Recent zbMATH articles in MSC 49https://www.zbmath.org/atom/cc/492021-04-16T16:22:00+00:00WerkzeugA spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations.https://www.zbmath.org/1456.490262021-04-16T16:22:00+00:00"Wang, Fangyuan"https://www.zbmath.org/authors/?q=ai:wang.fangyuan"Zhang, Zhongqiang"https://www.zbmath.org/authors/?q=ai:zhang.zhongqiang"Zhou, Zhaojie"https://www.zbmath.org/authors/?q=ai:zhou.zhaojieSummary: A spectral Galerkin approximation of a optimal control problem governed by a fractional advection-diffusion-reaction equation with integral fractional Laplacian is investigated in 1D. We first derive a first-order optimality condition and analyze the regularity of the solution based on this optimality condition. We present a spectral Galerkin scheme for the control problem using weighted Jacobi polynomials and prove optimal error estimates of the spectral method for state, adjoint state and control variables. We also propose a fast projected gradient algorithm of quasilinear complexity and present two numerical examples verifying our theoretical findings.A self-adaptive descent LQP alternating direction method for the structured variational inequalities.https://www.zbmath.org/1456.650472021-04-16T16:22:00+00:00"Bnouhachem, Abdellah"https://www.zbmath.org/authors/?q=ai:bnouhachem.abdellahSummary: In this paper, by combining the logarithmic-quadratic proximal (LQP) method and alternating direction method, we proposed an LQP alternating direction method for solving structured variational inequalities. The new iterate is generated by searching the optimal step size along a descent direction with a new step size \(\alpha_k\). The choice of the descent direction and the step size selection strategies are important for the algorithm's efficiency. The \(O(1/t)\) convergence rate of the proposed method is studied, and its efficiency is also verified by some numerical experiments.Improved canonical dual finite element method and algorithm for post-buckling analysis of nonlinear Gao beam.https://www.zbmath.org/1456.651512021-04-16T16:22:00+00:00"Ali, Elaf Jaafar"https://www.zbmath.org/authors/?q=ai:ali.elaf-jaafar"Gao, David Yang"https://www.zbmath.org/authors/?q=ai:gao.david-yangSummary: This paper deals a study on post-buckling problem of a large deformed elastic beam by using a canonical dual mixed finite element method (CD-FEM). The nonconvex total potential energy of this beam can be used to model post-buckling problems. To verify the triality theory, different types of dual stress interpolations are used. Applications are illustrated with different boundary conditions and different external loads using semi-definite programming (SDP) algorithm. The results show that the global minimizer of the total potential energy is stable buckled configuration, the local maximizer solution leads to the unbuckled state, and both of these two solutions are numerically stable. While the local minimizer is unstable buckled configuration and very sensitive.
For the entire collection see [Zbl 1387.49002].Existence, multiplicity and regularity of solutions for the fractional \(p\)-Laplacian equation.https://www.zbmath.org/1456.352162021-04-16T16:22:00+00:00"Kim, Yun-Ho"https://www.zbmath.org/authors/?q=ai:kim.yunho.1|kim.yunhoSummary: : We are concerned with the following elliptic equations:
\[
\begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \text{in }\Omega\\u= 0\quad\text{on }\mathbb{R}^N\backslash\Omega,\end{cases}
\]
where \(\lambda\) are real parameters, \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian operator, \(0 < s < 1 < p < +\infty, sp < N\), and \(f:\Omega\times\mathbb R\to\mathbb R\) satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters \(\lambda\) for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity \(f\) has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in \(L^{\infty}(\Omega)\) of any possible weak solution by applying the bootstrap argument.ADMM-type methods for generalized Nash equilibrium problems in Hilbert spaces.https://www.zbmath.org/1456.490252021-04-16T16:22:00+00:00"Börgens, Eike"https://www.zbmath.org/authors/?q=ai:borgens.eike"Kanzow, Christian"https://www.zbmath.org/authors/?q=ai:kanzow.christianEnergy decay and global solutions for a damped free boundary fluid-elastic structure interface model with variable coefficients in elasticity.https://www.zbmath.org/1456.490202021-04-16T16:22:00+00:00"Qin, Yizhao"https://www.zbmath.org/authors/?q=ai:qin.yizhao"Yao, Peng-Fei"https://www.zbmath.org/authors/?q=ai:yao.pengfeiSummary: We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier-Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.Subgradient projection methods extended to monotone bilevel equilibrium problems in Hilbert spaces.https://www.zbmath.org/1456.650452021-04-16T16:22:00+00:00"Anh, Pham Ngoc"https://www.zbmath.org/authors/?q=ai:pham-ngoc-anh.|anh.pham-ngoc"Tu, Ho Phi"https://www.zbmath.org/authors/?q=ai:tu.ho-phiSummary: In this paper, by basing on the inexact subgradient and projection methods presented by \textit{P. Santos} and \textit{S. Scheimberg} [Comput. Appl. Math. 30, No. 1, 91--107 (2011; Zbl 1242.90265)], we develop subgradient projection methods for solving strongly monotone equilibrium problems with pseudomonotone equilibrium constraints. The problem usually is called monotone bilevel equilibrium problems. We show that this problem can be solved by a simple and explicit subgradient method. The strong convergence for the proposed algorithms to the solution is guaranteed under certain assumptions in a real Hilbert space. Numerical illustrations are given to demonstrate the performances of the algorithms.Second-order necessary optimality conditions for a discrete optimal control problem.https://www.zbmath.org/1456.490212021-04-16T16:22:00+00:00"Toan, N. T."https://www.zbmath.org/authors/?q=ai:nguyen-thi-toan."Ansari, Q. H."https://www.zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Yao, J.-C."https://www.zbmath.org/authors/?q=ai:yao.jen-chihSummary: In this paper, we study second-order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function and control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order optimality conditions for a discrete optimal control problem.Stein's method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem.https://www.zbmath.org/1456.600592021-04-16T16:22:00+00:00"Bonis, Thomas"https://www.zbmath.org/authors/?q=ai:bonis.thomasSummary: We use Stein's method to bound the Wasserstein distance of order 2 between a measure \(\nu\) and the Gaussian measure using a stochastic process \((X_t)_{t \ge 0}\) such that \(X_t\) is drawn from \(\nu\) for any \(t > 0\). If the stochastic process \((X_t)_{t \ge 0}\) satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order \(p \ge 1\). Using our results, we provide convergence rates for the multi-dimensional central limit theorem in terms of Wasserstein distances of any order \(p \ge 2\) under simple moment assumptions.Optimum control laws in a non-autonomous dynamic model of a gas field.https://www.zbmath.org/1456.761122021-04-16T16:22:00+00:00"Kiselev, Yu. N."https://www.zbmath.org/authors/?q=ai:kiselev.yuri-n"Orlov, M. V."https://www.zbmath.org/authors/?q=ai:orlov.mikhail-vSummary: A model of gas field development described as a nonlinear optimum control problem with an infinite planning horizon is considered. The Pontryagin maximum principle is used to solve it. The theorem on sufficient optimumity conditions in terms of constructions of the Pontryagin maximum principles is used to substantiate the optimumity of the extremal solution. A procedure for constructing the optimum solution by dynamic programming is described and is of some methodological interest. The obtained optimum solution is used to construct the Bellman function. Reference is made to a work containing an economic interpretation of the problem.Nonconvex policy search using variational inequalities.https://www.zbmath.org/1456.681732021-04-16T16:22:00+00:00"Zhan, Yusen"https://www.zbmath.org/authors/?q=ai:zhan.yusen"Ammar, Haitham Bou"https://www.zbmath.org/authors/?q=ai:ammar.haitham-bou"Taylor, Matthew E."https://www.zbmath.org/authors/?q=ai:taylor.matthew-eSummary: Policy search is a class of reinforcement learning algorithms for finding optimal policies in control problems with limited feedback. These methods have been shown to be successful in high-dimensional problems such as robotics control. Though successful, current methods can lead to unsafe policy parameters that potentially could damage hardware units. Motivated by such constraints, we propose projection-based methods for safe policies.
