwastowski, J."https://www.zbmath.org/authors/?q=ai:chwastowski.j-j"Staszewski, R."https://www.zbmath.org/authors/?q=ai:staszewski.reiner|staszewski.robert-bogdan"Turnau, J."https://www.zbmath.org/authors/?q=ai:turnau.jean-remiSummary: A simple C++ class structure for construction of a Monte Carlo event generator which can produce unweighted events within relativistic phase space is presented. The generator is self-adapting to the provided matrix element and acceptance cuts. The program is designed specially for exclusive processes and includes, as an example of such an application, implementation of the model for exclusive production of meson pairs \(pp \to pM^+ M^- p\) in high energy proton-proton collisions.An analytic approximation to the Bayesian detection statistic for continuous gravitational waveshttps://www.zbmath.org/1475.830192022-01-14T13:23:02.489162Z"Bero, John J."https://www.zbmath.org/authors/?q=ai:bero.john-j"Whelan, John T."https://www.zbmath.org/authors/?q=ai:whelan.john-tPreface: Special issue dedicated to distance geometryhttps://www.zbmath.org/1475.900022022-01-14T13:23:02.489162ZFrom the text: This preface introduces the special issue of the Journal of Global Optimization dedicated to the workshop ``Distance Geometry Theory and Applications'' (DGTA16) which took place at the DIMACS Center, Rutgers University, Piscataway NJ, USA.Numerical solution of traffic flow modelshttps://www.zbmath.org/1475.900042022-01-14T13:23:02.489162Z"Vacek, Lukáš"https://www.zbmath.org/authors/?q=ai:vacek.lukas"Kučera, Václav"https://www.zbmath.org/authors/?q=ai:kucera.vaclavSummary: We describe the simulation of traffic flows on networks. On individual roads we use standard macroscopic traffic models. The discontinuous Galerkin method in space and a multistep method in time is used for the numerical solution. We introduce limiters to keep the density in an admissible interval as well as prevent spurious oscillations in the numerical solution. To simulate traffic on networks, one should construct suitable numerical fluxes at junctions.
For the entire collection see [Zbl 1471.65009].Tensor factorization with total variation for internet traffic data imputationhttps://www.zbmath.org/1475.900162022-01-14T13:23:02.489162Z"Yu, Gaohang"https://www.zbmath.org/authors/?q=ai:yu.gaohang"Wang, Liqin"https://www.zbmath.org/authors/?q=ai:wang.liqin"Wan, Shaochun"https://www.zbmath.org/authors/?q=ai:wan.shaochun"Qi, Liqun"https://www.zbmath.org/authors/?q=ai:qi.liqun"Xu, Yanwei"https://www.zbmath.org/authors/?q=ai:xu.yanweiSummary: Recovering network traffic data from incomplete observed data becomes increasingly critical in network engineering and management. To fully exploit the spatial-temporal features of the internet traffic data, this paper presents a new tensor completion model which combines the T-product-based tensor factorization with total variation (TV) regularization. To tackle the proposed mode, and effective proximal alternating minimization (PAM) algorithm with guaranteed convergence is designed. Extensive experiments on the real-world traffic datasets show that the proposed method has superiority over the existing state-of-the-art methods.Sensitivity analysis of queueing models based on polynomial chaos approachhttps://www.zbmath.org/1475.900172022-01-14T13:23:02.489162Z"Ameur, Lounes"https://www.zbmath.org/authors/?q=ai:ameur.lounes"Bachioua, Lahcene"https://www.zbmath.org/authors/?q=ai:bachioua.lahceneSummary: Queueing systems are modeled by equations which depend on a large number of input parameters. In practice, significant uncertainty is associated with estimates of these parameters, and this uncertainty must be considered in the analysis of the model. The objective of this paper is to propose a sensitivity analysis approach for a queueing model, presenting parameters that follow a Gaussian distribution. The approach consists in decomposing the output of the model (stationary distribution of the model) into a polynomial chaos. The sensitivity indices, allowing to quantify the contribution of each parameter to the variance of the output, are obtained directly from the coefficients of decomposition. The proposed approach is then applied to M/G/1/N queueing model. The most influential parameters are highlighted. Finally several numerical and data examples are sketched out to illustrate the accuracy of the proposed method and compare them with Monte Carlo simulation. The results of this work will be useful to practitioners in various fields of theoretical and applied sciences.On the unique solution of the generalized absolute value equationhttps://www.zbmath.org/1475.900352022-01-14T13:23:02.489162Z"Wu, Shiliang"https://www.zbmath.org/authors/?q=ai:wu.shiliang"Shen, Shuqian"https://www.zbmath.org/authors/?q=ai:shen.shuqianSummary: In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) \(Ax-B|x|=b\) with \(A, B\in\mathbb{R}^{n\times n}\) from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system \(Ax=b\) with \(A\in\mathbb{R}^{n\times n}\). Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.An efficient mixed conjugate gradient method for solving unconstrained optimisation problemshttps://www.zbmath.org/1475.900362022-01-14T13:23:02.489162Z"Mtagulwa, P."https://www.zbmath.org/authors/?q=ai:mtagulwa.peter"Kaelo, P."https://www.zbmath.org/authors/?q=ai:kaelo.proSummary: Conjugate gradient algorithms are most commonly used to solve large scale unconstrained optimisation problems. They are simple and do not require the computation and/or storage of the second derivative information about the objective function. We propose a new conjugate gradient method and establish its global convergence under suitable assumptions. Numerical examples demonstrate the efficiency and effectiveness of our method.Constraint reduction reformulations for projection algorithms with applications to wavelet constructionhttps://www.zbmath.org/1475.900642022-01-14T13:23:02.489162Z"Dao, Minh N."https://www.zbmath.org/authors/?q=ai:dao.minh-ngoc"Dizon, Neil D."https://www.zbmath.org/authors/?q=ai:dizon.neil-d"Hogan, Jeffrey A."https://www.zbmath.org/authors/?q=ai:hogan.jeffrey-a"Tam, Matthew K."https://www.zbmath.org/authors/?q=ai:tam.matthew-kSummary: We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra's classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the \textit{constraint reduction reformulation} and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas-Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas-Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas-Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants.Linearization of McCormick relaxations and hybridization