Recent zbMATH articles in MSC 65Dhttps://www.zbmath.org/atom/cc/65D2021-02-27T13:50:00+00:00WerkzeugAutomatic differentiation using operator overloading (ADOO) for implicit resolution of hyperbolic single phase and two-phase flow models.https://www.zbmath.org/1453.650492021-02-27T13:50:00+00:00"Fraysse, François"https://www.zbmath.org/authors/?q=ai:fraysse.francois"Saurel, Richard"https://www.zbmath.org/authors/?q=ai:saurel.richardSummary: Implicit time integration schemes are widely used in computational fluid dynamics to speed-up computations. Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit counterpart. The derivation of an implicit scheme is however a challenging and time-consuming task, increasing substantially with the model equations complexity since this method usually requires fairly accurate evaluation of the spatial scheme's matrix Jacobian. This article presents a flexible method to overcome the difficulties associated to the computation of the derivatives, based on the forward mode of automatic differentiation using operator overloading (ADOO). Flexibility and simplicity of the method are illustrated through implicit resolution of various flow models of increasing complexity such as the compressible Euler equations, a two-phase flow model in full equilibrium [\textit{S. Le Martelot} et al., ``Towards the direct numerical simulation of nucleate boiling flows'', Int. J. Multiphase Flow 66, 62--78 (2014; \url{doi:10.1016/j.ijmultiphaseflow.2014.06.010})] and a symmetric variant [the second author et al., J. Fluid Mech. 495, 283--321 (2003; Zbl 1080.76062)] of the two-phase flow model of \textit{M. R. Baer} and \textit{J. W. Nunziato} [Int. J. Multiphase Flow 12, 861--889 (1986; Zbl 0609.76114)] dealing with mixtures in total disequilibrium.The distance function from a real algebraic variety.https://www.zbmath.org/1453.650462021-02-27T13:50:00+00:00"Ottaviani, Giorgio"https://www.zbmath.org/authors/?q=ai:ottaviani.giorgio-maria"Sodomaco, Luca"https://www.zbmath.org/authors/?q=ai:sodomaco.lucaSummary: For any (real) algebraic variety \(X\) in a Euclidean space \(V\) endowed with a nondegenerate quadratic form \(q\), we introduce a polynomial \(\operatorname{EDpoly}_{X , u}( t^2)\) which, for any \(u \in V\), has among its roots the distance from \(u\) to \(X\). The degree of \(\operatorname{EDpoly}_{X , u}\) is the \textit{Euclidean Distance degree} of \(X\). We prove a duality property when \(X\) is a projective variety, namely \(\operatorname{EDpoly}_{X , u}( t^2) = \operatorname{EDpoly}_{X^\vee , u}(q(u) - t^2)\) where \(X^\vee\) is the dual variety of \(X\). When \(X\) is transversal to the isotropic quadric \(Q\), we prove that the ED polynomial of \(X\) is monic and the zero locus of its lower term is \(X \cup ( X^\vee \cap Q )^\vee \).A modified fifth order finite difference Hermite WENO scheme for hyperbolic conservation laws.https://www.zbmath.org/1453.652412021-02-27T13:50:00+00:00"Zhao, Zhuang"https://www.zbmath.org/authors/?q=ai:zhao.zhuang"Zhang, Yong-Tao"https://www.zbmath.org/authors/?q=ai:zhang.yongtao"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianThe authors present a new modified fifth-order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one-dimensional and two-dimensional hyperbolic conservation laws. One of the main features of this scheme is that the HWENO scheme is simpler and more robust. In particular, it is not required to add a positivity-preserving flux limiter, and larger CFL number can be applied than the usual to ensure the stability of schemes. The new HWENO scheme has fifth-order accuracy while based on the same reconstruction stencil the original finite difference HWENO scheme only has fourth-order accuracy when solving two-dimensional problems. Furthermore, the new scheme preserves the nice property of compactness, i.e., only immediate neighbor information is needed in the reconstruction. Several numerical tests for both one-dimensional and two-dimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed modified HWENO scheme.
Reviewer: Abdallah Bradji (Annaba)Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse.https://www.zbmath.org/1453.650352021-02-27T13:50:00+00:00"Khoromskij, Boris N."https://www.zbmath.org/authors/?q=ai:khoromskij.boris-nSummary: In this paper, we introduce the operator dependent range-separated (RS) tensor approximation of the discretized Dirac delta function (distribution) in \(\mathbb{R}^d\). It is constructed by application of the elliptic operator to the RS tensor representation of the associated Green kernel discretized on the \(d\)-dimensional Cartesian grid. The proposed local-global decomposition of the Dirac delta can be applied for solving the potential equations in a non-homogeneous medium when the density in the right-hand side is given by a large sum of pointwise singular charges. As an example of applications, we describe the regularization scheme for solving the Poisson-Boltzmann equation that models the electrostatics in bio-molecules. We show how the idea of the operator dependent RS tensor decomposition of the Dirac delta can be generalized to the closely related problem on range-separated tensor representation of the elliptic resolvent. This approach paves the way for application of tensor numerical methods to elliptic problems with non-regular data. Numerical tests confirm the expected localization properties of the RS tensor approximation of the Dirac delta represented on a tensor grid in 3D.A fast solution method for time dependent multidimensional Schrödinger equations.https://www.zbmath.org/1453.650542021-02-27T13:50:00+00:00"Lanzara, F."https://www.zbmath.org/authors/?q=ai:lanzara.flavia"Maz'ya, V."https://www.zbmath.org/authors/?q=ai:mazya.vladimir-gilelevich"Schmidt, G."https://www.zbmath.org/authors/?q=ai:schmidt.gunther.2Summary: In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.New approach to certain real hyper-elliptic integrals.https://www.zbmath.org/1453.330152021-02-27T13:50:00+00:00"Krasoń, Piotr"https://www.zbmath.org/authors/?q=ai:krason.piotr"Milewski, Jan"https://www.zbmath.org/authors/?q=ai:milewski.janThe purpose of this article is to present a new method for computation of certain real hyper-elliptic integrals of the form
\[
I_{ n,p} =\int\frac{(x-p)^ n dx}{\sqrt{Q(x)}},\ n\in\mathbb{Z},\tag{hyper}
\]
where \(Q(x)\) is a polynomial of degree \(M\) with \(M\) real zeroes of multiplicity 1. We find that
\[
I_{ n,p} =\sum_k(-1)^k\binom nkp^k I_{n-k},
\]
where \(I_k\) has a particularly simple form. In a similar way, the case \(n<-1\) in (hyper) can be treated. We assume no zeros in the polynomials in squareroots. The proofs use an invertible, upper triangular, infinite, transition matrix \(T_{l,n}\) and the Lauricella function. Finally, these two integrals can be transformed into the Riemann canonical form by using the projective line.