These methods, however, can handle only convex policy constraints. In this letter, we propose the first safe policy search reinforcement learner capable of operating under nonconvex policy constraints. This is achieved by observing, for the first time, a connection between nonconvex variational inequalities and policy search problems. We provide two algorithms, Mann and two-step iteration, to solve the above problems and prove convergence in the nonconvex stochastic setting. Finally, we demonstrate the performance of the algorithms on six benchmark dynamical systems and show that our new method is capable of outperforming previous methods under a variety of settings.Higher order strongly uniform convex functions.https://www.zbmath.org/1456.260142021-04-16T16:22:00+00:00"Noor, Muhammad Aslam"https://www.zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://www.zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: Some new concepts of the higher order strongly uniform convex functions with an increasing modulus \(\varphi(\cdot)\) vanishing only at 0 are considered in this paper. Some properties of the higher order strongly uniformly convex functions are investigated under suitable conditions. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine uniform convex functions as novel applications. It is shown that the minimum of the higher order strongly uniform convex functions can be characterized by the variational inequalities. Some important special cases as applications of our results are discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.Torus-like solutions for the Landau-de Gennes model. I: The Lyuksyutov regime.https://www.zbmath.org/1456.829092021-04-16T16:22:00+00:00"Dipasquale, Federico"https://www.zbmath.org/authors/?q=ai:dipasquale.federico"Millot, Vincent"https://www.zbmath.org/authors/?q=ai:millot.vincent"Pisante, Adriano"https://www.zbmath.org/authors/?q=ai:pisante.adrianoSummary: We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in [the first author et al., ``Torus-like solutions for the Landau-de Gennes model. II: Topology of \(\mathbb{S}^1\)-equivariant minimizers'', Preprint, \url{arxiv:2008.13676}; ``Torus-like solutions for the Landau-de Gennes model. III: Torus solutions vs split solutions'', in preparation], where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects.https://www.zbmath.org/1456.350912021-04-16T16:22:00+00:00"Garcke, Harald"https://www.zbmath.org/authors/?q=ai:garcke.harald"Lam, Kei Fong"https://www.zbmath.org/authors/?q=ai:lam.kei-fong"Signori, Andrea"https://www.zbmath.org/authors/?q=ai:signori.andreaThis article presents a Cahn-Hilliard model of tumor growth which exhibits mechanical effects due to cell-cell adhesion.
The authors consider a Cahn-Hilliard variant (e.g. cf. [\textit{A. Miranville}, The Cahn-Hilliard equation. Recent advances and applications. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1446.35001)]) coupled to a system of equations of linear elasticity, which communicates the proliferation rate, and a reaction-diffusion equation which describes nutrient concentration.
Well-possdness results are obtained as well as various continuous dependence estimates for different topologies.
The results here are important because of the difficulties that arise from the nonlinear coupling of the fourth-order Cahn-Hilliard equation and the quasi-static elasticity system.
Reviewer: Joseph Shomberg (Providence)Tomographic reconstruction from a few views: a multi-marginal optimal transport approach.https://www.zbmath.org/1456.490352021-04-16T16:22:00+00:00"Abraham, I."https://www.zbmath.org/authors/?q=ai:abraham.isabelle"Abraham, R."https://www.zbmath.org/authors/?q=ai:abraham.romain"Bergounioux, M."https://www.zbmath.org/authors/?q=ai:bergounioux.maitine"Carlier, G."https://www.zbmath.org/authors/?q=ai:carlier.guillaumeSummary: In this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We use a multi-marginal optimal transport approach. Preliminary numerical results are presented.Loss of double-integral character during relaxation.https://www.zbmath.org/1456.490142021-04-16T16:22:00+00:00"Kreisbeck, Carolin"https://www.zbmath.org/authors/?q=ai:kreisbeck.carolin"Zappale, Elvira"https://www.zbmath.org/authors/?q=ai:zappale.elviraPath integrals and symmetry breaking for optimal control theory.https://www.zbmath.org/1456.490272021-04-16T16:22:00+00:00"Kappen, H. J."https://www.zbmath.org/authors/?q=ai:kappen.hilbert-j\(V\)-prox-regular functions in smooth Banach spaces.https://www.zbmath.org/1456.490162021-04-16T16:22:00+00:00"Bounkhel, Messaoud"https://www.zbmath.org/authors/?q=ai:bounkhel.messaoud"Bachar, Mostafa"https://www.zbmath.org/authors/?q=ai:bachar.mostafaSummary: In this paper, we continue the study of the \(V\)-prox-regularity that we have started recently for sets. We define an appropriate concept of the \(V\)-prox-regularity for functions in reflexive smooth Banach spaces by adapting the one given in Hilbert spaces. Our main goal is to study the relationship between the \(V\)-prox-regularity of a given l.s.c. \(f\) and the \(V\)-prox-regularity of its epigraph.Fokker-Planck equations of jumping particles and mean field games of impulse control.https://www.zbmath.org/1456.490302021-04-16T16:22:00+00:00"Bertucci, Charles"https://www.zbmath.org/authors/?q=ai:bertucci.charlesAuthor's abstract: This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix the sets from which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for: the mean field game of impulse control. In both cases, we give a variational characterization of the densities of jumping particles.
Reviewer: Hector Jasso (Ciudad de México)Property (D) and the Lavrentiev phenomenon.https://www.zbmath.org/1456.490032021-04-16T16:22:00+00:00"Carlson, Dean A."https://www.zbmath.org/authors/?q=ai:carlson.dean-aSummary: \textit{M. A. Lavrentieff} [Ann. Mat. Pura Appl. (4) 4, 7--28 (1927; JFM 53.0481.02)] gave an example of a free problem in the calculus of variations for which the infimum over the class of functions in \(W^{1,1}[t_1,t_2]\) satisfying prescribed endpoint conditions was strictly less than the infimum over the dense subset \(W^{1,\infty}[t_1,t_2]\) of admissible functions in \(W^{1,1}[t_1,t_2]\). This property is now referred to as the Lavrentiev phenomenon. After Lavrentiev's discovery, [\textit{B. Mania}, Boll. Unione Mat. Ital. 13, 147--153 (1934; Zbl 0009.15803)] gave sufficient conditions under which this phenomenon does not arise. After these results, the study of the Lavrentiev phenomenon lay dormant until the 1980s when a series of papers by [\textit{J. M. Ball} and \textit{V. J. Mizel}, Bull. Am. Math. Soc., New Ser. 11, 143--146 (1984; Zbl 0541.49010)] and by \textit{F. H. Clarke} and \textit{R. B. Vinter} [Appl. Math. Optim. 12, 73--79 (1984; Zbl 0559.49012)] gave a number of new examples for which the Lavrentiev phenomenon occurred. Also \textit{T. S. Angell} [Rend. Circ. Mat. Palermo (2) 28, 258--272 (1979; Zbl 0445.49006)] showed that the Lavrentiev phenomenon did not occur if the integrands satisfy a certain analytic property known as property (D). Moreover, he showed that the conditions of \textit{L. Tonelli} [Rec. Math. Moscou 33, 87--98 (1926; JFM 52.0508.01)] and Mania [loc. cit.] insured that the analytic property (D) was satisfied. Since Angell's result there have been several papers that have discussed the nonoccurence of the Lavrentiev phenomenon for free problems in the calculus of variations. The purpose of this paper is twofold. First to present a general approach to the proofs of these later papers which unifies the results, and second to show that the extra conditions imposed on the integrands insure property (D) holds with respect to the relevant sequence.A class of differential inverse variational inequalities in finite dimensional spaces.https://www.zbmath.org/1456.490102021-04-16T16:22:00+00:00"Feng, Jun"https://www.zbmath.org/authors/?q=ai:feng.jun"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11"Chen, Hui"https://www.zbmath.org/authors/?q=ai:chen.hui"Chen, Yuanchun"https://www.zbmath.org/authors/?q=ai:chen.yuanchunSummary: In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Carathéodory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.Minimizers of convex functionals with small degeneracy set.https://www.zbmath.org/1456.490292021-04-16T16:22:00+00:00"Mooney, Connor"https://www.zbmath.org/authors/?q=ai:mooney.connorLet \(F:\mathbb R^n\to\mathbb R\) be convex and of class \(C^1\).
The paper studies the regularity of Lipschitz minimizers of
\[
E(u)=\int_{B_1}F(\nabla u)\,dx
\]
in \(\mathbb R^n\), i.e. functions \(u\in W^{1, \infty}(B_1)\) satisfying \(E(u+\varphi)\ge E(u)\) for all \(\varphi\in C^1_0(B_1)\).
In the extreme case that the graph of \(F\) contains a line segment, minimizers are no better than Lipschitz by simple examples. In the other extreme that \(F\) is smooth and uniformly convex, De Giorgi and Nash proved that Lipschitz minimizers are smooth and solve the Euler-Lagrange equation \(F_{ij}(\nabla u )u_{ij}=0\) classically [\textit{E. De Giorgi}, Mem. Accad. Sci. Torino, P. I., III. Ser. 3, 25--43 (1957; Zbl 0084.31901); \textit{J. F. Nash}, Am. J. Math. 80, 931--954 (1958; Zbl 0096.06902)]. The paper examines the intermediate case where \(F\) is strictly convex, but the eigenvalues of \(D^2F\) go to 0 or \(\infty\) on some set \(D_F\). Such functionals arise naturally in the study of anisotropic surface tensions [\textit{M. G. Delgadino} et al., Arch. Ration. Mech. Anal. 230, No. 3, 1131--1177 (2018; Zbl 1421.35076)], traffic flow [\textit{M. Colombo} and \textit{A. Figalli}, J. Math. Pures Appl. (9) 101, No. 1, 94--117 (2014; Zbl 1282.35175)], and statistical mechanics [\textit{H. Cohn} et al., J. Am. Math. Soc. 14, No. 2, 297--346 (2001; Zbl 1037.82016); \textit{R. Kenyon} et al., Ann. Math. (2) 163, No. 3, 1019--1056 (2006; Zbl 1154.82007)].