with the auxiliary variable methodhttps://www.zbmath.org/1475.900702022-01-14T13:23:02.489162Z"Najman, Jaromił"https://www.zbmath.org/authors/?q=ai:najman.jaromil"Bongartz, Dominik"https://www.zbmath.org/authors/?q=ai:bongartz.dominik"Mitsos, Alexander"https://www.zbmath.org/authors/?q=ai:mitsos.alexanderSummary: The computation of lower bounds via the solution of convex lower bounding problems depicts current state-of-the-art in deterministic global optimization. Typically, the nonlinear convex relaxations are further underestimated through linearizations of the convex underestimators at one or several points resulting in a lower bounding linear optimization problem. The selection of linearization points substantially affects the tightness of the lower bounding linear problem. Established methods for the computation of such linearization points, e.g., the sandwich algorithm, are already available for the auxiliary variable method used in state-of-the-art deterministic global optimization solvers. In contrast, no such methods have been proposed for the (multivariate) McCormick relaxations. The difficulty of determining a good set of linearization points for the McCormick technique lies in the fact that no auxiliary variables are introduced and thus, the linearization points have to be determined in the space of original optimization variables. We propose algorithms for the computation of linearization points for convex relaxations constructed via the (multivariate) McCormick theorems. We discuss alternative approaches based on an adaptation of Kelley's algorithm; computation of all vertices of an \(n\)-simplex; a combination of the two; and random selection. All algorithms provide substantial speed ups when compared to the single point strategy used in our previous works. Moreover, we provide first results on the hybridization of the auxiliary variable method with the McCormick technique benefiting from the presented linearization strategies resulting in additional computational advantages.On \(q\)-Newton's method for unconstrained multiobjective optimization problemshttps://www.zbmath.org/1475.900952022-01-14T13:23:02.489162Z"Mishra, Shashi Kant"https://www.zbmath.org/authors/?q=ai:mishra.shashi-kant"Panda, Geetanjali"https://www.zbmath.org/authors/?q=ai:panda.geetanjali"Ansary, Md Abu Talhamainuddin"https://www.zbmath.org/authors/?q=ai:ansary.md-abu-talhamainuddin"Ram, Bhagwat"https://www.zbmath.org/authors/?q=ai:ram.bhagwatSummary: In this paper, we present a method of so-called \(q\)-Newton's type descent direction for solving unconstrained multiobjective optimization problems. The algorithm presented in this paper is implemented by applying an independent parameter \(q\) (quantum) in an Armijo-like rule to compute the step length which guarantees that the value of the objective function decreases at every iteration. The search processes gradually shift from global in the beginning to local as the algorithm converges due to \(q\)-gradient. The algorithm is experimented on 41 benchmark/test functions which are unimodal and multi-modal with 1, 2, 3, 4, 5, 10 and 50 different dimensions. The performance of the proposed method is confirmed by comparing with three existing schemes.Iterative algorithm with structured diagonal Hessian approximation for solving nonlinear least squares problemshttps://www.zbmath.org/1475.901002022-01-14T13:23:02.489162Z"Awwal, Aliyu Muhammed"https://www.zbmath.org/authors/?q=ai:awwal.aliyu-muhammed"Kumam, Poom"https://www.zbmath.org/authors/?q=ai:kumam.poom"Mohammad, Hassan"https://www.zbmath.org/authors/?q=ai:mohammad.hassanSummary: Nonlinear least squares problems are special class of unconstrained optimization problems in which their gradient and Hessian have special structures. We exploit these structures and proposed a matrix free algorithm with diagonal Hessian approximation for solving nonlinear least squares problems. We devise appropriate safeguarding strategies to ensure the Hessian matrix is positive definite throughout the iteration process. The proposed algorithm generates descent direction and is globally convergent. Preliminary numerical experiments shows that the proposed method is competitive with a recently developed similar method.Analysis of stochastic gradient descent in continuous timehttps://www.zbmath.org/1475.901052022-01-14T13:23:02.489162Z"Latz, Jonas"https://www.zbmath.org/authors/?q=ai:latz.jonasSummary: Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time representation of stochastic gradient descent. The stochastic gradient process is a dynamical system that is coupled with a continuous-time Markov process living on a finite state space. The dynamical system -- a gradient flow -- represents the gradient descent part, the process on the finite state space represents the random subsampling. Processes of this type are, for instance, used to model clonal populations in fluctuating environments. After introducing it, we study theoretical properties of the stochastic gradient process: We show that it converges weakly to the gradient flow with respect to the full target function, as the learning rate approaches zero. We give conditions under which the stochastic gradient process with constant learning rate is exponentially ergodic in the Wasserstein sense. Then we study the case, where the learning rate goes to zero sufficiently slowly and the single target functions are strongly convex. In this case, the process converges weakly to the point mass concentrated in the global minimum of the full target function; indicating consistency of the method. We conclude after a discussion of discretisation strategies for the stochastic gradient process and numerical experiments.Existence of solution of constrained interval optimization problems with regularity concepthttps://www.zbmath.org/1475.901072022-01-14T13:23:02.489162Z"Roy, Priyanka"https://www.zbmath.org/authors/?q=ai:roy.priyanka"Panda, Geetanjali"https://www.zbmath.org/authors/?q=ai:panda.geetanjaliThe authors investigate properties of an interval minimization problem with interval inequality constraints. The objective function is interval valued, i.e. \(f: R^n \rightarrow I(R)\), where \(I(R)\) is the set of all real closed intervals and the set of feasible solutions is described by a system of interval inequality relations defined on \(I(R)\). A solution of such problems is an efficient point of the feasible set. Three types of solutions are considered: efficient points, weak efficient points and strong efficient points. Necessary and sufficient conditions for the existence of the solutions are derived. Theoretical results are illustrated on numerical examples.