Reviewer: Thomas Ernst (Uppsala)The nodal \(LTS_N\) solution in a rectangular domain: a new method to determine the outgoing angular flux at the boundary.https://www.zbmath.org/1453.821002021-02-27T13:50:00+00:00"Parigi, Aline R."https://www.zbmath.org/authors/?q=ai:parigi.aline-r"Segatto, Cynthia F."https://www.zbmath.org/authors/?q=ai:segatto.cynthia-f"Bodmann, Bardo E. J."https://www.zbmath.org/authors/?q=ai:bodmann.bardo-ernst-josefSummary: In the present contribution we discuss the neutron nodal \(S_N\) equation in a rectangular domain. The nodal method consists in transverse integration of the \(S_N\) equation and results in coupled one-dimensional \(S_N\) equations with unknown angular flux at the border. In the literature, the outgoing angular flux is considered a constant or exponential decreasing function, where the latter is used in this work. It is noteworthy that solutions found with these boundary conditions present unphysical results, i.e. negative angular fluxes in the border region, whereas the scalar flux is semi-positive definite. To overcome these shortcomings a new approach is proposed. The rectangular domain is covered by a finite discrete set of narrow rectangular sub-domains, so that in each rectangle the solution may be approximated by the one from a one-dimensional problem. Upon applying the \(LTS_N\) method combined with the DNI technique, i.e. interpolating the directions of the two-dimensional problem by means of one-dimensional directions, one obtains the angular flux at the border from the known one-dimensional \(LTS_N\) solution for any desired point. Numerical simulations and comparisons with results found in the literature are presented.
For the entire collection see [Zbl 1417.65006].A high resolution PDE approach to quadrilateral mesh generation.https://www.zbmath.org/1453.650452021-02-27T13:50:00+00:00"Marcon, Julian"https://www.zbmath.org/authors/?q=ai:marcon.julian"Kopriva, David A."https://www.zbmath.org/authors/?q=ai:kopriva.david-a"Sherwin, Spencer J."https://www.zbmath.org/authors/?q=ai:sherwin.spencer-j"Peiró, Joaquim"https://www.zbmath.org/authors/?q=ai:peiro.joaquimSummary: We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code \textit{Nektar++}. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation allows meshing of corners that are not multiples of \(\pi / 2\) in a discretization consistent manner, when needed. The high order field can then be exploited to accurately find irregular nodes, and can be accurately integrated using a high order separatrix integration method to avoid features like limit cycles. The result is a mesh with naturally curved quadrilateral elements that do not need to be curved \textit{a posteriori} to eliminate invalid elements. The mesh generation procedure is implemented in the open source mesh generation program \textit{NekMesh}.Boundary observability of semi-discrete second-order integro-differential equations derived from piecewise Hermite cubic orthogonal spline collocation method.https://www.zbmath.org/1453.654652021-02-27T13:50:00+00:00"Xu, Da"https://www.zbmath.org/authors/?q=ai:xu.daThe author studies the boundary observability of semi-discrete second-order integro-differential equations derived from a piecewise Hermite cubic orthogonal spline collocation method. First, the author analyzes the spectrum of the equation considered and gives a representation of the solution with a new finite sequence to the associated system. Then, he proves that the new sequence in the class \(C_{h}\) satisfies the theorems stated. Piecewise Hermite cubic orthogonal spline collocation semi-discretization is considered.
Reviewer: Seenith Sivasundaram (Daytona Beach)A method for dimensionally adaptive sparse trigonometric interpolation of periodic functions.https://www.zbmath.org/1453.650322021-02-27T13:50:00+00:00"Morrow, Zachary"https://www.zbmath.org/authors/?q=ai:morrow.zachary"Stoyanov, Miroslav"https://www.zbmath.org/authors/?q=ai:stoyanov.miroslav-kThe performance of higher moments estimators: an empirical study.https://www.zbmath.org/1453.625112021-02-27T13:50:00+00:00"Harun, H. F."https://www.zbmath.org/authors/?q=ai:harun.h-f"Abdullah, M. H."https://www.zbmath.org/authors/?q=ai:abdullah.m-kh|abdullah.mimi-hafizahSummary: This study investigates the performance of higher order moments, realised from the model-free Bakshi-Kapadia-Madan (MFBKM). We concentrate on investigating higher order option-implied moments â variance, skewness and kurtosis, chosen in relation to contracts dened in MFBKM, i.e. volatility, cubic, and quartic contract. The three approaches adopted in order to estimate the integrals of the dened MFBKM contracts are the basic (trapezoidal-rule), adapted (single-combined) and advanced method (cubic-spline). The sample data is extracted from DJIA index options data, which covers the period from January 2009 until December 2015. The results show that the advanced method performs poorly in estimating the MFBKM, especially in the case of skewness and kurtosis integrals estimation. The advanced method outperforms the other approaches in the case of the variance estimation. In estimating both model-free skewness and kurtosis, the adapted method is found to perform the best, instead.A nonintrusive reduced order modelling approach using proper orthogonal decomposition and locally adaptive sparse grids.https://www.zbmath.org/1453.650292021-02-27T13:50:00+00:00"Alsayyari, Fahad"https://www.zbmath.org/authors/?q=ai:alsayyari.fahad"Perkó, Zoltán"https://www.zbmath.org/authors/?q=ai:perko.zoltan"Lathouwers, Danny"https://www.zbmath.org/authors/?q=ai:lathouwers.danny"Kloosterman, Jan Leen"https://www.zbmath.org/authors/?q=ai:kloosterman.jan-leenSummary: Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems. To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.Generation of nested quadrature rules for generic weight functions via numerical optimization: application to sparse grids.https://www.zbmath.org/1453.650532021-02-27T13:50:00+00:00"Keshavarzzadeh, Vahid"https://www.zbmath.org/authors/?q=ai:keshavarzzadeh.vahid"Kirby, Robert M."https://www.zbmath.org/authors/?q=ai:kirby.robert-mike|kirby.robert-m-ii"Narayan, Akil"https://www.zbmath.org/authors/?q=ai:narayan.akil-cSummary: We present a numerical framework for computing nested quadrature rules for various weight functions. The well-known Kronrod method extends the Gauss-Legendre quadrature by adding new optimal nodes to the existing Gauss nodes for integration of higher order polynomials. Our numerical method generalizes the Kronrod rule for any continuous probability density function on real line with finite moments. We develop a bi-level optimization scheme to solve moment-matching conditions for two levels of main and nested rule and use a penalty method to enforce the constraints on the limits of the nodes and weights. We demonstrate our nested quadrature rule for probability measures on finite/infinite and symmetric/asymmetric supports. We generate Gauss-Kronrod-Patterson rules by slightly modifying our algorithm and present results associated with Chebyshev polynomials which are not reported elsewhere. We finally show the application of our nested rules in construction of sparse grids where we validate the accuracy and efficiency of such nested quadrature-based sparse grids on parameterized boundary and initial value problems in multiple dimensions.A continuation method for visualizing planar real algebraic curves with singularities.https://www.zbmath.org/1453.650402021-02-27T13:50:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuanSummary: We present a new method for visualizing planar real algebraic curves inside a bounding box based on numerical continuation and critical point methods. Since the topology of the curve near a singular point is not numerically stable, we trace the curve only outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is \(\epsilon\)-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the curve, which is important for applications such as solving bi-parametric polynomial systems.{
}The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small circles centered at singular points, regular critical points of every connected component of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters and tracing the curve by a try-and-resume strategy. The effectiveness of the algorithm is illustrated by numerous examples.