More precisely, the author assumes that there is a compact subset \(D_F\) of \( \mathbb R^n\) such that
\[
F\in C^2(\mathbb R^n\setminus D_F),\quad D_F=\mathbb R^n\setminus\left(\cup_k\{k^{-1}I<D^2F<kI\}\right).
\]
Theorem 2.1 states that if \(D_F\) is finite and is contained in a two-dimensional affine subspace of \(\mathbb R^n\), then \(u\in C^1(B_1)\). Theorem 2.3 shows that in general a Lipschitz minimizer of \(E\) may not be of class \(C^1\), by constructing a singular Lipschitz minimizer in \(\mathbb R^4\).
The above leave opens the possibility that Lipschitz minimizers are \(C^1\) in dimension \(n\ge 3\) in the case where \(D_F\) consists of finitely many points: the problem is connected to a result of Alexandrov in the classical differential geometry of convex surfaces and a related counterexample is conjectured (Conjecture 2.4).
Reviewer: Carlo Mariconda (Padova)Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDEs with singular drift.https://www.zbmath.org/1456.601402021-04-16T16:22:00+00:00"Baños, David"https://www.zbmath.org/authors/?q=ai:banos.david-r"Nilssen, Torstein"https://www.zbmath.org/authors/?q=ai:nilssen.torstein-k"Proske, Frank"https://www.zbmath.org/authors/?q=ai:proske.frank-norbertSummary: In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter \(H<\frac{1}{2}\). Furthermore, we prove the rather surprising result of the higher order Fréchet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a ``local time variational calculus''. We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.Strong convergence theorems of iterative algorithm for nonconvex variational inequalities.https://www.zbmath.org/1456.470262021-04-16T16:22:00+00:00"Inchan, Issara"https://www.zbmath.org/authors/?q=ai:inchan.issaraSummary: In this work, we suggest and analyze an iterative scheme for solving a system of nonconvex variational inequalities by using projection technique. We prove strong convergence of iterative scheme to a solution of asystem of nonconvex variational inequalities requires the modified mapping \(T\) be Lipschitz continuous but not strongly monotone. Our result can be viewed as improvement of the result of \textit{N. Petrot} [Abstr. Appl. Anal. 2010, Article ID 472760, 9 p. (2010; Zbl 1206.49012)].LQG online learning.https://www.zbmath.org/1456.681472021-04-16T16:22:00+00:00"Gnecco, Giorgio"https://www.zbmath.org/authors/?q=ai:gnecco.giorgio"Bemporad, Alberto"https://www.zbmath.org/authors/?q=ai:bemporad.alberto"Gori, Marco"https://www.zbmath.org/authors/?q=ai:gori.marco"Sanguineti, Marcello"https://www.zbmath.org/authors/?q=ai:sanguineti.marcelloSummary: Optimal control theory and machine learning techniques are combined to formulate and solve in closed form an optimal control formulation of online learning from supervised examples with regularization of the updates. The connections with the classical linear quadratic Gaussian (LQG) optimal control problem, of which the proposed learning paradigm is a nontrivial variation as it involves random matrices, are investigated. The obtained optimal solutions are compared with the Kalman filter estimate of the parameter vector to be learned. It is shown that the proposed algorithm is less sensitive to outliers with respect to the Kalman estimate (thanks to the presence of the regularization term), thus providing smoother estimates with respect to time. The basic formulation of the proposed online learning framework refers to a discrete-time setting with a finite learning horizon and a linear model. Various extensions are investigated, including the infinite learning horizon and, via the so-called
kernel trick, the case of nonlinear models.Lecture notes on variational mean field games.https://www.zbmath.org/1456.490312021-04-16T16:22:00+00:00"Santambrogio, Filippo"https://www.zbmath.org/authors/?q=ai:santambrogio.filippoThe author considers first-order Mean Field Games (MFG) with local couplings. The theory of MFG describe the evolution of a population of rational agents, each one choosing a path in a state space, according to some preferences which are affected by the presence of other agents nearby in a way physicists call mean-field effect. The main goal of the article is to prove that minimizers of a suitably expressed global energy are equilibria in the sense that a. e. trajectory solves a control problem with a running cost depending on the density of all the agents. Both the case of a cost penalizing high densities and of an \(L^{\infty}\) constraint on the same densities are considered.
For the entire collection see [Zbl 1456.49002].
Reviewer: Pavel Stoynov (Sofia)Lagrangian discretization of crowd motion and linear diffusion.https://www.zbmath.org/1456.651372021-04-16T16:22:00+00:00"Leclerc, Hugo"https://www.zbmath.org/authors/?q=ai:leclerc.hugo"Mérigot, Quentin"https://www.zbmath.org/authors/?q=ai:merigot.quentin"Santambrogio, Filippo"https://www.zbmath.org/authors/?q=ai:santambrogio.filippo"Stra, Federico"https://www.zbmath.org/authors/?q=ai:stra.federicoAn approximation scheme to solve evolution PDEs which have a gradient-flow structure is presented. The paper is organized as follows. Section 1 is an introduction. In Section 2 Lagrangian discretization of crowd motion is considered. The convergence of the discrete measures to a solution of the continuous PDE describing the crowd motion in dimension one is proved in this section. How a similar approach can be used to construct a Lagrangian discretization of a linear advection-diffusion equation is shown in Section 3. In Section 4 is proved that, for both the crowd motion and the linear diffusion discretizations, in one-dimension, there are bounds on the quantities which are relevant for the application of convergence theorems of the discrete schemes of sections 1 and 2. Computation of the Moreau-Yosida regularization, numerical experiments for crowd motion and for diffusion are given and discussed in Section 5. A numerical implementation in 2 dimensions to demonstrate the feasibility of the computations is also provided.
Reviewer: Temur A. Jangveladze (Tbilisi)On the best constant in the estimate related to \(H^1-\mathrm{BMO}\) duality.https://www.zbmath.org/1456.420012021-04-16T16:22:00+00:00"Osękowski, Adam"https://www.zbmath.org/authors/?q=ai:osekowski.adamSummary: Let \(I\subset\mathbb{R}\) be an interval and let \(f,\varphi\) be arbitrary elements of \(H^1(I)\) and \(\mathrm{BMO}(I)\), respectively, with \(\int_I\varphi=0\). The paper contains the proof of the estimate
\[
\int_If\varphi\leq\sqrt{2}\Vert f\Vert_{H^1(I)}\Vert\varphi\Vert_{\mathrm{BMO}(I)},
\]
and it is shown that \(\sqrt{2}\) cannot be replaced by a smaller universal constant. The argument rests on the existence of a special function enjoying appropriate size and concavity requirements.Bogolyubov's theorem for a controlled system related to a variational inequality.https://www.zbmath.org/1456.490152021-04-16T16:22:00+00:00"Tolstonogov, A. A."https://www.zbmath.org/authors/?q=ai:tolstonogov.alexander-aBook review of: B. S. Mordukhovich, Variational analysis and applications.https://www.zbmath.org/1456.000182021-04-16T16:22:00+00:00"Khan, Akhtar A."https://www.zbmath.org/authors/?q=ai:khan.akhtar-aliReview of [Zbl 1402.49003].Main features of non-linear optimization. 2nd revised and expanded edition.https://www.zbmath.org/1456.900022021-04-16T16:22:00+00:00"Stein, Oliver"https://www.zbmath.org/authors/?q=ai:stein.oliver.2|stein.oliver-t|stein.oliver.1Publisher's description: Das vorliegende Lehrbuch ist eine Einführung in die nichtlineare Optimierung, die mathematische Sachverhalte einerseits stringent behandelt, sie aber andererseits auch sehr ausführlich motiviert und mit 42 Abbildungen illustriert. Das Buch richtet sich daher nicht nur an Mathematiker, sondern auch an Natur-, Ingenieur- und Wirtschaftswissenschaftler, die mathematisch fundierte Verfahren in ihrem Gebiet verstehen und anwenden möchten.
Mit etwas mehr als zweihundert Seiten stellt das Buch genügend Auswahlmöglichkeiten zur Verfügung, um es als Grundlage für unterschiedlich angelegte Vorlesungen zur nichtlinearen Optimierung zu verwenden. Viele geometrische Ansätze für das Verständnis sowohl von Optimalitätsbedingungen als auch von numerischen Verfahren setzen dabei einen neuen Akzent, der den Bestand der bisherigen Lehrbücher zur Optimierung bereichert. Dies betrifft insbesondere die ausführliche Behandlung der Probleme, die durch verschiedene funktionale Beschreibungen derselben Geometrie der Menge zulässiger Punkte entstehen können, und die dadurch motivierte Einführung von Constraint Qualifications für die Herleitung ableitungsbasierter Optimalitätsbedingungen.
Die vorliegende zweite Auflage wurde überarbeitet und um einige Passagen ergänzt.
See the review of the first edition in [Zbl 1386.90004].Unconditional convergence for discretizations of dynamical optimal transport.https://www.zbmath.org/1456.650462021-04-16T16:22:00+00:00"Lavenant, Hugo"https://www.zbmath.org/authors/?q=ai:lavenant.hugoSummary: The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or computational fluid dynamics formulation, amounts to writing the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several discretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space.