Reviewer: Karel Zimmermann (Praha)An interior point method for \(P_*(\kappa)\)-horizontal linear complementarity problem based on a new proximity functionhttps://www.zbmath.org/1475.901092022-01-14T13:23:02.489162Z"Fathi Hafshejani, Sajad"https://www.zbmath.org/authors/?q=ai:fathi-hafshejani.sajad"Fakharzadeh Jahromi, Alireza"https://www.zbmath.org/authors/?q=ai:fakharzadehjahromi.alireza|fakharzadeh-jahromi.alirezaSummary: Kernel functions play an important role in the design and complexity analysis of interior point algorithms for solving convex optimization problems. They determine both search directions and the proximity measure between the iterate and the central path. In this paper, we introduce a primal-dual interior point algorithm for solving \(P_*(\kappa)\)-horizontal linear complementarity problems based on a new kernel function that has a trigonometric function in its barrier term. By using some simple analysis tools, we present some properties of the new kernel function. Our analysis shows that the algorithm meets the best known complexity bound i.e., \(O\left((1+2\kappa)\sqrt{n}\log n\log \frac{n}{\varepsilon}\right)\) for large-update methods. Finally, we present some numerical results illustrating the performance of the algorithm.A system of differential set-valued variational inequalities in finite dimensional spaceshttps://www.zbmath.org/1475.901132022-01-14T13:23:02.489162Z"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11"Wang, Xing"https://www.zbmath.org/authors/?q=ai:wang.xing"Huang, Nan-Jing"https://www.zbmath.org/authors/?q=ai:huang.nan-jingSummary: A system of differential set-valued variational inequalities is introduced and studied in finite dimensional Euclidean spaces. An existence theorem of weak solutions for the system of differential set-valued variational inequalities in the sense of Carathéodory is proved under some suitable conditions. Furthermore, a convergence result on Euler time-dependent procedure for solving the system of differential set-valued variational inequalities is also given.Non-interior-point smoothing Newton method for CP revisited and its application to support vector machineshttps://www.zbmath.org/1475.901142022-01-14T13:23:02.489162Z"Ni, Tie"https://www.zbmath.org/authors/?q=ai:ni.tieSummary: Non-interior-point smoothing Newton method (SNM) for optimization have been widely studied for over three decades. SNM is a popular approach for solving small- and medium-scale complementarity problem (CP) and many optimization problems. The main purpose of this paper is to revisit the SNM and show that the Hessian matrix in SNM becomes increasingly ill-conditioned while smoothing parameter approaches to zero, which leads to their practical use remains limited due to computational difficulties in solving large-scale CP. To tackle this, we redesign a new smoothing method, called accelerated preconditioned smoothing method (APSM) for the efficient solution of regularized support vector machines in machine learning. With the help of suitable preconditioner, we can correct the ill-conditioning of associated smoothing Hessian matrix and thereby the associated smoothing Hessian equation can be solved in a few of iterations by using iterative methods in linear algebra. Two accelerated techniques are designed in the paper to reduce our computation time. Finally we present numerical experiments to support our theoretical guarantees and test accelerated convergence obtained by APSM. The result showed that APSM computes faster than state of art algorithm without reducing classification accuracy.A nonmonotone smoothing Newton algorithm for weighted complementarity problemhttps://www.zbmath.org/1475.901152022-01-14T13:23:02.489162Z"Tang, Jingyong"https://www.zbmath.org/authors/?q=ai:tang.jingyong"Zhang, Hongchao"https://www.zbmath.org/authors/?q=ai:zhang.hongchaoSummary: The weighted complementarity problem (denoted by WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this paper, by introducing a one-parametric class of smoothing functions which includes the weight vector, we propose a smoothing Newton algorithm with nonmonotone line search to solve WCP. We show that any accumulation point of the iterates generated by this algorithm, if exists, is a solution of the considered WCP. Moreover, when the solution set of WCP is nonempty, under assumptions weaker than the Jacobian nonsingularity assumption, we prove that the iteration sequence generated by our algorithm is bounded and converges to one solution of WCP with local superlinear or quadratic convergence rate. Promising numerical results are also reported.An alternate minimization method beyond positive definite proximal regularization: convergence and complexityhttps://www.zbmath.org/1475.901162022-01-14T13:23:02.489162Z"Wang, Xianfu"https://www.zbmath.org/authors/?q=ai:wang.xianfu"Ou, Xiaoqing"https://www.zbmath.org/authors/?q=ai:ou.xiaoqing"Zhang, Tao"https://www.zbmath.org/authors/?q=ai:zhang.tao.5|zhang.tao.6"Chen, Jiawei"https://www.zbmath.org/authors/?q=ai:chen.jiaweiSummary: In this paper, an alternate minimization method beyond positive definite proximal regularization is introduced for solving linearly constrained separable convex optimization problems. The proposed method can be interpreted as the prediction-correction method from the perspective of variational inequalities. The convergence of the proposed method is established without strong convexity. Finally, the iteration complexity of the proposed method is also derived in the ergodic sense.A regularization Newton method based on the generalized Fischer-Burmeister smoothing function for the NCPhttps://www.zbmath.org/1475.901172022-01-14T13:23:02.489162Z"Zhang, Ping"https://www.zbmath.org/authors/?q=ai:zhang.ping|zhang.ping.6|zhang.ping.2|zhang.ping.3|zhang.ping.1|zhang.ping.5Summary: Based on the generalized Fischer-Burmeister smoothing function, we propose a regularization Newton method for solving the nonlinear complementarity problem. The proposed method views the regularization parameter as an independent variable. Moreover, it solves a perturbed Newton equation to obtain the search direction and adopts a simple nonmonotone line search scheme to improve the numerical results. Under suitable assumptions, we prove that our method has global and local quadratic convergence and the regularization parameter converges to zero globally Q-linearly. Numerical results shows that there does exist new smoothing function which is better than the Fischer-Burmeister smoothing function.Regularization parameter selection for the low rank matrix recoveryhttps://www.zbmath.org/1475.901232022-01-14T13:23:02.489162Z"Shang, Pan"https://www.zbmath.org/authors/?q=ai:shang.pan"Kong, Lingchen"https://www.zbmath.org/authors/?q=ai:kong.lingchenSummary: A popular approach to recover low rank matrices is the nuclear norm regularized minimization (NRM) for which the selection of the regularization parameter is inevitable. In this paper, we build up a novel rule to choose the regularization parameter for NRM, with the help of the duality theory. Our result provides a safe set for the regularization parameter when the rank of the solution has an upper bound. Furthermore, we apply this idea to NRM with quadratic and Huber functions, and establish simple formulae for the regularization parameters. Finally, we report numerical results on some signal shapes by embedding our rule into the cross validation, which state that our rule can reduce the computational time for the selection of the regularization parameter. To the best of our knowledge, this is the first attempt to select the regularization parameter for the low rank matrix recovery.A robust numerical method for pricing American options under Kou's jump-diffusion models based on penalty methodhttps://www.zbmath.org/1475.913992022-01-14T13:23:02.489162Z"Gan, Xiaoting"https://www.zbmath.org/authors/?q=ai:gan.xiaoting"Yang, Ying"https://www.zbmath.org/authors/?q=ai:yang.ying"Zhang, Kun"https://www.zbmath.org/authors/?q=ai:zhang.kunSummary: We develop a novel numerical method for pricing American options under Kou's jump-diffusion model which governed by a partial integro-differential complementarity problem (PIDCP). By using a penalty approach, the PIDCP results in a nonlinear partial integro-differential equation (PIDE). To numerically solve this nonlinear penalized PIDE, a fitted finite volume method is introduced for the spatial discretization and the Backward Euler and Crank-Nicolson schemes for the time discretization. We show that these schemes are consistent, stable and monotone, hence convergence to the solution of continuous problem. Numerical experiments are performed to verify the effectiveness of this new method.Stability of numerical methods under the regime-switching jump-diffusion model with variable coefficientshttps://www.zbmath.org/1475.914002022-01-14T13:23:02.489162Z"Lee, Sunju"https://www.zbmath.org/authors/?q=ai:lee.sunju"Lee, Younhee"https://www.zbmath.org/authors/?q=ai:lee.younheeRegime-switching models incorporate into financial models the possibility to consider different states of the economy [\textit{Y. Lee}, Comput. Math. Appl. 68, No. 3, 392--404 (2014; Zbl 1369.91192)]. In this paper, a regime-switching jump-diffusion price model with variable coefficients is considered. Under a risk-neutral regime-switching model European options can be evaluated by solving a partial integro-differential equation (PIDE), and American options can be evaluated solving a linear complementarity problem. To solve this type of problems, numerical methods are needed.
In this paper three numerical methods are proposed and developed: the implicit method (IM), the Crank-Nicolson method (CN) and the second-step backward differentiation formula method (BDF2). It is proved that the three methods are stable with the second-order convergence rate in the discrete \(\ell_2\)-norm.
The paper is recommendable for researchers and practitioners interested in regime-switching models, and specially, for those who work in pricing financial derivatives.