For the entire collection see [Zbl 1396.68014].Computation of an infinite integral using integration by parts.https://www.zbmath.org/1453.260042021-02-27T13:50:00+00:00"Tang, Jian-guo"https://www.zbmath.org/authors/?q=ai:tang.jianguoSummary: In this paper, an infinite integral concerning numerical computation in crystallography is investigated, which was studied in two recent articles, and integration by parts is employed for calculating this typical integral. A variable transformation and a single integration by parts lead to a new formula for this integral, and at this time, it becomes a completely definite integral. Using integration by parts iteratively, the singularity at the points near three points \(a = 0,1,2\) can be eliminated in terms containing obtained integrals, and the factors of amplifying round-off error are released into two simple fractions independent of the integral. Series expansions for this integral are obtained, and estimations of its remainders are given, which show that accuracy \(2^{-n}\) is achieved in about \(2n\) operations for every value in a given domain. Finally, numerical results are given to verify error analysis, which coincide well with the theoretical results.Construction of periodic adapted orthonormal frames on closed space curves.https://www.zbmath.org/1453.650362021-02-27T13:50:00+00:00"Farouki, Rida T."https://www.zbmath.org/authors/?q=ai:farouki.rida-t"Kim, Soo Hyun"https://www.zbmath.org/authors/?q=ai:kim.soohyun"Moon, Hwan Pyo"https://www.zbmath.org/authors/?q=ai:pyo-moon.hwanSummary: The construction of continuous adapted orthonormal frames along \(C^1\) closed-loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid-body motions along smooth closed paths. The construction is illustrated through the simplest non-trivial context -- namely, \(C^1\) closed loops defined by a single Pythagorean-hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two-parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of \(\pi\). The desired frame is constructed through a rotation applied to the normal-plane vectors of the \textit{Euler-Rodrigues frame}, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on \(C^1\) closed-loop PH curves is possible, although this incurs transcendental terms. However, the \(C^1\) closed-loop PH quintics admit particularly simple rational periodic adapted frames.Multiscale texture orientation analysis using spectral total-variation decomposition.https://www.zbmath.org/1453.940092021-02-27T13:50:00+00:00"Horesh, Dikla"https://www.zbmath.org/authors/?q=ai:horesh.dikla"Gilboa, Guy"https://www.zbmath.org/authors/?q=ai:gilboa.guySummary: Multi-level texture separation can considerably improve texture analysis, a significant component in many computer vision tasks. This paper aims at obtaining precise local texture orientations of images in a multiscale manner, characterizing the main obvious ones as well as the very subtle ones. We use the total variation spectral framework to decompose the image into its different textural scales. Gabor filter banks are then employed to detect prominent orientations within the multiscale representation. A necessary condition for perfect texture separation is given, based on the spectral total-variation theory. We show that using this method we can detect and differentiate a mixture of overlapping textures and obtain with high fidelity a multi-valued orientation representation of the image.
For the entire collection see [Zbl 1362.68008].Multi-degree B-splines: algorithmic computation and properties.https://www.zbmath.org/1453.650342021-02-27T13:50:00+00:00"Toshniwal, Deepesh"https://www.zbmath.org/authors/?q=ai:toshniwal.deepesh"Speleers, Hendrik"https://www.zbmath.org/authors/?q=ai:speleers.hendrik"Hiemstra, René R."https://www.zbmath.org/authors/?q=ai:hiemstra.rene-r"Manni, Carla"https://www.zbmath.org/authors/?q=ai:manni.carla"Hughes, Thomas J. R."https://www.zbmath.org/authors/?q=ai:hughes.thomas-j-rSummary: This paper addresses theoretical considerations behind the algorithmic computation of polynomial multi-degree spline basis functions as presented in [\textit{D. Toshniwal} et al., Comput. Methods Appl. Mech. Eng. 316, 1005--1061 (2017; Zbl 1439.65016)]. The approach in [Toshniwal et al., loc. cit.] breaks from the reliance on computation of integrals recursively for building B-spline-like basis functions that span a given multi-degree spline space. The gains in efficiency are indisputable; however, the theoretical robustness needs to be examined. In this paper, we show that the construction of Toshniwal et al. [loc. cit.] yields linearly independent functions with the minimal support property that span the entire multi-degree spline space and form a convex partition of unity.A RBFWENO finite difference scheme for Hamilton-Jacobi equations.https://www.zbmath.org/1453.652072021-02-27T13:50:00+00:00"Abedian, Rooholah"https://www.zbmath.org/authors/?q=ai:abedian.rooholah"Salehi, Rezvan"https://www.zbmath.org/authors/?q=ai:salehi.rezvanSummary: The aim of this paper is to study the numerical application of radial basis functions (RBFs) approximation in the reconstruction process of well known ENO/WENO schemes. The resulted schemes are employed for approximating the viscosity solution of Hamilton-Jacobi (H-J) equations. The accuracy in the smooth area is enhanced by locally optimizing the shape parameter according to the results. It is revealed that the proposed schemes in this research prepare more accurate reconstructions and sharper solution near singularities by comparing the RBFENO/RBFWENO schemes and the classical ENO/WENO schemes for some benchmark examples. Looking at the several numerical examples in 1D, 2D and 3D illustrate that the proposed schemes in this paper perform better than the traditional ENO/WENO schemes for solving H-J equations.Using Oshima splines to produce accurate numerical results and high quality graphical output.https://www.zbmath.org/1453.682252021-02-27T13:50:00+00:00"Takato, Setsuo"https://www.zbmath.org/authors/?q=ai:takato.setsuo"Vallejo, José A."https://www.zbmath.org/authors/?q=ai:vallejo.jose-antonioSummary: We illustrate the use of Oshima splines in producing high-quality \LaTeX output in two cases: first, the numerical computation of derivatives and integrals, and second, the display of silhouettes and wireframe surfaces, using the macros package KeTCindy. Both cases are of particular interest for college and university teachers wanting to create handouts to be used by students, or drawing figures for a research paper. When dealing with numerical computations, KeTCindy can make a call to the CAS Maxima to check for accuracy; in the case of surface graphics, it is particularly important to be able to detect intersections of projected curves, and we show how to do it in a seamlessly manner using Oshima splines in KeTCindy. A C compiler can be called in this case to speed up computations.A high order method for pricing of financial derivatives using radial basis function generated finite differences.https://www.zbmath.org/1453.911082021-02-27T13:50:00+00:00"Milovanović, Slobodan"https://www.zbmath.org/authors/?q=ai:milovanovic.slobodan"von Sydow, Lina"https://www.zbmath.org/authors/?q=ai:von-sydow.linaSummary: In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher accuracy. Still, the discretization matrix is fairly sparse. As a model problem, we consider the pricing of European options in 2D. Since such options have a discontinuity in the first derivative of the payoff function which prohibits high order convergence, we smooth this function using an established technique for Cartesian grids. Numerical experiments show that we acquire a fourth order scheme in space, both for the uniform and the nonuniform node layouts that we use. The high order method with the nonuniform node layout achieves very high accuracy with relatively few nodes. This renders the potential for solving pricing problems in higher spatial dimensions since the computational memory and time demand become much smaller with this method compared to standard techniques.An improved lower bound for general position subset selection.https://www.zbmath.org/1453.650472021-02-27T13:50:00+00:00"Rudi, Ali Gholami"https://www.zbmath.org/authors/?q=ai:rudi.ali-gholamiSummary: In the general position subset selection (GPSS) problem, the goal is to find the largest possible subset of a set of points, such that no three of its members are collinear. If \(s\) is the size the optimal solution, the square root of \(s\) is the current best guarantee for the size of the solution obtained using a polynomial time algorithm. In this paper, we present an algorithm for GPSS to improve this bound based on the number of collinear pairs of points. We experimentally evaluate this and few other GPSS algorithms; the result of these experiments suggests further opportunities for obtaining tighter lower bounds for GPSS.Solving electrical impedance tomography with deep learning.https://www.zbmath.org/1453.650412021-02-27T13:50:00+00:00"Fan, Yuwei"https://www.zbmath.org/authors/?q=ai:fan.yuwei"Ying, Lexing"https://www.zbmath.org/authors/?q=ai:ying.lexingSummary: This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map. Both the forward map from the electrical conductivity to the DtN map and the inverse map are high-dimensional and nonlinear. Motivated by the linear perturbative analysis of the forward map and based on a numerically low-rank property, we propose compact neural network architectures for the forward and inverse maps for both 2D and 3D problems. Numerical results demonstrate the efficiency of the proposed neural networks.Singularly perturbed convection-diffusion boundary value problems with two small parameters using nonpolynomial spline technique.https://www.zbmath.org/1453.651722021-02-27T13:50:00+00:00"Khandelwal, Pooja"https://www.zbmath.org/authors/?q=ai:khandelwal.pooja"Khan, Arshad"https://www.zbmath.org/authors/?q=ai:khan.arshad-ali|khan.arshad-alam|khan.arshad-m|khan.arshad-ahmadSummary: In this paper, a new nonpolynomial cubic spline method is developed for solving two-parameter singularly perturbed boundary value problems. Convergence analysis is briefly discussed. Numerical examples and computational results illustrate and guarantee a higher accuracy by this technique. Comparisons are made to confirm the reliability and accuracy of the proposed technique.Adaptive activation functions accelerate convergence in deep and physics-informed neural networks.https://www.zbmath.org/1453.681652021-02-27T13:50:00+00:00"Jagtap, Ameya D."https://www.zbmath.org/authors/?q=ai:jagtap.ameya-d"Kawaguchi, Kenji"https://www.zbmath.org/authors/?q=ai:kawaguchi.kenji"Karniadakis, George Em"https://www.zbmath.org/authors/?q=ai:karniadakis.george-emSummary: We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima. Furthermore, the proposed adaptive activation functions are shown to accelerate the minimization process of the loss values in standard deep learning benchmarks using CIFAR-10, CIFAR-100, SVHN, MNIST, KMNIST, Fashion-MNIST, and Semeion datasets with and without data augmentation.Structure-preserving function approximation via convex optimization.https://www.zbmath.org/1453.901242021-02-27T13:50:00+00:00"Zala, Vidhi"https://www.zbmath.org/authors/?q=ai:zala.vidhi"Kirby, Mike"https://www.zbmath.org/authors/?q=ai:kirby.mike"Narayan, Akil"https://www.zbmath.org/authors/?q=ai:narayan.akil-cSparse polynomial arithmetic with the BPAS library.https://www.zbmath.org/1453.654702021-02-27T13:50:00+00:00"Asadi, Mohammadali"https://www.zbmath.org/authors/?q=ai:asadi.mohammadali"Brandt, Alexander"https://www.zbmath.org/authors/?q=ai:brandt.alexander"Moir, Robert H. C."https://www.zbmath.org/authors/?q=ai:moir.robert-h-c"Moreno Maza, Marc"https://www.zbmath.org/authors/?q=ai:moreno-maza.marcSummary: We discuss algorithms for pseudo-division and division with remainder of multivariate polynomials with sparse representation. This work is motivated by the computations of normal forms and pseudo-remainders with respect to regular chains. We report on the implementation of those algorithms with the BPAS library.
For the entire collection see [Zbl 1396.68014].A mixed quadrature rule blending Lobatto and Gauss-Legendre three-point rule for approximate evaluation of real definite integrals.https://www.zbmath.org/1453.650572021-02-27T13:50:00+00:00"Tripathy, Arun Kumar"https://www.zbmath.org/authors/?q=ai:tripathy.arun-kumar"Dash, Rajani Ballav"https://www.zbmath.org/authors/?q=ai:dash.rajani-ballav"Baral, Amarendra"https://www.zbmath.org/authors/?q=ai:baral.amarendraSummary: In this paper, a mixed quadrature rule of precision seven has been designed by the linear combination of two rules of precision five. The error analysis of these two formulas has been incorporated. Through some numerical examples the effectiveness of the mixed quadrature rule over its constituent ones has been shown.The structural topology optimisation based on parameterised level-set method in isogeometric analysis.https://www.zbmath.org/1453.653472021-02-27T13:50:00+00:00"Wu, Zijun"https://www.zbmath.org/authors/?q=ai:wu.zijun"Wang, Shuting"https://www.zbmath.org/authors/?q=ai:wang.shutingSummary: The isogeometric analysis (IGA), which establishes a bridge between CAD and CAE, offers a new convenient framework of optimisation. This article develops an approach to apply the non-uniform rational basis splines (NURBS)-based IGA to the topology optimisation using parameterised level set method. Here, the objective function is evaluated according to the basis function of NURBS and the level set function is constructed through collocation points using the compactly supported radial basis function. To evaluate the equivalent strain energy for each element, the level-set function value of every node is calculated from the corresponding value of collocation points. We have compared the numerical result accuracy as well as the time cost of the proposed method, and it turns out to be very promising.Sparse approximation of fitting surface by elastic net.https://www.zbmath.org/1453.650312021-02-27T13:50:00+00:00"Hao, Yong-Xia"https://www.zbmath.org/authors/?q=ai:hao.yongxia"Lu, Dianchen"https://www.zbmath.org/authors/?q=ai:lu.dianchenSummary: The goal of this paper is to develop a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant space and the balanced \(l_{1}, l_{2}\) norm minimization (named elastic net). The elastic net can be solved efficiently by an adapted split Bregman iteration algorithm. Numerical experiments indicate that by choosing appropriate regularization parameters, the model can efficiently provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution.On variational and PDE-based methods for accurate distance function estimation.https://www.zbmath.org/1453.650422021-02-27T13:50:00+00:00"Fayolle, P.-A."https://www.zbmath.org/authors/?q=ai:fayolle.pierre-alain"Belyaev, A. G."https://www.zbmath.org/authors/?q=ai:belyaev.alexander-gSummary: A new variational problem for accurate approximation of the distance from the boundary of a domain is proposed and studied. It is shown that the problem can be efficiently solved by the alternating direction method of multipliers. Links between this problem and \(p\)-Laplacian diffusion are established and studied. Advantages of the proposed distance function estimation method are demonstrated by numerical experiments.Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points.https://www.zbmath.org/1453.652872021-02-27T13:50:00+00:00"Peherstorfer, Benjamin"https://www.zbmath.org/authors/?q=ai:peherstorfer.benjamin"Drmač, Zlatko"https://www.zbmath.org/authors/?q=ai:drmac.zlatko"Gugercin, Serkan"https://www.zbmath.org/authors/?q=ai:gugercin.serkanThis work is motivated by the fact that the empirical interpolation is widely used for approximating nonlinear terms in reduced models and for recovering state fields from few spatial measurements but stability issues have been remarked in the presence of noise and other perturbations. The authors present a probabilistic analysis which shows that the particular instability that arises because of perturbations such as noise can be avoided by employing GappyPOD (gappy proper orthogonal decomposition) and taking more sampling points than dimensions of the reduced space. Numerical results of reconstructing velocity fields from noisy measurements of combustion processes and model reduction in the presence of noise demonstrate the instability of empirical interpolation and the stability of GappyPOD with oversampling.