In this paper, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional problem for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by \textit{P. Gladbach} et al. [SIAM J. Math. Anal. 52, No. 3, 2759--2802 (2020; Zbl 1447.49062)], as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien, and Solomon, fit in this framework.Extended Newton-type method and its convergence analysis for nonsmooth generalized equations.https://www.zbmath.org/1456.490242021-04-16T16:22:00+00:00"Rashid, M. H."https://www.zbmath.org/authors/?q=ai:rashid.malik-h-m|rashid.mohammed-harunor|rashid.muhammad-hSummary: Let \(X\) and \(Y\) be Banach spaces and \(\Omega \) be an open subset of \(X\). Suppose that \(f:{\Omega \subseteq X}\rightarrow {Y}\) is a single-valued function which is nonsmooth and it has point based approximations on \(\Omega \) and \(F:X\rightrightarrows 2^Y\) is a set-valued mapping with closed graph. An extended Newton-type method is introduced in the present paper for solving the nonsmooth generalized equation \(0\in {f(x)+F(x)}\). Semilocal and local convergence of this method are analyzed based on the concept of point-based approximation.Variational and optimal control representations of conditioned and driven processes.https://www.zbmath.org/1456.930072021-04-16T16:22:00+00:00"Chetrite, Raphaël"https://www.zbmath.org/authors/?q=ai:chetrite.raphael"Touchette, Hugo"https://www.zbmath.org/authors/?q=ai:touchette.hugoA posteriori modeling error estimates in the optimization of two-scale elastic composite materials.https://www.zbmath.org/1456.651592021-04-16T16:22:00+00:00"Conti, Sergio"https://www.zbmath.org/authors/?q=ai:conti.sergio"Geihe, Benedict"https://www.zbmath.org/authors/?q=ai:geihe.benedict"Lenz, Martin"https://www.zbmath.org/authors/?q=ai:lenz.martin"Rumpf, Martin"https://www.zbmath.org/authors/?q=ai:rumpf.martinSummary: The a posteriori analysis of the discretization error and the modeling error is studied for a compliance cost functional in the context of the optimization of composite elastic materials and a two-scale linearized elasticity model. A mechanically simple, parametrized microscopic supporting structure is chosen and the parameters describing the structure are determined minimizing the compliance objective. An a posteriori error estimate is derived which includes the modeling error caused by the replacement of a nested laminate microstructure by this considerably simpler microstructure. Indeed, nested laminates are known to realize the minimal compliance and provide a benchmark for the quality of the microstructures. To estimate the local difference in the compliance functional the dual weighted residual approach is used. Different numerical experiments show that the resulting adaptive scheme leads to simple parametrized microscopic supporting structures that can compete with the optimal nested laminate construction. The derived a posteriori error indicators allow to verify that the suggested simplified microstructures achieve the optimal value of the compliance up to a few percent. Furthermore, it is shown how discretization error and modeling error can be balanced by choosing an optimal level of grid refinement. Our two scale results with a single scale microstructure can provide guidance towards the design of a producible macroscopic fine scale pattern.Maximizing the total population with logistic growth in a patchy environment.https://www.zbmath.org/1456.490412021-04-16T16:22:00+00:00"Nagahara, Kentaro"https://www.zbmath.org/authors/?q=ai:nagahara.kentaro"Lou, Yuan"https://www.zbmath.org/authors/?q=ai:lou.yuan"Yanagida, Eiji"https://www.zbmath.org/authors/?q=ai:yanagida.eijiSummary: This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the population of a single species with logistic growth residing in a patchy environment and study the effects of dispersal and spatial heterogeneity of patches on the total population at equilibrium. Our objective is to maximize the total population by redistributing the resources among the patches under the constraint that the total amount of resources is limited. It is shown that the global maximizer can be characterized for any number of patches when the diffusion rate is either sufficiently small or large. To show this, we compute the first variation of the total population with respect to resources in the two patches case. In the case of three or more patches, we compute the asymptotic expansion of all patches by using the Taylor expansion with respect to the diffusion rate. To characterize the shape of the global maximizer, we use a recurrence relation to determine all coefficients of all patches.Hamilton-Jacobi equations for neutral-type systems: inequalities for directional derivatives of minimax solutions.https://www.zbmath.org/1456.490092021-04-16T16:22:00+00:00"Lukoyanov, Nikolai Yu."https://www.zbmath.org/authors/?q=ai:lukoyanov.nikolai-yu"Plaksin, Anton R."https://www.zbmath.org/authors/?q=ai:plaksin.anton-romanovichThe authors consider the Hamilton-Jacobi (HJ) equations for differential systems of neutral type. They continue a series of study of the dynamical system from the point of view of directional derivatives in optimization context. They give an infinitesimal criterion of the minimax (generalized) solutions for the HJ equations arising in optimal control problems and differential games, using a suitable definition and techniques of directional derivatives.
Reviewer: A. Omrane (Cayenne)Regularity of area minimizing currents mod \(p\).https://www.zbmath.org/1456.490322021-04-16T16:22:00+00:00"De Lellis, Camillo"https://www.zbmath.org/authors/?q=ai:de-lellis.camillo"Hirsch, Jonas"https://www.zbmath.org/authors/?q=ai:hirsch.jonas"Marchese, Andrea"https://www.zbmath.org/authors/?q=ai:marchese.andrea"Stuvard, Salvatore"https://www.zbmath.org/authors/?q=ai:stuvard.salvatoreRegularity of area minimizing currents mod(p) have had an important role in several classical problems of geometric measure theory and mathematical physics. Several authors studied the regularity of area [\textit{C. De Lellis} and \textit{E.N. Spadaro}, Ann. of Math. (2), 183, No. 2, 499--575 (2016; Zbl 1345.49052); \textit{C. De Lellis} and \textit{E.N. Spadaro}, Geom. Funct. Anal., 24, No. 6, 1831--1884 (2014; Zbl 1307.49043); \textit{C. De Lellis} and \textit{E. N. Spadaro}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 4, 1239--1269 (2015; Zbl 1343.49073); \textit{T. De Pauw} and \textit{R. Hardt}, Am. J. Math., 134, No. 1, 1--69 (2012; Zbl 1252.49070); \textit{T. De Pauw} and \textit{R. Hardt}, J. Math. Anal. Appl., 418, No. 2, 1047--1061 (2014; Zbl 1347.49073); \textit{L. Simon}, J. Diff. Geom. 38, No. 3, 585--652 (1993; Zbl 0819.53029); \textit{L. Spolaor}, Adv. Math., 350, 747--815 (2019; Zbl 1440.49048)].
The principal objective in this paper is to establish a first general partial regularity theorem for area minimizing currents mod(p), for every p, in any dimension and codimension. More precisely, the authors prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current mod(p) cannot be larger than \(m-1\).
Reviewer: Lakehal Belarbi (Mostaganem)On the minimax inequality of Brézis-Nirenberg-Stampacchia.https://www.zbmath.org/1456.470182021-04-16T16:22:00+00:00"Park, Sehie"https://www.zbmath.org/authors/?q=ai:park.sehieSummary: Since the celebrated Knaster-Kuratowski-Mazurkiewicz (simply KKM) theorem appeared in [\textit{B. Knaster} et al., Fundam. Math. 14, 132--137 (1929; JFM 55.0972.01)], a large number of its generalizations and modifications followed. Based on a lemma which generalizes the KKM theorem, Brézis-Nirenberg-Stampacchia (simply BNS) [\textit{H. Brézis} et al., Boll. Unione Mat. Ital., IV. Ser. 6, 293--300 (1972; Zbl 0264.49013); ibid. (9) 1, No. 2, 257--264 (2008; Zbl 1225.49014)] obtained a slightly more general result than the 1961 KKM lemma of \textit{K. Fan} [Math. Ann. 142, 305--310 (1961; Zbl 0093.36701)]. Then they obtained a generalization of the 1972 minimax inequality of \textit{K. Fan} [in: Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 103--113 (1972; Zbl 0302.49019)] and some of its applications.
In the present article, we show that one of our previous KKM type theorems for abstract convex spaces [the author, ``On the von Neumann type minimax theorems in abstract convex spaces'', J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 50, No. 2, 1--24 (2011), \url{http://parksehie.com/datafiles/research/2011L=JNAS.pdf}] can be applied to generalize the KKM type lemma and the minimax inequality due to Brézis-Nirenberg-Stampacchia [loc.\,cit.]. Using our results, we can correct certain results of Brézis-Nirenberg-Stampacchia [loc.\,cit.].Bounds on optimal transport maps onto log-concave measures.https://www.zbmath.org/1456.490362021-04-16T16:22:00+00:00"Colombo, Maria"https://www.zbmath.org/authors/?q=ai:colombo.maria"Fathi, Max"https://www.zbmath.org/authors/?q=ai:fathi.maxIn this paper the authors provide quantitative bounds on the regularity of transport maps sending a standard Gaussian distribution onto a \(\log\)-concave probability measures on \(\mathbb{R}^d\).