Reviewer: Josep Vives (Barcelona)CTMC integral equation method for American options under stochastic local volatility modelshttps://www.zbmath.org/1475.914012022-01-14T13:23:02.489162Z"Ma, Jingtang"https://www.zbmath.org/authors/?q=ai:ma.jingtang"Yang, Wensheng"https://www.zbmath.org/authors/?q=ai:yang.wensheng"Cui, Zhenyu"https://www.zbmath.org/authors/?q=ai:cui.zhenyuSummary: In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the \(4/2\) model and the \(\alpha\)-hypergeometric models. They illustrate the high accuracy and efficiency of the method.A second order numerical scheme for fractional option pricing modelshttps://www.zbmath.org/1475.914022022-01-14T13:23:02.489162Z"Zhang, Ling-Xi"https://www.zbmath.org/authors/?q=ai:zhang.lingxi"Peng, Ren-Feng"https://www.zbmath.org/authors/?q=ai:peng.ren-feng"Yin, Jun-Feng"https://www.zbmath.org/authors/?q=ai:yin.junfengSummary: A number of fractional option models (FMLS, CGMY, KoBol) are proposed and studied under assumption that the motion of the underlying assets follows a Lévy process. Numerical methods for these option pricing models are based on solution of fractional partial differential equations. To discretise them, we employ a second order numerical scheme and study its stability and convergence. Numerical experiments show the efficiency of the method and its convergence. Simulations related to practical stock markets, further confirm the robustness of the scheme and show that KoBol model has advantage over the classical Black-Scholes model.Numerical solution of a degenerate, diffusion reaction based biofilm growth model on structured non-orthogonal gridshttps://www.zbmath.org/1475.920072022-01-14T13:23:02.489162Z"Ali, Md. Afsar"https://www.zbmath.org/authors/?q=ai:ali.mohamed-afsar"Eberl, Hermann J."https://www.zbmath.org/authors/?q=ai:eberl.hermann-j"Sudarsan, Rangarajan"https://www.zbmath.org/authors/?q=ai:sudarsan.rangarajanSummary: A previously developed semi-implicit method to solve a density dependent diffusion-reaction biofilm growth model on uniform Cartesian grids is extended to accommodate non-orthogonal grids in order to allow simulation on more complicated domains. The model shows two non-linear diffusion effects: it degenerates where the dependent solution vanishes, and a super-diffusion singularity where it approaches its upper bound. The governing equation is transformed to a general non-orthogonal \(\xi\)-\(\eta\) curvilinear coordinate system and then discretized spatially using a cell centered finite volume method. The nonlinear biomass fluxes at the faces of the control volume cell are split into orthogonal and non-orthogonal components. The orthogonal component is handled in a conventional manner, while the non-orthogonal component is treated as a part of the source term. Extensive tests showed that this treatment of the non-orthogonal flux component on the control volume face works well if the maximum deviation from orthogonality in the region of the grid where the biomass is growing is within 15--20 degrees. This range of validity is smaller than the one obtained with the same method for the simpler porous medium equation which is the standard test problem for degenerate diffusion equation but does not have all of the features of the biofilm model.A novel model for the contamination of a system of three artificial lakeshttps://www.zbmath.org/1475.920082022-01-14T13:23:02.489162Z"Hatipoğlu, Veysel Fuat"https://www.zbmath.org/authors/?q=ai:hatipoglu.veysel-fuatSummary: In this study, a new model has been developed to monitor the contamination in connected three lakes. The model has been motivated by two biological models, i.e. cell compartment model and lake pollution model. Haar wavelet collocation method has been proposed for the numerical solutions of the model containing a system of three linear differential equations. In addition to the solutions of the system, convergence analysis has been briefly given for the proposed method. The contamination in each lake has been investigated by considering three different pollutant input cases, namely impulse imposed pollutant source, exponentially decaying imposed pollutant source, and periodic imposed pollutant source. Each case has been illustrated with a numerical example and results are compared with the exact ones. Regarding the results in each case it has been seen that, Haar wavelet collocation method is an efficient algorithm to monitor the contamination of a system of lakes problem.Neural field models with transmission delays and diffusionhttps://www.zbmath.org/1475.920202022-01-14T13:23:02.489162Z"Spek, Len"https://www.zbmath.org/authors/?q=ai:spek.len"Kuznetsov, Yuri A."https://www.zbmath.org/authors/?q=ai:kuznetsov.yuri-a"van Gils, Stephan A."https://www.zbmath.org/authors/?q=ai:van-gils.stephan-aSummary: A neural field models the large scale behaviour of large groups of neurons. We extend previous results for these models by including a diffusion term into the neural field, which models direct, electrical connections. We extend known and prove new sun-star calculus results for delay equations to be able to include diffusion and explicitly characterise the essential spectrum. For a certain class of connectivity functions in the neural field model, we are able to compute its spectral properties and the first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical example, we find that the addition of diffusion suppresses non-synchronised steady-states while favouring synchronised oscillatory modes.Determining parastichy numbers using discrete Fourier transformshttps://www.zbmath.org/1475.920222022-01-14T13:23:02.489162Z"Negishi, Riichirou"https://www.zbmath.org/authors/?q=ai:negishi.riichirou"Sekiguchi, Kumiko"https://www.zbmath.org/authors/?q=ai:sekiguchi.kumiko"Totsuka, Yuichi"https://www.zbmath.org/authors/?q=ai:totsuka.yuichi"Uchida, Masaya"https://www.zbmath.org/authors/?q=ai:uchida.masayaSummary: We report a practical method to assign parastichy numbers to spiral patterns formed by sunflower seeds and pineapple ramenta using a discrete Fourier transform. We designed various simulation models of sunflower seeds and pineapple ramenta and simulated their point patterns. The parastichy numbers can be directly and accurately assigned using the discrete Fourier transform method to analyze point patterns even when the parastichy numbers contain a divergence angle that results in two or more generalized Fibonacci numbers. The presented method can be applied to extract the structural features of any spiral pattern.Some mathematical properties of morphoelasticityhttps://www.zbmath.org/1475.920232022-01-14T13:23:02.489162Z"Egberts, Ginger"https://www.zbmath.org/authors/?q=ai:egberts.ginger"Smits, Daan"https://www.zbmath.org/authors/?q=ai:smits.daan"Vermolen, Fred"https://www.zbmath.org/authors/?q=ai:vermolen.fred-j"van Zuijlen, Paul"https://www.zbmath.org/authors/?q=ai:van-zuijlen.paul-p-mSummary: We consider a morphoelastic framework that models permanent deformations. The text treats a stability assessment in one dimension and a preservation of symmetry in multiple dimensions. Next, we treat the influence of uncertainty in some of the field variables onto the predicted behaviour of tissue.