Reviewer: Abdallah Bradji (Annaba)Coercing machine learning to output physically accurate results.https://www.zbmath.org/1453.681642021-02-27T13:50:00+00:00"Geng, Zhenglin"https://www.zbmath.org/authors/?q=ai:geng.zhenglin"Johnson, Daniel"https://www.zbmath.org/authors/?q=ai:johnson.daniel-cowan|johnson.daniel-d|johnson.daniel-h|johnson.daniel-p"Fedkiw, Ronald"https://www.zbmath.org/authors/?q=ai:fedkiw.ronald-pSummary: Many machine/deep learning artificial neural networks are trained to simply be interpolation functions that map input variables to output values interpolated from the training data in a linear/nonlinear fashion. Even when the input/output pairs of the training data are physically accurate (e.g. the results of an experiment or numerical simulation), interpolated quantities can deviate quite far from being physically accurate. Although one could project the output of a network into a physically feasible region, such a postprocess is not captured by the energy function minimized when training the network; thus, the final projected result could incorrectly deviate quite far from the training data. We propose folding any such projection or postprocess directly into the network so that the final result is correctly compared to the training data by the energy function. Although we propose a general approach, we illustrate its efficacy on a specific convolutional neural network that takes in human pose parameters (joint rotations) and outputs a prediction of vertex positions representing a triangulated cloth mesh. While the original network outputs vertex positions with erroneously high stretching and compression energies, the new network trained with our physics ``prior'' remedies these issues producing highly improved results.Imposing boundary conditions to match a CAD virtual geometry for the mesh curving problem.https://www.zbmath.org/1453.653022021-02-27T13:50:00+00:00"Ruiz-Gironés, Eloi"https://www.zbmath.org/authors/?q=ai:ruiz-girones.eloi"Roca, Xevi"https://www.zbmath.org/authors/?q=ai:roca.xeviSummary: We present a high-order mesh curving method where the mesh boundary is enforced to match a target virtual geometry. Our method has the unique capability to allow curved elements to span and slide on top of several CAD entities during the mesh curving process. The main advantage is that small angles or small patches of the CAD model do not compromise the topology, quality and size of the boundary elements. We associate each high-order boundary node to a unique group of either curves (virtual wires) or surfaces (virtual shell). Then, we deform the volume elements to accommodate the boundary curvature, while the boundary condition is enforced with a penalty method. At each iteration of the penalty method, the boundary condition is updated by projecting the boundary interpolative nodes of the previous iteration on top of the corresponding virtual entities. The method is suitable to curve meshes featuring non-uniform isotropic and highly stretched elements while matching a given virtual geometry.
For the entire collection see [Zbl 1417.65007].Quadrature formulas of Gauss type for a sphere with nodes characterized by regular prism symmetry.https://www.zbmath.org/1453.650522021-02-27T13:50:00+00:00"Voloshchenko, A. M."https://www.zbmath.org/authors/?q=ai:voloshchenko.a-m"Russkov, A. A."https://www.zbmath.org/authors/?q=ai:russkov.a-aSummary: When the transport equation is solved by the discrete ordinate method, the problem arises of constructing quadrature formulas on a sphere characterized by the required accuracy and making it possible to use the quadrature nodes to approximate the transport equation in \(r\), \(\vartheta\), \(z\) geometry, in which quadrature nodes are simultaneously used to approximate the derivative with respect to the azimuth angle \(\varphi\) of the transport equation, that is, must be located in levels on the sphere with the same values of the polar angle \(\theta \). An algorithm is considered to construct quadrature formulas of the needed form that are characterized by regular prism (dihedron) symmetry and exact for all spherical polynomials of degree not exceeding some maximal value \(L\). This study is a development of the work of \textit{A. N. Kazakov} and \textit{V. I. Lebedev} [Proc. Steklov Inst. Math. 203, 89--99 (1995; Zbl 1126.41302); translation from Tr. Mat. Inst. Steklova 203, 100--112 (1994)]. The constructed family of quadratures, unlike that in the above work, does not contain nodes with \(\varphi = 0,\pi/2, \pi 3\pi/2\), at the poles \(\theta = \pm \pi/2\), and on the equator \(\theta = 0\) of the sphere. It is shown that this family ensures a significant computational gain when radiation transport problems are solved in three-dimensional geometry.Block basis factorization for scalable kernel evaluation.https://www.zbmath.org/1453.650912021-02-27T13:50:00+00:00"Wang, Ruoxi"https://www.zbmath.org/authors/?q=ai:wang.ruoxi"Li, Yingzhou"https://www.zbmath.org/authors/?q=ai:li.yingzhou"Mahoney, Michael W."https://www.zbmath.org/authors/?q=ai:mahoney.michael-w"Darve, Eric"https://www.zbmath.org/authors/?q=ai:darve.ericTowards simulation-driven optimization of high-order meshes by the target-matrix optimization paradigm.https://www.zbmath.org/1453.652982021-02-27T13:50:00+00:00"Dobrev, Veselin"https://www.zbmath.org/authors/?q=ai:dobrev.veselin-a"Knupp, Patrick"https://www.zbmath.org/authors/?q=ai:knupp.patrick-m"Kolev, Tzanio"https://www.zbmath.org/authors/?q=ai:kolev.tzanio-v"Tomov, Vladimir"https://www.zbmath.org/authors/?q=ai:tomov.vladimir-zSummary: We present a method for simulation-driven optimization of high-order curved meshes. This work builds on the results of the first author et al. [SIAM J. Sci. Comput. 41, No. 1, B50--B68 (2019; Zbl 1450.65109)], where we described a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of \textit{P. Knupp} [``Introducing the target-matrix paradigm for mesh optimization via node-movement'', Eng. Comput. 28, No. 4, 419--429 (2012; \url{doi:10.1007/s00366-011-0230-1})]. In contrast to Dobrev et al. [loc. cit.], where all targets were based strictly on geometric information, in this work we blend physical information into the high-order mesh optimization process. The construction of target-matrices is enhanced by using discrete fields of interest, e.g., proximity to a particular region. As these discrete fields are defined only with respect to the initial mesh, their values on the intermediate meshes (produced during the optimization process) must be computed. We present two approaches for obtaining values on the intermediate meshes, namely, interpolation in physical space, and advection remap on the intermediate meshes. Our algorithm allows high-order applications to have precise control over local mesh quality, while still improving the mesh globally. The benefits of the new high-order TMOP methods are illustrated on examples from a high-order arbitrary Lagrangian-Eulerian application [``BLAST: High-order curvilinear finite elements for shock hydrodynamics'', LLNL code (2018), \url{http://www.llnl.gov/CASC/blast}].
For the entire collection see [Zbl 1417.65007].Cheney-Sharma-type operators on a triangle with two or three curved edges.https://www.zbmath.org/1453.410022021-02-27T13:50:00+00:00"Baboş, A."https://www.zbmath.org/authors/?q=ai:babos.alinaIn this paper the author constructs operators of Cheney-Sharma-type with several interpolation properties on triangles with curved edges (two and three, respectively). He studies the properties of this type of interpolation as well as its degree of accuracy. Some examples are given.