Caffarelli contraction theorem states that, when the target measure \(\mu\) is uniformly \(\log\)-concave, i.e. \(\mu=e^{-V} \mathrm{d} x\) with \(D^2V \ge \alpha\), the optimal transport map is \(\alpha^{-1/2}\)-Lipschitz continuous.
It is thus evident that such regularity degenerates if \(\mu\) is only \(\log\)-concave, i.e. \(\mu=e^{-V} \mathrm{d} x\) with \(D^2V \ge 0\) only. The first main result deals with this case:
the authors prove the quadratic growth at infinity, i.e.
Theorem.
Let \(\mu\) be a centered, isotropic, \(\log\)-concave probability measure on \(\mathbb{R}^d\). Then there exists a universal numerical constant \(C\) such that the Brenier map sending the standard Gaussiandistribution onto \(\mu\) satisfies
\[|T(x)| \le C(d+|x|^2).\]
If instead we assume that \(\mu\) is centered and satisfies a Gaussian concentration property with constant \(\beta\), then
\[|T(x)| \le 12\beta^{-1/2}(17d+|x|^2)^{1/2}.\]
As the authors noted, the left hand side behaves like \( d^{1/2}\), while the right hand side scales like \(d\), thus the above estimates are a bit off-average.
The second result proves some a priori regularity estimates on derivatives of \(T\). The strategy is to revisit Kolesnikov's proof of Sobolev estimates in the uniformly \(\log\)-concave case, to allow for non-uniform lower bounds on the Hessian of the potential. More precisely, the authors obtain the following bound:
Theorem.
Let \(T = \nabla\varphi\) be the Brenier map sending the standard Gaussian measure onto \(\mu = e^{-V} \mathrm{d} x\). Assume that \(\mu\) is centered, isotropic, and that for all \(x\in \mathbb{R}^d\)
\[ c_1 Id \ge D^2 V(x) \ge \frac{c_2}{d+|x|}Id \]
for some \(c_1,c_2>0\). Then
\[ \Big\| \frac{\partial_{ee}^2\varphi}{\sqrt{d+|x|^2}} \Big\|_{p+2,\gamma} \le \frac{C}{c_2} \Big(1+p\frac{\sqrt{c_1}}{4\sqrt{d}}\Big). \]
That is, the authors obtain a bound of the form
\[\partial_{ee}^2\varphi \le Cr \sqrt{d+|x|^2}\]
on the complement of a set with very small Gaussian measure.
Finally, in the spirit of the Caffarelli contraction theorem, the authors obtain a bound on the growth of the eigenvalues in \(L^\infty\):
Theorem.
Let \(T = \nabla\varphi\) be the Brenier map sending the standard Gaussian measure onto \(\mu = e^{-V} \mathrm{d} x\). Assume that \(\mu\) is centered, isotropic, and that for all \(x\in \mathbb{R}^d\)
\[ c_1 Id \ge D^2 V(x) \ge \frac{c_2}{d+|x|}Id \]
for some \(c_1,c_2>0\). Then
\[\|\nabla T(x)\|_{op} \le \max\Big(C\frac{c_1^2}{c_2^2},1\Big)(d+|x|^2)^2. \]
If moreover
\[c_3\ge |\nabla V(x)|,\]
then
\[\|\nabla T(x)\|_{op} \le \max\Big(C\frac{c_1^2}{c_2^2},1\Big)(d^{4/3}+|x|^2). \]
Reviewer: Xin Yang Lu (Thunder Bay)Semidefinite relaxation of multimarginal optimal transport for strictly correlated electrons in second quantization.https://www.zbmath.org/1456.490372021-04-16T16:22:00+00:00"Khoo, Yuehaw"https://www.zbmath.org/authors/?q=ai:khoo.yuehaw"Lin, Lin"https://www.zbmath.org/authors/?q=ai:lin.lin"Lindsey, Michael"https://www.zbmath.org/authors/?q=ai:lindsey.michael"Ying, Lexing"https://www.zbmath.org/authors/?q=ai:ying.lexingConvergence and quasi-optimality of \(L^2\)-norms based an adaptive finite element method for nonlinear optimal control problems.https://www.zbmath.org/1456.490062021-04-16T16:22:00+00:00"Lu, Zuliang"https://www.zbmath.org/authors/?q=ai:lu.zuliang"Huang, Fei"https://www.zbmath.org/authors/?q=ai:huang.fei"Wu, Xiankui"https://www.zbmath.org/authors/?q=ai:wu.xiankui"Li, Lin"https://www.zbmath.org/authors/?q=ai:li.lin.2|li.lin|li.lin.1"Liu, Shang"https://www.zbmath.org/authors/?q=ai:liu.shangSummary: This paper aims at investigating the convergence and quasi-optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by \(L^2\)-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.A non-iterative reconstruction method for an inverse problem modeled by a Stokes-Brinkmann equations.https://www.zbmath.org/1456.650942021-04-16T16:22:00+00:00"Hassine, Maatoug"https://www.zbmath.org/authors/?q=ai:hassine.maatoug"Hrizi, Mourad"https://www.zbmath.org/authors/?q=ai:hrizi.mourad"Malek, Rakia"https://www.zbmath.org/authors/?q=ai:malek.rakiaSummary: : This work is concerned with a geometric inverse problem in fluid mechanics. The aim is to reconstruct an unknown obstacle immersed in a Newtonian and incompressible fluid flow from internal data. We assume that the fluid motion is governed by the Stokes-Brinkmann equations in the two dimensional case. We propose a simple and efficient reconstruction method based on the topological sensitivity concept. The geometric inverse problem is reformulated as a topology optimization one minimizing a least-square functional. The existence and stability of the optimization problem solution are discussed. A topological sensitivity analysis is derived with the help of a straightforward approach based on a penalization technique without using the classical truncation method. The theoretical results are exploited for building a non-iterative reconstruction algorithm. The unknown obstacle is reconstructed using a level-set curve of the topological gradient. The accuracy and the robustness of the proposed method are justified by some numerical examples.Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints.https://www.zbmath.org/1456.352362021-04-16T16:22:00+00:00"Safdari, Mohammad"https://www.zbmath.org/authors/?q=ai:safdari.mohammadSummary: We prove the optimal \(W^{2, \infty}\) regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be \(C^1\) or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal \(W^{2, \infty}\) regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.An extremal problem in uniform distribution theory.https://www.zbmath.org/1456.111332021-04-16T16:22:00+00:00"Baláž, Vladimír"https://www.zbmath.org/authors/?q=ai:balaz.vladimir"Iacò, Maria Rita"https://www.zbmath.org/authors/?q=ai:iaco.maria-rita"Strauch, Oto"https://www.zbmath.org/authors/?q=ai:strauch.oto"Thonhauser, Stefan"https://www.zbmath.org/authors/?q=ai:thonhauser.stefan"Tichy, Robert F."https://www.zbmath.org/authors/?q=ai:tichy.robert-franzLet $x_n$ and $y_n$, $n= 1,2,\dots$ be uniformly distributed sequences in the unit interval and $F$ be a given continuous function on $[0,1]^2$. A classical problem is the study of extremal limits of the form $\frac1N\sum_{n=1}^NF(x_n, y_n)$, where $N\to\infty$. It is equivalent to find optimal bounds for Riemann-Stieltjes integrals of the form $\int_0^1\int_0^1 F(x, y)dC(x, y)$, where $C$ is the asymptotic distribution function of the sequence $(x_n, y_n)$ and it is usually referred to as copula. As pointed out in [the reviewer and \textit{O. Strauch}, Unif. Distrib. Theory 6, No. 1, 101--125 (2011; Zbl 1313.11089)] the solution of this problem depends on the sign of the partial derivative $\frac{\partial^2F(x,y)}{\partial x\partial y}$.
The main result of the paper is the following: Let $0< x_1< x_2<1$ and
\[
F(x, y) =\begin{cases}
F_1(x, y) &\text{if }x\in (0, x_1),\frac{\partial^2F_1(x,y)}{\partial x\partial y}>0,\\
F_2(x, y) &\text{if }x\in (x_1, x_2),\frac{\partial^2F_2(x,y)}{\partial x\partial y}<0,\\
F_3(x, y) &\text{if }x\in (x_2,1),\frac{\partial^2F_3(x,y)}{\partial x\partial y}>0.\end{cases}
\]
Then the copula maximizing $\int_0^1\int_0^1 F(x, y)d\tilde{C}(x, y)$ has the form
\[
C(x, y) =\begin{cases}
\min(x, h_1(y))&\text{if }x\in [0, x_1],\\
\max(x+h_2(y)-x_2, h_1(y)) &\text{if }x\in [x_1, x_2],\\
\min(x-x_2+h_2(y), y)&\text{if }x\in [x_2,1],\end{cases}
\]
where $h_1(y) =C(x_1, y)$, $h_2(y) =C(x_2, y)$ and $(h_1, h_2)$ satisfy the Euler-Lagrange differential equations. The authors also discuss connections of extremal limits of couples to the theory of optimal transport. In the final section, they solve the example $F(x, y) = \sin(\pi(x+y))$ and relate this problem to combinatorial optimization based on the work of \textit{L. Uckelmann} [in: Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 275--281 (1997; Zbl 0907.60022)].