For the entire collection see [Zbl 1471.65009].Pattern formation and transition to chaos in a chemotaxis model of acute inflammationhttps://www.zbmath.org/1475.920322022-01-14T13:23:02.489162Z"Giunta, Valeria"https://www.zbmath.org/authors/?q=ai:giunta.valeria"Lombardo, Maria Carmela"https://www.zbmath.org/authors/?q=ai:lombardo.maria-carmela"Sammartino, Marco"https://www.zbmath.org/authors/?q=ai:sammartino.marco-maria-luigiTrial on low-pass filter design for bio-signal based on nonlinear analysishttps://www.zbmath.org/1475.920652022-01-14T13:23:02.489162Z"Tanimura, Toru"https://www.zbmath.org/authors/?q=ai:tanimura.toru"Jono, Yusuke"https://www.zbmath.org/authors/?q=ai:jono.yusuke"Hirata, Takayuki"https://www.zbmath.org/authors/?q=ai:hirata.takayuki"Matsuura, Yasuyuki"https://www.zbmath.org/authors/?q=ai:matsuura.yasuyuki"Takada, Hiroki"https://www.zbmath.org/authors/?q=ai:takada.hirokiSummary: The nonlinearity of the mathematical model describing the cerebral blood flow dynamics and the body sway in the prefrontal cortex was investigated experimentally. The measured bio-signal data were smoothed with each low-pass filter. The signal was set to 0.1--2 Hz for the cerebral blood flow dynamics and to 0.1--20 Hz for the body sway. Nonlinearity was observed in the biological signal when the cut-off frequency of the low-pass filtering was 0.2 Hz or less, and while the body sway was 0.5 Hz or less, and was considered as a stochastic differential equations.A learning-based formulation of parametric curve fitting for bioimage analysishttps://www.zbmath.org/1475.920972022-01-14T13:23:02.489162Z"Mandal, Soham"https://www.zbmath.org/authors/?q=ai:mandal.soham"Uhlmann, Virginie"https://www.zbmath.org/authors/?q=ai:uhlmann.virginieSummary: Parametric curve models are convenient to describe and quantitatively characterize the contour of objects in bioimages. Unfortunately, designing algorithms to fit smoothly such models onto image data classically requires significant domain expertise. Here, we propose a convolutional neural network-based approach to predict a continuous parametric representation of the outline of biological objects. We successfully apply our method on the Kaggle 2018 Data Science Bowl dataset composed of a varied collection of images of cell nuclei. This work is a first step towards user-friendly bioimage analysis tools that extract continuously-defined representations of objects.
For the entire collection see [Zbl 1471.65009].An unconditionally positivity-preserving implicit-explicit scheme for evolutionary stable distribution modelhttps://www.zbmath.org/1475.921192022-01-14T13:23:02.489162Z"Zhang, Chun-Hua"https://www.zbmath.org/authors/?q=ai:zhang.chunhua|zhang.chunhua.1"Chen, Guang-Ze"https://www.zbmath.org/authors/?q=ai:chen.guang-ze"Fang, Zhi-Wei"https://www.zbmath.org/authors/?q=ai:fang.zhiwei"Lin, Xue-lei"https://www.zbmath.org/authors/?q=ai:lin.xuelei"Sun, Hai-Wei"https://www.zbmath.org/authors/?q=ai:sun.haiweiSummary: In this paper, we study the numerical solution for the evolutionary stable distribution (ESD) model. A new implicit-explicit scheme is proposed to discretize the ESD model, which can keep the numerical solution unconditionally positive at each time step. Moreover, it is theoretically proved that the proposed scheme satisfies the property of the entropy dissipation. Numerical experiments are carried out to demonstrate the numerical accuracy and the effectiveness of the proposed scheme.On the dynamics of a hyperbolic-exponential model of growth with density dependencehttps://www.zbmath.org/1475.921222022-01-14T13:23:02.489162Z"Cánovas, Jose S."https://www.zbmath.org/authors/?q=ai:canovas.jose-s"Muñoz-Guillermo, María"https://www.zbmath.org/authors/?q=ai:munoz-guillermo.mariaSummary: In this paper we consider a hyperbolic-exponential model of growth with density regulation and two different stages, following the scheme proposed in \textit{D. J. Rodriguez} [Theor. Popul. Biol. 34, No. 2, 93--117 (1988; Zbl 0648.92015)]. We analyze the dynamics and the complexity of the system, in particular, we study the existence and stability of the fixed points in terms of the W Lambert function and the existence of chaos is proved for a range of parameter values. The model also exhibits dynamic Parrondo's paradox, obtaining complex dynamics when two simple maps are combined.Traveling wave solutions for the dispersive models describing population dynamics of \textit{Aedes aegypti}https://www.zbmath.org/1475.921402022-01-14T13:23:02.489162Z"Yamashita, William M. S."https://www.zbmath.org/authors/?q=ai:yamashita.william-m-s"Takahashi, Lucy T."https://www.zbmath.org/authors/?q=ai:takahashi.lucy-tiemi"Chapiro, Grigori"https://www.zbmath.org/authors/?q=ai:chapiro.grigoriIn this work, the authors deal with the study of mathematical models dealing with the life cycle of the mosquito using partial differential equations. They investigated the existence of traveling wave solutions using semi-analytical method combining dynamical systems techniques and numerical integration. Obtained solutions are validated through direct numerical simulations using finite difference schemes. They also present initial study concerning structural stability of traveling wave solution. The derived results are novel and have great importance for public health in countries where climatic and environmental conditions are favorable for the propagation of this disease. Also, the research idea enriches the dynamical system theory of delayed differential equation to some degree.