Reviewer: Antonio López-Carmona (Granada)Adaptive space-time isogeometric analysis for parabolic evolution problems.https://www.zbmath.org/1453.653302021-02-27T13:50:00+00:00"Langer, Ulrich"https://www.zbmath.org/authors/?q=ai:langer.ulrich"Matculevich, Svetlana"https://www.zbmath.org/authors/?q=ai:matculevich.svetlana"Repin, Sergey"https://www.zbmath.org/authors/?q=ai:repin.sergey-iSummary: The paper proposes new locally stabilized space-time isogeometric analysis approximations to initial boundary value problems of the parabolic type. Previously, similar schemes (but weighted with a global mesh parameter) have been presented and studied by U. Langer, M. Neumüller, and S. Moore [\textit{U. Langer} et al., Comput. Methods Appl. Mech. Eng. 306, 342--363 (2016; Zbl 1436.76027)]. The current work devises a localized version of this scheme, which is suited for adaptive mesh refinement. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with standard approximation error estimates for B-splines and NURBS, we show that the space-time isogeometric analysis solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. Error indicators used for mesh refinement are based on a posteriori error estimates of the functional type that have been introduced by \textit{S. I. Repin} and \textit{M. E. Frolov} [Zh. Vychisl. Mat. Mat. Fiz. 42, No. 12, 1774--1787 (2002; Zbl 1116.65324); translation in Comput. Math. Math. Phys. 42, No. 12, 1704--1716 (2002)], and later rigorously studied in the context of isogeometric analysis by U. Langer, S. Matculevich, and S. Repin [\textit{U. Langer} et al., Radon Ser. Comput. Appl. Math. 25, 141--183 (2019; Zbl 07224809)]. Numerical results discussed in the paper illustrate an improved convergence of global approximation errors and respective error majorants. They also confirm the local efficiency of the error indicators produced by the error majorants.
For the entire collection see [Zbl 1425.65008].A surface moving mesh method based on equidistribution and alignment.https://www.zbmath.org/1453.650432021-02-27T13:50:00+00:00"Kolasinski, Avary"https://www.zbmath.org/authors/?q=ai:kolasinski.avary"Huang, Weizhang"https://www.zbmath.org/authors/?q=ai:huang.weizhang.1|huang.weizhangSummary: A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples are presented.Numerical approximation of the Schrödinger equation with concentrated potential.https://www.zbmath.org/1453.653762021-02-27T13:50:00+00:00"Banjai, L."https://www.zbmath.org/authors/?q=ai:banjai.lehel|banyai.ladislaus-alexander"López-Fernández, Maria"https://www.zbmath.org/authors/?q=ai:lopez-fernandez.mariaSummary: We present a family of algorithms for the numerical approximation of the Schrödinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms. These algorithms are special implementations of Lubich's Convolution Quadrature which allow, for certain applications in particular parabolic problems, to significantly reduce the computational cost and memory requirements. Recently it has been noticed that their use can be extended to some hyperbolic problems. Here we propose a new family of such efficient algorithms tailored to the features of the Green's function for Schrödinger equations. In this way, we are able to keep the computational cost and the storage requirements significantly below existing approaches. These features allow us to perform reliable numerical simulations for longer times even in cases where the solution becomes highly oscillatory or seems to develop finite time blow-up. We illustrate our new algorithm with several numerical experiments.Mesh-independent streamline tracing.https://www.zbmath.org/1453.650382021-02-27T13:50:00+00:00"Batista, David"https://www.zbmath.org/authors/?q=ai:batista.davidSummary: Following the theory of R-functions, a mesh cell of any type can be implicitly described by one single operator constructed from several distance-like functions which, combined with the transfinite interpolation technique and Darcy's velocities specified at the grid nodes, allows us to compute a velocity vector everywhere in the domain, producing a globally continuous vector field with locally performed calculations. Then, knowing a particle's initial position and using these velocities, one can compute its trajectory, obtaining simple equations to determine the exit point of the particle from a given cell. This alternative approach differs from standard schemes in that there is little dependence on the Eulerian topology for tracing the paths and all computations can be made directly at the physical space for any kind of mesh. Numerical experiments show the applicability of our method for streamline modeling on problems with hybrid grids, domains with non-trivial inner structures, and highly variating flows, among others.The image-based multiscale multigrid solver, preconditioner, and reduced order model.https://www.zbmath.org/1453.654312021-02-27T13:50:00+00:00"Yushu, Dewen"https://www.zbmath.org/authors/?q=ai:yushu.dewen"Matouš, Karel"https://www.zbmath.org/authors/?q=ai:matous.karelSummary: We present a novel image-based multiscale multigrid solver that can efficiently address the computational complexity associated with highly heterogeneous systems. This solver is developed based on an image-based, multiresolution model that enables reliable data flow between corresponding computational grids and provides large data compression. A set of inter-grid operators is constructed based on the microstructural data which remedies the issue of missing coarse grid information. Moreover, we develop an image-based multiscale preconditioner from the multiscale coarse images which does not traverse through any intermediate grid levels and thus leads to a faster solution process. Finally, an image-based reduced order model is designed by prolongating the coarse-scale solution to approximate the fine-scale one with improved accuracy. The numerical robustness and efficiency of this image-based computational framework is demonstrated on a two-dimensional example with high degrees of data heterogeneity and geometrical complexity.A super-smooth \(C^1\) spline space over planar mixed triangle and quadrilateral meshes.https://www.zbmath.org/1453.650332021-02-27T13:50:00+00:00"Grošelj, Jan"https://www.zbmath.org/authors/?q=ai:groselj.jan"Kapl, Mario"https://www.zbmath.org/authors/?q=ai:kapl.mario"Knez, Marjeta"https://www.zbmath.org/authors/?q=ai:knez.marjeta"Takacs, Thomas"https://www.zbmath.org/authors/?q=ai:takacs.thomas"Vitrih, Vito"https://www.zbmath.org/authors/?q=ai:vitrih.vitoSummary: In this paper we introduce a \(C^1\) spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis. In this context, a mesh is considered to be a partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle, cf. [\textit{J. H. Argyris} et al., ``The TUBA family of plate elements for the matrix displacement method'', Aeronaut. J. 72, No. 692, 701--709 (1968; \url{doi:10.1017/S000192400008489X})], with the \(C^1\) quadrilateral element introduced in [\textit{S. C. Brenner} and \textit{L.-Y. Sung}, J. Sci. Comput. 22--23, 83--118 (2005; Zbl 1071.65151); \textit{M. Kapl} et al., ``A family of \(C^1\) quadrilateral finite elements'', Preprint, \url{arXiv:2005.04251}] for polynomial degrees \(p \geq 5\). The space is assumed to be \(C^2\) at all vertices and \(C^1\) across edges, and the splines are uniquely determined by \(C^2\)-data at the vertices, values and normal derivatives at chosen points on the edges, and values at some additional points in the interior of the elements.
The motivation for combining the Argyris triangle element with a recent \(C^1\) quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a \(C^1\) fashion is non-trivial and of theoretical interest. We provide not only approximation error bounds but also numerical tests verifying the results. On the other hand, the construction facilitates the meshing process by allowing more flexibility while remaining \(C^1\) everywhere. This is for instance relevant when trimming of tensor-product B-splines is performed.
In the presented construction we assume to have (bi)linear element mappings and piecewise polynomial function spaces of arbitrary degree \(p\geq 5\). The basis is simple to implement and the obtained results are optimal with respect to the mesh size for \(L^\infty\), \(L^2\) as well as Sobolev norms \(H^1\) and \(H^2\).The complexity of cylindrical algebraic decomposition with respect to polynomial degree.https://www.zbmath.org/1453.130792021-02-27T13:50:00+00:00"England, Matthew"https://www.zbmath.org/authors/?q=ai:england.matthew"Davenport, James H."https://www.zbmath.org/authors/?q=ai:davenport.james-haroldSummary: Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors [Proceedings of the 40th international symposium on symbolic and algebraic computation. New York, NY: ACM, 165--172 (2015; Zbl 1346.68283)] proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged.{
} In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Gröbner bases to move towards the intuitive bound for polynomial degree.