Reviewer: Jana Fialová (Trnava)Numerical analysis of a family of optimal distributed control problems governed by an elliptic variational inequality.https://www.zbmath.org/1456.490122021-04-16T16:22:00+00:00"Olguin, Mariela C."https://www.zbmath.org/authors/?q=ai:olguin.mariela-c"Tarzia, Domingo A."https://www.zbmath.org/authors/?q=ai:tarzia.domingo-albertoSummary: The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter \(\alpha > 0\)) is obtained by considering the finite element method with parameter \(h>0\). A commutative diagram for two continuous optimal control problems and the corresponding two discrete optimal control problems is obtained when \(h \rightarrow 0\), \(\alpha \rightarrow \infty\), and \((h,\alpha)\rightarrow (0,\infty)\).Spectral element methods a priori and a posteriori error estimates for penalized unilateral obstacle problem.https://www.zbmath.org/1456.651092021-04-16T16:22:00+00:00"Djeridi, Bochra"https://www.zbmath.org/authors/?q=ai:djeridi.bochra"Ghanem, Radouen"https://www.zbmath.org/authors/?q=ai:ghanem.radouen"Sissaoui, Hocine"https://www.zbmath.org/authors/?q=ai:sissaoui.hocineSummary: The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre-Gauss-Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.Iterative scheme of strongly nonlinear general nonconvex variational inequalities problem.https://www.zbmath.org/1456.470312021-04-16T16:22:00+00:00"Sudsukh, Chanan"https://www.zbmath.org/authors/?q=ai:sudsukh.chanan"Inchan, Issara"https://www.zbmath.org/authors/?q=ai:inchan.issaraSummary: In this work, we suggest and analyze an iterative scheme for solving strongly nonlinear general nonconvex variational inequalities by using the projection technique and the Wiener-Hopf technique. We prove that strong convergence of the iterative scheme to the solution of the strongly nonlinear general nonconvex variational inequalities requires the modified mapping $T$ to be Lipschitz continuous but not strongly monotone. Our result can be viewed as an improvement of the result of \textit{E. Al-Shemas} [``General nonconvex Wiener-Hopf equations and general nonconvex variational inequalities'', J. Math. Sci., Adv. Appl. 19, No. 1, 1--11 (2013), \url{http://scientificadvances.co.in/admin/img_data/602/images/[1]%20JMSAA%207100121122%20Enab%20Al-Shemas%20[1-11].pdf}].Pattern formation for a local/nonlocal interaction functional arising in colloidal systems.https://www.zbmath.org/1456.822102021-04-16T16:22:00+00:00"Daneri, Sara"https://www.zbmath.org/authors/?q=ai:daneri.sara"Runa, Eris"https://www.zbmath.org/authors/?q=ai:runa.erisThe ability of matter to arrange itself in periodic structures
is often referred to as spontaneous pattern formation. This phenomenon is of fundamental
importance in science, technology, engineering, and mathematics, and it
is often caused by the interaction between local attractive and nonlocal repulsive
forces.
In this paper, the authors study pattern formation for a physical local/nonlocal interaction
functional where the local attractive term is given by the 1-perimeter and the nonlocal repulsive
term is the Yukawa (or screened Coulomb) potential [\textit{B. Derjaguin} and \textit{L. Landau}, ``Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes'', Acta Phys. Chem. USSR, 14, 633--662 (1941)]. This model is physically interesting as it is
the \(\Gamma\)-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal
systems. The authors prove that in a suitable regime minimizers are periodic stripes in any
space dimension.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Lower semicontinuity for functionals defined on piecewise rigid functions and on \(GSBD\).https://www.zbmath.org/1456.490132021-04-16T16:22:00+00:00"Friedrich, Manuel"https://www.zbmath.org/authors/?q=ai:friedrich.manuel"Perugini, Matteo"https://www.zbmath.org/authors/?q=ai:perugini.matteo"Solombrino, Francesco"https://www.zbmath.org/authors/?q=ai:solombrino.francescoSummary: In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called \(BD\)-ellipticity, which is in the spirit of \(BV\)-ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its \(BV\) analog. We further provide a sufficient condition implying \(BD\)-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of \(GSBD^p\) functions.Stochastic conditional gradient++: (Non)convex minimization and continuous submodular maximization.https://www.zbmath.org/1456.490232021-04-16T16:22:00+00:00"Hassani, Hamed"https://www.zbmath.org/authors/?q=ai:hassani.hamed"Karbasi, Amin"https://www.zbmath.org/authors/?q=ai:karbasi.amin"Mokhtari, Aryan"https://www.zbmath.org/authors/?q=ai:mokhtari.aryan"Shen, Zebang"https://www.zbmath.org/authors/?q=ai:shen.zebangRobust preconditioners for multiple saddle point problems and applications to optimal control problems.https://www.zbmath.org/1456.490042021-04-16T16:22:00+00:00"Beigl, Alexander"https://www.zbmath.org/authors/?q=ai:beigl.alexander"Sogn, Jarle"https://www.zbmath.org/authors/?q=ai:sogn.jarle"Zulehner, Walter"https://www.zbmath.org/authors/?q=ai:zulehner.walterThe Aronsson equation, Lyapunov functions, and local Lipschitz regularity of the minimum time function.https://www.zbmath.org/1456.490222021-04-16T16:22:00+00:00"Soravia, Pierpaolo"https://www.zbmath.org/authors/?q=ai:soravia.pierpaoloSummary: We define and study \(C^1\)-solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that \(C^1\)-solutions are absolutely minimizing functions. We discuss how \(C^1\)-supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct \(C^1\)-solutions (even classical) explicitly.An adaptive correction approach for tensor completion.https://www.zbmath.org/1456.901042021-04-16T16:22:00+00:00"Bai, Minru"https://www.zbmath.org/authors/?q=ai:bai.minru"Zhang, Xiongjun"https://www.zbmath.org/authors/?q=ai:zhang.xiongjun"Ni, Guyan"https://www.zbmath.org/authors/?q=ai:ni.guyan"Cui, Chunfeng"https://www.zbmath.org/authors/?q=ai:cui.chunfengThe isotropic Cosserat shell model including terms up to \(O(h^5)\). II: Existence of minimizers.https://www.zbmath.org/1456.490402021-04-16T16:22:00+00:00"Ghiba, Ionel-Dumitrel"https://www.zbmath.org/authors/?q=ai:ghiba.ionel-dumitrel"Bîrsan, Mircea"https://www.zbmath.org/authors/?q=ai:birsan.mircea"Lewintan, Peter"https://www.zbmath.org/authors/?q=ai:lewintan.peter"Neff, Patrizio"https://www.zbmath.org/authors/?q=ai:neff.patrizioSummary: We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including \(O(h^5)\) terms. The form of the energy allows us to show the coercivity for terms up to order \(O(h^5)\) and the convexity of the energy. Secondly, we consider only that part of the energy including \(O(h^3)\) terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including \(O(h^5)\) the conditions on the thickness \(h\) are those considered in the modelling process and they are independent of the constitutive parameter, in the \(O(h^3)\)-case the coercivity is proven under some more restrictive conditions on the thickness \(h\).