Reviewer: Changjin Xu (Guiyang)Solutions of differential equations for prediction of COVID-19 cases by homotopy perturbation methodhttps://www.zbmath.org/1475.921602022-01-14T13:23:02.489162Z"Fatima, Nahid"https://www.zbmath.org/authors/?q=ai:fatima.nahid"Dhariwal, Monika"https://www.zbmath.org/authors/?q=ai:dhariwal.monikaSummary: In this chapter we will have three differential equations for COVID-19 by the introduction of the SIR model. We have taken into account the number of people who became sick from coronavirus in India before September 29, 2020. With the help of this data, we have formulated differential equations for COVID-19 with the help of the SIR model and solved these equations by using the homotopy perturbation method (HPM). HPM solves the most difficult equations with ease and requires minimal calculations. In the current research work we shall discuss the impact of the novel coronavirus on the education sector. In this paper we have taken into account 74,196,729 susceptible persons, and the total confirmed cases of COVID-19 in India on September 29, 2020 was 6,223,519, recovered was 5,184,723, active cases 940,384, and deaths 97,527. We can extend this work by taking data from November 29 and predicted susceptible, infected, and recovered on December 14. Hence, we can give more predictions for COVID-19.
For the entire collection see [Zbl 1472.92005].Adaptive mesh refinement and coarsening for diffusion-reaction epidemiological modelshttps://www.zbmath.org/1475.921672022-01-14T13:23:02.489162Z"Grave, Malú"https://www.zbmath.org/authors/?q=ai:grave.malu"Coutinho, Alvaro L. G. A."https://www.zbmath.org/authors/?q=ai:coutinho.alvaro-l-g-aSummary: The outbreak of COVID-19 in 2020 has led to a surge in the interest in the mathematical modeling of infectious diseases. Disease transmission may be modeled as compartmental models, in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another. Usually, they are composed of a system of ordinary differential equations in time. A class of such models considers the susceptible, exposed, infected, recovered, and deceased populations, the SEIRD model. However, these models do not always account for the movement of individuals from one region to another. In this work, we extend the formulation of SEIRD compartmental models to diffusion-reaction systems of partial differential equations to capture the continuous spatio-temporal dynamics of COVID-19. Since the virus spread is not only through diffusion, we introduce a source term to the equation system, representing exposed people who return from travel. We also add the possibility of anisotropic non-homogeneous diffusion. We implement the whole model in \texttt{libMesh}, an open finite element library that provides a framework for multiphysics, considering adaptive mesh refinement and coarsening. Therefore, the model can represent several spatial scales, adapting the resolution to the disease dynamics. We verify our model with standard SEIRD models and show several examples highlighting the present model's new capabilities.Numerical solutions of the fractional SIS epidemic model via a novel techniquehttps://www.zbmath.org/1475.921732022-01-14T13:23:02.489162Z"Khalouta, Ali"https://www.zbmath.org/authors/?q=ai:khalouta.ali"Kadem, Abdelouahab"https://www.zbmath.org/authors/?q=ai:kadem.abdelouahabSummary: This article introduces a novel technique called modified fractional Taylor series method (MFTSM) to find numerical solutions for the fractional SIS epidemic model. The fractional derivative is considered in the sense of Caputo. The most important feature of the MFTSM is that it is very effective, accurate, simple, and more computational than the methods found in literature. The validity and effectiveness of the proposed technique are investigated and verified through numerical example.A partitioned finite element method for power-preserving discretization of open systems of conservation lawshttps://www.zbmath.org/1475.930512022-01-14T13:23:02.489162Z"Cardoso-Ribeiro, Flávio Luiz"https://www.zbmath.org/authors/?q=ai:cardoso-ribeiro.flavio-luiz"Matignon, Denis"https://www.zbmath.org/authors/?q=ai:matignon.denis"Lefèvre, Laurent"https://www.zbmath.org/authors/?q=ai:lefevre.laurentSummary: This paper presents a structure-preserving spatial discretization method for distributed parameter port-Hamiltonian systems. The class of considered systems are hyperbolic systems of two conservation laws in arbitrary spatial dimension and geometries. For these systems, a partitioned finite element method (PFEM) is derived, based on the integration by parts of one of the two conservation laws written in weak form. The non-linear one-dimensional shallow-water equation (SWE) is first considered as a motivation example. Then, the method is investigated on the example of the non-linear two-dimensional SWE. Complete derivation of the reduced finite-dimensional port-Hamiltonian system (pHs) is provided and numerical experiments are performed. Extensions to curvilinear (polar) coordinate systems, space-varying coefficients and higher-order pHs (Euler-Bernoulli