For the entire collection see [Zbl 1346.68010].A numerical method for computing border curves of bi-parametric real polynomial systems and applications.https://www.zbmath.org/1453.650392021-02-27T13:50:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuanSummary: For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation.{
} The border curve enables us to construct a so-called ``solution map''. For a given value \(u\) of the parameters inside the rectangle but not on the border, the solution map tells the subset that \(u\) belongs to together with a connected path from the corresponding sample point \(w\) to \(u\). Consequently, all the real solutions of the system at \(u\) (which are isolated) can be obtained by tracking a real homotopy starting from all the real roots at \(w\) throughout the path. The effectiveness of the proposed method is illustrated by some examples.
For the entire collection see [Zbl 1346.68010].Numerical computation of integral of analytic functions in complex plane.https://www.zbmath.org/1453.650512021-02-27T13:50:00+00:00"Hota, Manoj Kumar"https://www.zbmath.org/authors/?q=ai:hota.manoj-kumar"Dash, Biranchi Narayan"https://www.zbmath.org/authors/?q=ai:dash.biranchi-narayan"Mohanty, Prasanta Kumar"https://www.zbmath.org/authors/?q=ai:mohanty.prasanta-kumarSummary: A family of interpolatory type of quadrature rules of degree of precision at least seven has been constructed for the approximate evaluation of contour integrals of analytic functions along a directed line segment in the complex plane. The relative accuracies of rules have been studied in their respective classes and also numerically verified by integrating some standard test integrals.A benchmark of selected algorithmic differentiation tools on some problems in computer vision and machine learning.https://www.zbmath.org/1453.650502021-02-27T13:50:00+00:00"Srajer, Filip"https://www.zbmath.org/authors/?q=ai:srajer.filip"Kukelova, Zuzana"https://www.zbmath.org/authors/?q=ai:kukelova.zuzana"Fitzgibbon, Andrew"https://www.zbmath.org/authors/?q=ai:fitzgibbon.andrew-wSummary: Algorithmic differentiation (AD) allows exact computation of derivatives given only an implementation of an objective function. Although many AD tools are available, a proper and efficient implementation of AD methods is not straightforward. The existing tools are often too different to allow for a general test suite. In this paper, we compare 15 ways of computing derivatives including 11 automatic differentiation tools implementing various methods and written in various languages (C++, F\#, MATLAB, Julia and Python), 2 symbolic differentiation tools, finite differences and hand-derived computation.
We look at three objective functions from computer vision and machine learning. These objectives are for the most part simple, in the sense that no iterative loops are involved, and conditional statements are encapsulated in functions such as abs or logsumexp. However, it is important for the success of AD that such `simple' objective functions are handled efficiently, as so many problems in computer vision and machine learning are of this form.Evaluation of Abramowitz functions in the right half of the complex plane.https://www.zbmath.org/1453.650482021-02-27T13:50:00+00:00"Gimbutas, Zydrunas"https://www.zbmath.org/authors/?q=ai:gimbutas.zydrunas"Jiang, Shidong"https://www.zbmath.org/authors/?q=ai:jiang.shidong"Luo, Li-Shi"https://www.zbmath.org/authors/?q=ai:luo.lishiSummary: A numerical scheme is developed for the evaluation of Abramowitz functions \(J_n\) in the right half of the complex plane. For \(n=-1,\dots,2\), the scheme utilizes series expansions for \(|z|<1\), asymptotic expansions for \(|z|>R\) with \(R\) determined by the required precision, and least squares Laurent polynomial approximations on each sub-region in the intermediate region \(1\leq |z|\leq R\). For \(n>2\), \(J_n\) is evaluated via a forward recurrence relation. The scheme achieves nearly machine precision for \(n=-1,\dots,2\) at a cost that is competitive as compared with software packages for the evaluation of other special functions in the complex domain.Kinematics and dynamics motion planning by polar piecewise interpolation and geometric considerations.https://www.zbmath.org/1453.700012021-02-27T13:50:00+00:00"Dupac, Mihai"https://www.zbmath.org/authors/?q=ai:dupac.mihaiSummary: The importance of numerical methods in science and engineering [\textit{S. C. Chapra} and \textit{R. P. Canale}, Numerical methods for engineers. 6. ed. New York, NY: McGraw-Hill (2010)] was long recognised and considered a fundamental factor in improving productivity and reducing production costs. The ability to model flexible systems and describe their trajectories [\textit{A. Gasparetto} et al., ``Trajectory planning in robotics'', Math. Comput. Sci. 6, 269--279 (2012; \url{doi:10.1007/s11786-012-0123-8})] involves usually the study of nonlinear coupled partial differential equations. Since their exact solutions are not normally feasible in practice, computational methods [\textit{V. Kumar} et al., ``Motion planning and control of robots'', in: Handbook of industrial robotics. 2nd ed. Hoboken, NJ: Wiley and sons. 295--315 (2007)] can be considered.
For the entire collection see [Zbl 1392.00002].Optimized quasiconformal parameterization with user-defined area distortions.https://www.zbmath.org/1453.650442021-02-27T13:50:00+00:00"Lam, Ka Chun"https://www.zbmath.org/authors/?q=ai:lam.ka-chun"Lui, Lok Ming"https://www.zbmath.org/authors/?q=ai:lui.lok-mingSummary: Parameterization, a process of mapping a complicated domain onto a simple canonical domain, is crucial in different areas such as computer graphics, medical imaging and scientific computing. Conformal parameterization has been widely used since it preserves the local geometry well. However, a major drawback is the area distortion introduced by the conformal parameterization, causing inconvenience in many applications such as texture mapping in computer graphics or visualization in medical imaging. This work proposes a remedy to construct a parameterization that balances between conformality and area distortions. We present a variational algorithm to compute the optimized quasiconformal parameterization with controllable area distortions. The distribution of the area distortion can be prescribed by users according to the application. The main strategy is to minimize a combined energy functional consisting of an area mismatching term and a regularization term involving the Beltrami coefficient of the map. The Beltrami coefficient controls the conformality of the parameterization. Landmark constraints can be incorporated into the model to obtain landmark-aligned parameterization. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to compute the optimized parameterization with controllable area distortion while preserving the local geometry as well as possible.Quasi Monte Carlo integration and kernel-based function approximation on Grassmannians.https://www.zbmath.org/1453.650072021-02-27T13:50:00+00:00"Breger, Anna"https://www.zbmath.org/authors/?q=ai:breger.anna"Ehler, Martin"https://www.zbmath.org/authors/?q=ai:ehler.martin"Gräf, Manuel"https://www.zbmath.org/authors/?q=ai:graf.manuelSummary: Numerical integration and function approximation on compact Riemannian manifolds based on eigenfunctions of the Laplace-Beltrami operator have been widely studied in the recent literature. The standard example in numerical experiments is the Euclidean sphere. Here, we derive numerically feasible expressions for the approximation schemes on the Grassmannian manifold, and we present the associated numerical experiments on the Grassmannian. Indeed, our experiments illustrate and match the corresponding theoretical results in the literature.