For Part I, see [the authors, ibid. 142, No. 2, 201--262 (2020; Zbl 1456.74116)].On the existence and uniqueness of an inverse problem in epidemiology.https://www.zbmath.org/1456.490392021-04-16T16:22:00+00:00"Coronel, Aníbal"https://www.zbmath.org/authors/?q=ai:coronel.anibal"Friz, Luis"https://www.zbmath.org/authors/?q=ai:friz.luis"Hess, Ian"https://www.zbmath.org/authors/?q=ai:hess.ian"Zegarra, María"https://www.zbmath.org/authors/?q=ai:zegarra.mariaSummary: In this paper, we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising from the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction-diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the final time. The unknowns of the inverse problem are the coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. Moreover, we establish the uniqueness up an additive constant of the identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations.On uniform distributions on metric compacta.https://www.zbmath.org/1456.490332021-04-16T16:22:00+00:00"Ivanov, A. V."https://www.zbmath.org/authors/?q=ai:ivanov.aleksandr-vladimirovich|ivanov.aleksei-valerevich|ivanov.anton-valerevich|ivanov.aleksandr-vladimirovich.1|ivanov.aleksei-vladimirovich|ivanov.aleksei-valerevich.1|ivanov.aleksandr-valentinovich|ivanov.alexey-v|ivanov.andrey-vSummary: We introduce the notion of uniform distribution on a metric compactum. The desired distribution is defined as the limit of a sequence of the classical uniform distributions on finite sets which are uniformly distributed on the compactum in the geometric sense. We show that a uniform distribution exists on the metrically homogeneous compacta and the canonically closed subsets of a Euclidean space whose boundary has Lebesgue measure zero. If a compactum (satisfying some metric constraints) admits a uniform distribution then so does its every canonically closed subset that has zero uniform measure of the boundary. We prove that compacta, admitting a uniform distribution, are dimensionally homogeneous in the sense of box-dimension.A new low-cost double projection method for solving variational inequalities.https://www.zbmath.org/1456.651902021-04-16T16:22:00+00:00"Gibali, Aviv"https://www.zbmath.org/authors/?q=ai:gibali.aviv"Thong, Duong Viet"https://www.zbmath.org/authors/?q=ai:duong-viet-thong.Summary: In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of [\textit{Y. Censor} et al., J. Optim. Theory Appl. 148, No. 2, 318--335 (2011; Zbl 1229.58018)] and Popov's method. The proposed scheme combines some of the advantages of the methods mentioned above, first it requires only one orthogonal projection onto the feasible set of the problem while the next computation has a closed formula. Second, only one mapping evaluation is required per each iteration and there is also a usage of an adaptive step size rule that avoids the need to know the Lipschitz constant of the associated mapping. We present two convergence theorems of the proposed method, weak convergence result which requires pseudomonotonicity, Lipschitz and sequentially weakly continuity of the associated mapping and strong convergence theorem with rate of convergence which requires Lipschitz continuity and strongly pseudomonotone only. Primary numerical experiments and comparisons demonstrate the advantages and potential applicability of the new scheme.Existence of solutions for a class of nonlinear Choquard equations with critical growth.https://www.zbmath.org/1456.351862021-04-16T16:22:00+00:00"Ao, Yong"https://www.zbmath.org/authors/?q=ai:ao.yongSummary: In this paper, we consider the nonlinear Choquard equation of the form
\[
- \Delta u+u = (I_\alpha * |u|^p)|u|^{p-2}u + |u|^{q-2}u \text{ in } \mathbb{R}^N,
\] where \(I_\alpha\) is a Riesz potential, \(\alpha \in (0,1) \), \(N \geq 4\), \(p=\frac{N+\alpha}{N-2}\), \(2<q<2^*\). We show the existence of a nontrivial solution of the equation. Moreover, we consider the corresponding minimizing problem and obtain a nonnegative minimizer.Mean field games. Cetraro, Italy, June 10--14, 2019. Lecture notes given at the summer school.https://www.zbmath.org/1456.490022021-04-16T16:22:00+00:00"Cardaliaguet, Pierre (ed.)"https://www.zbmath.org/authors/?q=ai:cardaliaguet.pierre"Porretta, Alessio (ed.)"https://www.zbmath.org/authors/?q=ai:porretta.alessioPublisher's description: This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School ``Mean Field Games'' held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio.
These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Cardaliaguet, Pierre; Porretta, Alessio}, An introduction to mean field game theory, 1-158 [Zbl 1457.91058]
\textit{Santambrogio, Filippo}, Lecture notes on variational mean field games, 159-201 [Zbl 1456.49031]
\textit{Delarue, François}, Master equation for finite state mean field games with additive common noise, 203-248 [Zbl 1457.91059]
\textit{Achdou, Yves; Laurière, Mathieu}, Mean field games and applications: numerical aspects, 249-307 [Zbl 1457.91057]The geometry of synchronization problems and learning group actions.https://www.zbmath.org/1456.051052021-04-16T16:22:00+00:00"Gao, Tingran"https://www.zbmath.org/authors/?q=ai:gao.tingran"Brodzki, Jacek"https://www.zbmath.org/authors/?q=ai:brodzki.jacek"Mukherjee, Sayan"https://www.zbmath.org/authors/?q=ai:mukherjee.sayanSummary: We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group \(G\) on connected graph \(\Gamma\) with a flat principal \(G\)-bundle over \(\Gamma\), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma\) into \(G\). We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal \(G\)-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham-Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions -- partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations -- and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities.https://www.zbmath.org/1456.490112021-04-16T16:22:00+00:00"Hieu, Vu Trung"https://www.zbmath.org/authors/?q=ai:hieu.vu-trungSummary: In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities (VVIs). We show that if the weak Pareto solution set of a monotone VVI is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone VVIs are raised at the end of the paper.Convex optimization. Introductory course.https://www.zbmath.org/1456.490012021-04-16T16:22:00+00:00"Moklyachuk, Mikhail"https://www.zbmath.org/authors/?q=ai:moklyachuk.mykhailo-pPublisher's description: This book provides easy access to the basic principles and methods for solving constrained and unconstrained convex optimization problems. Included are sections that cover: basic methods for solving constrained and unconstrained optimization problems with differentiable objective functions; convex sets and their properties; convex functions and their properties and generalizations; and basic principles of sub-differential calculus and convex programming problems. The book provides detailed proofs for most of the results presented in the book and also includes many figures and exercises for a better understanding of the material. Exercises are given at the end of each chapter, with solutions and hints to selected exercises given at the end of the book. Undergraduate and graduate students, researchers in different disciplines, as well as practitioners will all benefit from this accessible approach to convex optimization methods.Partial smoothness of the numerical radius at matrices whose fields of values are disks.https://www.zbmath.org/1456.490172021-04-16T16:22:00+00:00"Lewis, A. S."https://www.zbmath.org/authors/?q=ai:lewis.adrian-s"Overton, M. L."https://www.zbmath.org/authors/?q=ai:overton.michael-lImplicit parametrizations and applications in optimization and control.https://www.zbmath.org/1456.490082021-04-16T16:22:00+00:00"Tiba, Dan"https://www.zbmath.org/authors/?q=ai:tiba.danThe subject is the characterization (with numerical applications in mind) of the manifold \(V\) of solutions of the nonlinear system
\[
F_j(x_1, x_2, \dots, x_d) = 0 \quad (1 \le j \le l )\quad l \le d - 1 \tag{1}
\]
in the vicinity of \(x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \in V,\) under the Jacobian assumption
\[
\frac{\partial (F_1, F_2, \dots, F_l)}{\partial (x_1, x_2, \dots, x_l)} \ne 0
\quad \hbox{in} \ x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \, .
\]
The first step involves the underdetermined linear system
\[
v(x) \cdot \nabla F_j(x) = 0 \quad (1 \le j \le l)
\]
which is used to obtain bases \((v_1(x), v_2(x), \dots, v_{d - l}(x))\) for the tangent spaces of \(V.\) Next, the chain of differential equations
\begin{align*}
\frac{\partial y_1(t_1)}{\partial t_1} &= v_1(y_1(t_1)), y_1(0) = x^0
\cr
\frac{\partial y_2(t_1, t_2)}{\partial t_2} &= v_2(y_2(t_1, t_2)),\quad y_2(t_1, 0) = y(t_1)
\cr
& \hskip 2em \dots \dots \dots \dots
\cr
\frac{\partial y_{d - l}(t_1, t_2, \dots, t_{d - l})}{ \partial t_{d - l}}
&= v_{d - l}(y_{d - l}(t_1, t_2, \dots, t_{d - l})) \, ,
\cr
& \hskip 2.7em y_{d - l}(t_1, \dots , t_{d - l - 1}, 0) = y_{d - l - 1}(t_1, t_2, \dots , t_{d - l - 1})
\end{align*}
is set up, thus constructing a parametrization of \(V\) which may be considered as an explicit form of the implicit function theorem. The result is used to construct an algorithm for the solution of the problem of minimizing a function
\(g(x_0, x_2, \dots , x_d)\) subject to (1). Some generalizations are covered, such as the case where (1) includes inequality constraints and/or regularity is relaxed. In the last section the results are applied to the control problem
of minimizing \(l(x(0), x(1))\) among the trajectories of the system
\(x'(t) = f(t, x(t), u(t))\) subject to the state-control constraint \(h(x(t), u(t)) = 0.\) There are several numerical implementation of the algorithms and the author notes that computations can be carried out using standard Matlab routines.
Reviewer: Hector O. Fattorini (Los Angeles)Existence results for a super-Liouville equation on compact surfaces.https://www.zbmath.org/1456.580162021-04-16T16:22:00+00:00"Jevnikar, Aleks"https://www.zbmath.org/authors/?q=ai:jevnikar.aleks"Malchiodi, Andrea"https://www.zbmath.org/authors/?q=ai:malchiodi.andrea"Wu, Ruijun"https://www.zbmath.org/authors/?q=ai:wu.ruijunLet \((M,g)\) be a closed Riemannian surface endowed with a genus bigger than one and \(K_g\) stands for the Gauss curvature of \(M\). The authors consider the functional energy defined by
\[\displaystyle J_\rho(u,\psi)=\int_M\left(|\nabla_g u|^2+2K_gu+\exp(2u)+2\langle({D}_g-\rho\exp(u))\psi,\psi\rangle\right)dv_g,\]
such that \(u\in C^\infty(M)\), \(\rho\) is a positive parameter, \(\psi\) is a spinor field on \(M\), and \({D_g}\) represents the Dirac operator on spinors. The Euler-Lagrange equation associated to \(J_\rho\) is defined by
\[ (*):\ \Delta_gu=\exp(2u)+K_g-\rho\exp(u)|\psi|^2\text{ and }{D}_g\psi=\rho\exp(u)\psi.\]
Then the authors state that \((*)\) has a non-zero solution whenever zero and \(\rho\) do not belong to the spectrum of \({D}_{g_0}\) (where \(g_0\) is a conformal metric to \(g\)) and \(K_{g_0}=-1\) (Theorem 1.1). The proof is essentially based on looking for a critical point of \(J_\rho\).
