Foundations of Computational Mathematics最新文献

{"title":"Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions","authors":"Benjamin Dörich","doi":"10.1007/s10208-024-09639-w","DOIUrl":"https://doi.org/10.1007/s10208-024-09639-w","url":null,"abstract":"<p>In the present paper, we consider a class of quasilinear wave equations on a smooth, bounded domain. We discretize it in space with isoparametric finite elements and apply a semi-implicit Euler and midpoint rule as well as the exponential Euler and midpoint rule to obtain four fully discrete schemes. We derive rigorous error bounds of optimal order for the semi-discretization in space and the fully discrete methods in norms which are stronger than the classical <span>(H^1times L^2)</span> energy norm under weak CFL-type conditions. To confirm our theoretical findings, we also present numerical experiments.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"84 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139733608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of a Modified Regularity-Preserving Euler Scheme for Parabolic Semilinear SPDEs: Total Variation Error Bounds for the Numerical Approximation of the Invariant Distribution","authors":"Charles-Edouard Bréhier","doi":"10.1007/s10208-024-09644-z","DOIUrl":"https://doi.org/10.1007/s10208-024-09644-z","url":null,"abstract":"<p>We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. This new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein–Uhlenbeck process obtained when the nonlinearity is removed, for any time-step size. The weak order of convergence of the proposed method is shown to be equal to 1/2 in a general setting, like for the standard Euler scheme. A stronger weak approximation result is obtained when considering the approximation of a Gibbs invariant distribution, when the nonlinearity is a gradient: one obtains an approximation in total variation distance of order 1/2, which does not hold for the standard method. This is the first result of this type in the literature and this is the major and most original result of this article.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"98 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139710666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Phaseless Sampling on Square-Root Lattices","authors":"Philipp Grohs, Lukas Liehr","doi":"10.1007/s10208-024-09640-3","DOIUrl":"https://doi.org/10.1007/s10208-024-09640-3","url":null,"abstract":"<p>Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions <span>(g in {L^2({mathbb R}^d)})</span> and which sampling sets <span>(Lambda subseteq {mathbb R}^{2d})</span> is every <span>(f in {L^2({mathbb R}^d)})</span> uniquely determined (up to a global phase factor) by phaseless samples of the form </p><span>$$begin{aligned} |V_gf(Lambda )| = left{ |V_gf(lambda )|: lambda in Lambda right} , end{aligned}$$</span><p>where <span>(V_gf)</span> denotes the STFT of <i>f</i> with respect to <i>g</i>. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if <span>(Lambda )</span> is a lattice, i.e <span>(Lambda = A{mathbb Z}^{2d}, A in textrm{GL}(2d,{mathbb R}))</span>. Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form </p><span>$$begin{aligned} Lambda = A left( sqrt{{mathbb Z}} right) ^{2d}, sqrt{{mathbb Z}} = { pm sqrt{n}: n in {mathbb N}_0 }, end{aligned}$$</span><p>guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"38 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139710668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low-Dimensional Invariant Embeddings for Universal Geometric Learning","authors":"Nadav Dym, Steven J. Gortler","doi":"10.1007/s10208-024-09641-2","DOIUrl":"https://doi.org/10.1007/s10208-024-09641-2","url":null,"abstract":"<p>This paper studies separating invariants: mappings on <i>D</i>-dimensional domains which are invariant to an appropriate group action and which separate orbits. The motivation for this study comes from the usefulness of separating invariants in proving universality of equivariant neural network architectures. We observe that in several cases the cardinality of separating invariants proposed in the machine learning literature is much larger than the dimension <i>D</i>. As a result, the theoretical universal constructions based on these separating invariants are unrealistically large. Our goal in this paper is to resolve this issue. We show that when a continuous family of semi-algebraic separating invariants is available, separation can be obtained by randomly selecting <span>(2D+1 )</span> of these invariants. We apply this methodology to obtain an efficient scheme for computing separating invariants for several classical group actions which have been studied in the invariant learning literature. Examples include matrix multiplication actions on point clouds by permutations, rotations, and various other linear groups. Often the requirement of invariant separation is relaxed and only generic separation is required. In this case, we show that only <span>(D+1)</span> invariants are required. More importantly, generic invariants are often significantly easier to compute, as we illustrate by discussing generic and full separation for weighted graphs. Finally we outline an approach for proving that separating invariants can be constructed also when the random parameters have finite precision.