For the entire collection see [Zbl 1373.42002].Optimization of fast algorithms for global quadrature by expansion using target-specific expansions.https://www.zbmath.org/1453.654642021-02-27T13:50:00+00:00"Wala, Matt"https://www.zbmath.org/authors/?q=ai:wala.matt"Klöckner, Andreas"https://www.zbmath.org/authors/?q=ai:klockner.andreasSummary: We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX (`GIGAQBX') and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion (`QBX') using local expansions formed via the Fast Multipole Method (FMM). Target-specific expansions serve to lower the cost of the formation and evaluation of QBX local expansions, reducing the associated computational effort from \(O((p + 1)^2)\) to \(O(p + 1)\) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a `global' QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for `local' QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels.On the hexagonal Shepard method.https://www.zbmath.org/1453.650302021-02-27T13:50:00+00:00"Dell'Accio, Francesco"https://www.zbmath.org/authors/?q=ai:dellaccio.francesco"Di Tommaso, Filomena"https://www.zbmath.org/authors/?q=ai:di-tommaso.filomenaThe authors consider the problem of scattered data interpolation on functions of two variables. More precisely, they introduce the so-called hexagonal Shepard method extending the Shepard and triangular Shepard methods to the case of six points. In doing so, the authors use the multinode basis functions [\textit{F. Dell'Accio} et al., Appl. Math. Comput. 318, 51--69 (2018; Zbl 1426.65012)] based on six points and local quadratic Lagrange polynomials that interpolate on the six points of each basis function. The global interpolant has quadratic precision and reaches cubic approximation order. Numerical experiments show performance of the new interpolation method.
Reviewer: Roberto Cavoretto (Torino)Spectral collocation solutions to problems on unbounded domains.https://www.zbmath.org/1453.650042021-02-27T13:50:00+00:00"Gheorghiu, Călin-Ioan"https://www.zbmath.org/authors/?q=ai:gheorghiu.calin-ioanA usual procedure in discretizing problems on unbounded domains is to cut off an exterior part of the domain, to impose suitable boundary conditions on the newly obtained boundary and then to apply methods designed for problems on bounded domains. The author instead discretizes the problems directly as given by spectral approximation with trial functions defined on the entire domain combined with collocation in a finite number of points. As trial function he considers Hermite functions, Laguerre functions (with both the roots of the Laguerre polynomials and the Gauss-Radau nodes as collocation points) and sinc functions and also a mapping technique. The trial functions depend on a scaling parameter which is also adapted for obtaining good results.
The author concentrates on presenting a large number of numerical results, an analysis of the methods is not in the focus. The book contains 105 figures and 23 tables (a list of problems considered can be found under the above keywords). The calculations are performed using MATLAB (for some representative problems MATLAB scripts are given in the fifth chapter of the book). The bibliography comprises 195 items.
The book is for sale at 19.99 Euro. But I think that this affordable price is not the explanation for the not so successful layout of the figures. They seem to be placed at random in the center, at the left margin, colored or not, with weakly or firmly printed labels and lines.
Reviewer: Rolf Dieter Grigorieff (Berlin)A new simultaneous extension method for B-spline curves blending with \(G^2\)-continuity.https://www.zbmath.org/1453.650372021-02-27T13:50:00+00:00"Yu, Hongying"https://www.zbmath.org/authors/?q=ai:yu.hongying"Lyu, Xuegeng"https://www.zbmath.org/authors/?q=ai:lyu.xuegengSummary: Curve blending is an extremely common problem in CAD systems. The current blending methods were looking for a third curve to join curves and some of the methods needed to distinguish the transition curve were C-or S-shaped. In this paper, we study a simultaneous extension method to blend curves with \(G^2\)-continuity. The method simultaneously extends the two curves at one of their endpoints and makes them intersect at a common joint under geometric constraints. The basic concept of B-spline curves and its extension theory is presented firstly. Then we propose the blending algorithm of simultaneous extension. This method does not need to prejudge the shape of transition curves, which is, without considering the placement of two original curves and reduces the number of blending joints from two to one. Four curve blending examples are presented to verify the validity of the new method.A note on generalized averaged Gaussian formulas for a class of weight functions.https://www.zbmath.org/1453.650562021-02-27T13:50:00+00:00"Spalević, Miodrag M."https://www.zbmath.org/authors/?q=ai:spalevic.miodrag-mSummary: In the recent paper [\textit{S. E. Notaris}, Numer. Math. 142, No. 1, 129--147 (2019; Zbl 1411.41022)] it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss-Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see [\textit{M. M. Spalević}, Math. Comput. 76, No. 259, 1483--1492 (2007; Zbl 1113.65025)]. Moreover, as almost immediate consequence of our results from [Spalević, loc. cit.] and that theory, we prove the main statements in [Notaris, loc. cit.] in a different manner, by means of the Jacobi tridiagonal matrix approach.Enforcing constraints for interpolation and extrapolation in generative adversarial networks.https://www.zbmath.org/1453.681592021-02-27T13:50:00+00:00"Stinis, Panos"https://www.zbmath.org/authors/?q=ai:stinis.panos"Hagge, Tobias"https://www.zbmath.org/authors/?q=ai:hagge.tobias-j"Tartakovsky, Alexandre M."https://www.zbmath.org/authors/?q=ai:tartakovsky.alexandre-m"Yeung, Enoch"https://www.zbmath.org/authors/?q=ai:yeung.enochSummary: Generative Adversarial Networks (GANs) are becoming popular machine learning choices for training generators. At the same time there is a concerted effort in the machine learning community to expand the range of tasks in which learning can be applied as well as to utilize methods from other disciplines to accelerate learning. With this in mind, in the current work we suggest ways to enforce given constraints in the output of a GAN generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently. For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary game-theoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments. The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions.Quadrature formulae of Euler-Maclaurin type based on generalized Euler polynomials of level \(m\).https://www.zbmath.org/1453.650552021-02-27T13:50:00+00:00"Quintana, Yamilet"https://www.zbmath.org/authors/?q=ai:quintana.yamilet"Urieles, Alejandro"https://www.zbmath.org/authors/?q=ai:urieles.alejandroSummary: This article deals with some properties -- which are, to the best of our knowledge, new -- of the generalized Euler polynomials of level \(m\). These properties include a new recurrence relation satisfied by these polynomials and quadrature formulae of Euler-Maclaurin type based on them. Numerical examples are also given.The Gaussian quadrature after Gauss.https://www.zbmath.org/1453.010112021-02-27T13:50:00+00:00"Sanz-Serna, J. M."https://www.zbmath.org/authors/?q=ai:sanz-serna.jesus-mariaThis is a Spanish translation, summarized and provided with extensive comments, of the 1815 memoir \textit{Methodus nova integralium valores per approximationem inveniendi}, in which Gauss introduced the quadrature rules that bear his name. Gauss's approach differs significantly from what is presented nowadays as Gaussian quadrature in textbooks. The original memoir displays a mastery of work with series, in which the problem is rephrased as one of functional approximation, solved with the help of continuous fractions.
Reviewer: Victor V. Pambuccian (Glendale)Lanczos-type algorithms with embedded interpolation and extrapolation models for solving large-scale systems of linear equations.https://www.zbmath.org/1453.650602021-02-27T13:50:00+00:00"Maharani, Maharani"https://www.zbmath.org/authors/?q=ai:maharani.maharani"Larasati, Niken"https://www.zbmath.org/authors/?q=ai:larasati.niken"Salhi, Abdellah"https://www.zbmath.org/authors/?q=ai:salhi.abdellah"Khan, Wali Mashwani"https://www.zbmath.org/authors/?q=ai:khan.wali-mashwaniSummary: The new approach to combating instability in Lanczos-type algorithms for large-scale problems is proposed. It is a modification of so called the embedded interpolation and extrapolation model in Lanczos-type algorithms (EIEMLA), which enables us to interpolate the sequence of vector solutions generated by a Lanczos-type algorithm entirely, without rearranging the position of the entries of the vector solutions. The numerical results show that the new approach performs more effectively than the EIEMLA. In fact, we extend this new approach on the use of a restarting framework to obtain the convergence of Lanczos algorithms accurately. This kind of restarting challenges other existing restarting strategies in Lanczos-type algorithms.