Reviewer: Mohammed El Aïdi (Bogotá)A singular radial connection over \(\mathbb B^5\) minimizing the Yang-Mills energy.https://www.zbmath.org/1456.580152021-04-16T16:22:00+00:00"Petrache, Mircea"https://www.zbmath.org/authors/?q=ai:petrache.mirceaSummary: We prove that the pullback of the \(\mathrm{SU}(n)\)-soliton of Chern number \(c_2=1\) over \(\mathbb S^4\) via the radial projection \(\pi :\mathbb B^5{\setminus }\{0\}\to \mathbb S^4\) minimizes the Yang-Mills energy under a topologically fixed boundary trace constraint. In particular this shows that stationary Yang-Mills connections in high dimension can have singular sets of codimension 5.Depth-optimal distribution of drilling meterage under uncertainty.https://www.zbmath.org/1456.650412021-04-16T16:22:00+00:00"Gorbiychuk, M. I."https://www.zbmath.org/authors/?q=ai:gorbiychuk.m-i"Lazoriv, O. T."https://www.zbmath.org/authors/?q=ai:lazoriv.o-t"Zaiachuk, Y. I."https://www.zbmath.org/authors/?q=ai:zaiachuk.y-iSummary: The problem of depth-optimal distribution of drilling meterage is solved for the case where parameters of the optimality criterion are fuzzy numbers. This assumption made it possible to transform a deterministic nonlinear programming problem into a fuzzy nonlinear programming problem. Efficiency of the proposed method is confirmed by a simulation example.The Liouville theorem for \(p\)-harmonic functions and quasiminimizers with finite energy.https://www.zbmath.org/1456.350552021-04-16T16:22:00+00:00"Björn, Anders"https://www.zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://www.zbmath.org/authors/?q=ai:bjorn.jana"Shanmugalingam, Nageswari"https://www.zbmath.org/authors/?q=ai:shanmugalingam.nageswariSummary: We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite \(p\)th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global \(p\)-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line \(\mathbf{R}\), we characterize all locally doubling measures, supporting a local \(p\)-Poincaré inequality, for which there exist nonconstant quasiminimizers of finite \(p\)-energy, and show that a quasiminimizer is of finite \(p\)-energy if and only if it is bounded. As \(p\)-harmonic functions are quasiminimizers they are covered by these results.Dynamics and optimal control of a Monod-Haldane predator-prey system with mixed harvesting.https://www.zbmath.org/1456.921222021-04-16T16:22:00+00:00"Liu, Xinxin"https://www.zbmath.org/authors/?q=ai:liu.xinxin"Huang, Qingdao"https://www.zbmath.org/authors/?q=ai:huang.qingdaoLipschitz regularity results for a class of obstacle problems with nearly linear growth.https://www.zbmath.org/1456.490282021-04-16T16:22:00+00:00"Bertazzoni, Giacomo"https://www.zbmath.org/authors/?q=ai:bertazzoni.giacomo"Riccò, Samuele"https://www.zbmath.org/authors/?q=ai:ricco.samueleThis paper deals with the regularity of variational obstacle problems with nearly linear growth. The authors prove the Lipschitz continuity of solutions for a large class of problems and then the Lavrentiev phenomenon does not occur.
The main tool used here is a new higher differentiability result which reveals to be crucial because it allows to transform the constrained problem in an unconstrained one, by means a linearization procedure.
It is interesting to note that the same Sobolev regularity both for the gradient of the obstacle and for the coefficients is assumed.
Reviewer: Elvira Mascolo (Firenze)The isoperimetric problem of a complete Riemannian manifold with a finite number of \(C^0\)-asymptotically Schwarzschild ends.https://www.zbmath.org/1456.530302021-04-16T16:22:00+00:00"Muñoz Flores, Abraham Enrique"https://www.zbmath.org/authors/?q=ai:munoz-flores.abraham-enrique"Nardulli, Stefano"https://www.zbmath.org/authors/?q=ai:nardulli.stefanoSummary: We show existence and we give a geometric characterization of isoperimetric regions for large volumes, in \(C^2\)-locally asymptotically Euclidean Riemannian manifolds with a finite number of \(C^0\)-asymptotically Schwarzschild ends. This work extends previous results contained in
[\textit{M. Eichmair} and \textit{J. Metzger}, Invent. Math. 194, No. 3, 591--630 (2013; Zbl 1297.49078); J. Differ. Geom. 94, No. 1, 159--186 (2013; Zbl 1269.53071); \textit{S. Brendle} and \textit{M. Eichmair}, J. Differ. Geom. 94, No. 3, 387--407 (2013; Zbl 1282.53053)]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named \(C^0\)-strongly asymptotic Schwarzschild, extending results of [Zbl 1282.53053]. Such results are of interest in the field of mathematical general relativity.A new projection-type method for solving multi-valued mixed variational inequalities without monotonicity.https://www.zbmath.org/1456.490192021-04-16T16:22:00+00:00"Wang, Zhong-bao"https://www.zbmath.org/authors/?q=ai:wang.zhongbao"Chen, Zhang-you"https://www.zbmath.org/authors/?q=ai:chen.zhangyou"Xiao, Yi-bin"https://www.zbmath.org/authors/?q=ai:xiao.yibin"Zhang, Cong"https://www.zbmath.org/authors/?q=ai:zhang.congSummary: In this paper, a new projection-type algorithm for solving multi-valued mixed variational inequalities without monotonicity is presented. Under some suitable assumptions, it is showed that the sequence generated by the proposed algorithm converges globally to a solution of the multi-valued mixed variational inequality considered. The algorithm exploited in this paper is based on the generalized \(f\)-projection operator due to \textit{K. Wu} and \textit{N. Huang} [Bull. Aust. Math. Soc. 73, No. 2, 307--317 (2006; Zbl 1104.47053)] rather than the well-known resolvent operator. Preliminary computational experience is also reported. The results presented in this paper generalize and improve some known results given in the literature.Multilevel optimal transport: a fast approximation of Wasserstein-1 distances.https://www.zbmath.org/1456.490382021-04-16T16:22:00+00:00"Liu, Jialin"https://www.zbmath.org/authors/?q=ai:liu.jialin"Yin, Wotao"https://www.zbmath.org/authors/?q=ai:yin.wotao"Li, Wuchen"https://www.zbmath.org/authors/?q=ai:li.wuchen"Chow, Yat Tin"https://www.zbmath.org/authors/?q=ai:chow.yat-tinAn interior-point approach for solving risk-averse PDE-constrained optimization problems with coherent risk measures.https://www.zbmath.org/1456.490052021-04-16T16:22:00+00:00"Garreis, Sebastian"https://www.zbmath.org/authors/?q=ai:garreis.sebastian"Surowiec, Thomas M."https://www.zbmath.org/authors/?q=ai:surowiec.thomas"Ulbrich, Michael"https://www.zbmath.org/authors/?q=ai:ulbrich.michaelGeneralized subdifferentials of spectral functions over Euclidean Jordan algebras.https://www.zbmath.org/1456.490182021-04-16T16:22:00+00:00"Lourenço, Bruno F."https://www.zbmath.org/authors/?q=ai:lourenco.bruno-f"Takeda, Akiko"https://www.zbmath.org/authors/?q=ai:takeda.akikoOn the quantitative isoperimetric inequality in the plane.https://www.zbmath.org/1456.490342021-04-16T16:22:00+00:00"Bianchini, Chiara"https://www.zbmath.org/authors/?q=ai:bianchini.chiara"Croce, Gisella"https://www.zbmath.org/authors/?q=ai:croce.gisella"Henrot, Antoine"https://www.zbmath.org/authors/?q=ai:henrot.antoineSummary: In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set \(\Omega\), different from a ball, which minimizes the ratio \(\delta(\Omega)/\lambda^{2}(\Omega)\), where \(\delta\) is the isoperimetric deficit and \(\lambda\) the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.On the optimal control of rate-independent soft crawlers.https://www.zbmath.org/1456.490072021-04-16T16:22:00+00:00"Colombo, Giovanni"https://www.zbmath.org/authors/?q=ai:colombo.giovanni"Gidoni, Paolo"https://www.zbmath.org/authors/?q=ai:gidoni.paoloSummary: Existence of optimal solutions and necessary optimality conditions for a controlled version of Moreau's sweeping process are derived. The control is a measurable ingredient of the dynamics and the constraint set is a polyhedron. The novelty consists in considering time periodic trajectories, adding the requirement that the control has zero average, and considering an integral functional that lacks weak semicontinuity. A model coming from the locomotion of a soft-robotic crawler, that motivated our setting, is analysed in detail. In obtaining necessary conditions, a variant of the method of discrete approximations is used.