beam equation) are provided.Approximating eigenvectors with fixed-point arithmetic: a step towards secure spectral clusteringhttps://www.zbmath.org/1475.940152022-01-14T13:23:02.489162Z"Steverink, Lisa"https://www.zbmath.org/authors/?q=ai:steverink.lisa"Veugen, Thijs"https://www.zbmath.org/authors/?q=ai:veugen.thijs"van Gijzen, Martin B."https://www.zbmath.org/authors/?q=ai:van-gijzen.martin-bSummary: We investigate the adaptation of the spectral clustering algorithm to the privacy preserving domain. Spectral clustering is a data mining technique that divides points according to a measure of connectivity in a data graph. When the matrix data are privacy sensitive, cryptographic techniques can be applied to protect the data. A pivotal part of spectral clustering is the partial eigendecomposition of the graph Laplacian. The Lanczos algorithm is used to approximate the eigenvectors of the Laplacian. Many cryptographic techniques are designed to work with positive integers, whereas the numerical algorithms are generally applied in the real domain. To overcome this problem, the Lanczos algorithm is adapted to be performed with fixed-point arithmetic. Square roots are eliminated and floating-point computations are transformed to fixed-point computations. The effects of these adaptations on the accuracy and stability of the algorithm are investigated using standard datasets. The performance of the original and the adapted algorithm is similar when few eigenvectors are needed. For a large number of eigenvectors loss of orthogonality affects the results.
For the entire collection see [Zbl 1471.65009].Exact camera location recovery by least unsquared deviationshttps://www.zbmath.org/1475.940192022-01-14T13:23:02.489162Z"Lerman, Gilad"https://www.zbmath.org/authors/?q=ai:lerman.gilad"Shi, Yunpeng"https://www.zbmath.org/authors/?q=ai:shi.yunpeng"Zhang, Teng"https://www.zbmath.org/authors/?q=ai:zhang.tengGalaxy image restoration with shape constrainthttps://www.zbmath.org/1475.940232022-01-14T13:23:02.489162Z"Nammour, Fadi"https://www.zbmath.org/authors/?q=ai:nammour.fadi"Schmitz, Morgan A."https://www.zbmath.org/authors/?q=ai:schmitz.morgan-a"Mboula, Fred Maurice Ngolè"https://www.zbmath.org/authors/?q=ai:mboula.fred-maurice-ngole"Starck, Jean-Luc"https://www.zbmath.org/authors/?q=ai:starck.jean-luc"Girard, Julien N."https://www.zbmath.org/authors/?q=ai:girard.julien-nSummary: Images acquired with a telescope are blurred and corrupted by noise. The blurring is usually modelled by a convolution with the Point Spread Function and the noise by Additive Gaussian Noise. Recovering the observed image is an ill-posed inverse problem. Sparse deconvolution is well known to be an efficient deconvolution technique, leading to optimized pixel Mean Square Errors, but without any guarantee that the shapes of objects (e.g. galaxy images) contained in the data will be preserved. In this paper, we introduce a new shape constraint and exhibit its properties. By combining it with a standard sparse regularization in the wavelet domain, we introduce the Shape COnstraint REstoration algorithm (SCORE), which performs a standard sparse deconvolution, while preserving galaxy shapes. We show through numerical experiments that this new approach leads to a reduction of galaxy ellipticitiy measurement errors by at least 44\%.Kalman-wavelet combined filteringhttps://www.zbmath.org/1475.940362022-01-14T13:23:02.489162Z"La Mura, Guillermo"https://www.zbmath.org/authors/?q=ai:la-mura.guillermo"Sirne, Ricardo"https://www.zbmath.org/authors/?q=ai:sirne.ricardo-o"Fabio, Marcela A."https://www.zbmath.org/authors/?q=ai:fabio.marcela-aSummary: In this chapter we propose to combine two well-known techniques suitable for the analysis of nonstationary process: Kalman filtering and discrete wavelet transform. Signal filtering is an inverse problem in the sense that, based on the noisy observations obtained from measurements, it intends to estimate the state variables knowing the model of the system and the statistical behavior of the intervening noises. This technique performs simultaneously estimation and decomposition of random signals through a filter bank based on the Kalman filtering approach using wavelets. The algorithm introduced in the following pages takes advantage of the relative benefits of both methods. We present some numerical results considering two special cases of wavelets: Haar and Daubechies of four coefficients.
For the entire collection see [Zbl 1464.65005].Constructing pseudorandom sequences by means of 2-linear shift registerhttps://www.zbmath.org/1475.940812022-01-14T13:23:02.489162Z"Kozlitin, O. A."https://www.zbmath.org/authors/?q=ai:kozlitin.oleg-aSummary: We describe the periodicity properties for almost all 2-linear recurrent sequences generated by 2-linear shift register with identical connection polynomials of maximal period. A class of self-control nonlinear functions are suggested such that the existence of maximally possible cycles in a transition graph of states is guaranteed. Linear output functions preserving the period of sequence are described.