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"25 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139710669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling","authors":"Eugen Bronasco","doi":"10.1007/s10208-023-09638-3","DOIUrl":"https://doi.org/10.1007/s10208-023-09638-3","url":null,"abstract":"<p>B-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-Series and S-Series. Precisely, we prove the relationship between the Grossman–Larson algebras over exotic and grafted forests and the corresponding duals to the Connes–Kreimer coalgebras and use it to study the natural composition laws on exotic S-Series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge–Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"6 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extremal Points and Sparse Optimization for Generalized Kantorovich–Rubinstein Norms","authors":"Marcello Carioni, José A. Iglesias, Daniel Walter","doi":"10.1007/s10208-023-09634-7","DOIUrl":"https://doi.org/10.1007/s10208-023-09634-7","url":null,"abstract":"<p>A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover, it enables the application of fully corrective generalized conditional gradient methods for their efficient solution. In this manuscript, this program is adapted to the minimization of a smooth convex fidelity term which is augmented with an unbalanced transport regularization term given in the form of a generalized Kantorovich–Rubinstein norm for Radon measures. More precisely, we show that the extremal points associated to the latter are given by all Dirac delta functionals supported in the spatial domain as well as certain dipoles, i.e., pairs of Diracs with the same mass but with different signs. Subsequently, this characterization is used to derive precise first-order optimality conditions as well as an efficient solution algorithm for which linear convergence is proved under natural assumptions. This behavior is also reflected in numerical examples for a model problem.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138571239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computational Complexity of Decomposing a Symmetric Matrix as a Sum of Positive Semidefinite and Diagonal Matrices","authors":"Levent Tunçel, Stephen A. Vavasis, Jingye Xu","doi":"10.1007/s10208-023-09637-4","DOIUrl":"https://doi.org/10.1007/s10208-023-09637-4","url":null,"abstract":"<p>We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive-semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis, and they have been studied for many decades. On the one hand, we prove that when the rank of the positive-semidefinite matrix in the decomposition is bounded above by an absolute constant, the problem can be solved in polynomial time. On the other hand, we prove that, in general, these problems as well as their certain approximation versions are all NP-hard. Finally, we prove that many of these low-rank decomposition problems are complete in the first-order theory of the reals, i.e., given any system of polynomial equations, we can write down a low-rank decomposition problem in polynomial time so that the original system has a solution iff our corresponding decomposition problem has a feasible solution of certain (lowest) rank.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"17 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138559325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Efficient Random Walks on Riemannian Manifolds","authors":"Simon Schwarz, Michael Herrmann, Anja Sturm, Max Wardetzky","doi":"10.1007/s10208-023-09635-6","DOIUrl":"https://doi.org/10.1007/s10208-023-09635-6","url":null,"abstract":"<p>According to a version of Donsker’s theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge in distribution to the correct Brownian motion as long as the geodesic equation is approximated up to second order. As a result, we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" 24","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138473494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fast Optimistic Gradient Descent Ascent (OGDA) Method in Continuous and Discrete Time","authors":"Radu Ioan Boţ, Ernö Robert Csetnek, Dang-Khoa Nguyen","doi":"10.1007/s10208-023-09636-5","DOIUrl":"https://doi.org/10.1007/s10208-023-09636-5","url":null,"abstract":"<p>In the framework of real Hilbert spaces, we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator <i>V</i>. The starting point of our investigations is a second-order dynamical system that combines a vanishing damping term with the time derivative of <i>V</i> along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates of order <span>(o left( frac{1}{tbeta (t)} right) )</span> for <span>(Vert V(z(t))Vert )</span>, where <span>(z(cdot ))</span> denotes the generated trajectory and <span>(beta (cdot ))</span> is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of <i>V</i>. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that the generated sequence of iterates <span>((z_k)_{k ge 0})</span> shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order <span>(o left( frac{1}{kbeta _k} right) )</span> for <span>(Vert V(z^k)Vert )</span> and the restricted gap function, where <span>((beta _k)_{k ge 0})</span> is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm, we show by additionally assuming that the operator <i>V</i> is Lipschitz continuous convergence rates of order <span>(o left( frac{1}{k} right) )</span> for <span>(Vert V(z^k)Vert )</span> and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition to these, we prove for both algorithms the convergence of the iterates to a zero of <i>V</i>. To our knowledge, our study exhibits the best-known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods designed to solve monotone equations governed by monotone and Lipschitz continuous operators.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"122 34","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138468746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Representation and Learning of Monotone Triangular Transport Maps","authors":"Ricardo Baptista, Youssef Marzouk, Olivier Zahm","doi":"10.1007/s10208-023-09630-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09630-x","url":null,"abstract":"<p>Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement—are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"65 8","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}