Foundations of Computational Mathematics最新文献_第8页

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields 分片常数和分片平分向量场和张量场的离散亥姆霍兹分解

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-03-01 DOI: 10.1007/s10208-024-09642-1

引用次数: 0

Polynomial Factorization Over Henselian Fields 汉塞尔域上的多项式因式分解

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-21 DOI: 10.1007/s10208-024-09646-x

Maria Alberich-Carramiñana, Jordi Guàrdia, Enric Nart, Adrien Poteaux, Joaquim Roé, Martin Weimann

引用次数: 0

Stable Spectral Methods for Time-Dependent Problems and the Preservation of Structure 时变问题的稳定谱方法与结构保持

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-15 DOI: 10.1007/s10208-024-09647-w

Arieh Iserles

{"title":"Stable Spectral Methods for Time-Dependent Problems and the Preservation of Structure","authors":"Arieh Iserles","doi":"10.1007/s10208-024-09647-w","DOIUrl":"https://doi.org/10.1007/s10208-024-09647-w","url":null,"abstract":"This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any (text {C}^1(a,b)) weight function such that (w(a)=w(b)=0), we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case (a=-infty ), (b=+infty ), only few powers of that matrix are bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function (x^alpha textrm{e}^{-x}) for (x>0) and (alpha >0) and the ultraspherical weight function ((1-x^2)^alpha ), (xin (-1,1)), (alpha >0), and establish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sensitive to the choice of (alpha ), and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"240 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750190","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}

引用次数: 0

On the Existence of Monge Maps for the Gromov–Wasserstein Problem 论格罗莫夫-瓦瑟斯坦问题的蒙格映射的存在性

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-15 DOI: 10.1007/s10208-024-09643-0

Théo Dumont, Théo Lacombe, François-Xavier Vialard

{"title":"On the Existence of Monge Maps for the Gromov–Wasserstein Problem","authors":"Théo Dumont, Théo Lacombe, François-Xavier Vialard","doi":"10.1007/s10208-024-09643-0","DOIUrl":"https://doi.org/10.1007/s10208-024-09643-0","url":null,"abstract":"The Gromov–Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between points. Akin to the standard optimal transportation problem, it is natural to ask for conditions guaranteeing some structure on the optimizers, for instance, if these are induced by a (Monge) map. We study this question in Euclidean spaces when the cost functions are either given by (i) inner products or (ii) squared distances, two standard choices in the literature. We establish the existence of an optimal map in case (i) and of an optimal 2-map (the union of the graphs of two maps) in case (ii), both under an absolute continuity condition on the source measure. Additionally, in case (ii) and in dimension one, we numerically design situations where optimizers of the Gromov–Wasserstein problem are 2-maps but are not maps. This suggests that our result cannot be improved in general for this cost. Still in dimension one, we additionally establish the optimality of monotone maps under some conditions on the measures, thereby giving insight into why such maps often appear to be optimal in numerical experiments.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"100 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750239","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}

引用次数: 0

Discrete Pseudo-differential Operators and Applications to Numerical Schemes 离散伪微分算子及其在数值计算中的应用

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-15 DOI: 10.1007/s10208-024-09645-y

Erwan Faou, Benoît Grébert

引用次数: 0

Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions 弱 CFL 型条件下准线性波方程的强规范误差约束

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-13 DOI: 10.1007/s10208-024-09639-w

Benjamin Dörich

引用次数: 0

Analysis of a Modified Regularity-Preserving Euler Scheme for Parabolic Semilinear SPDEs: Total Variation Error Bounds for the Numerical Approximation of the Invariant Distribution 抛物半线性 SPDEs 的修正正则保全欧拉方案分析：不变分布数值逼近的总变化误差边界

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-08 DOI: 10.1007/s10208-024-09644-z

Charles-Edouard Bréhier

{"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":"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.","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}

引用次数: 0

Phaseless Sampling on Square-Root Lattices 方根网格上的无相采样

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-08 DOI: 10.1007/s10208-024-09640-3

Philipp Grohs, Lukas Liehr

{"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":"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 (g in {L^2({mathbb R}^d)}) and which sampling sets (Lambda subseteq {mathbb R}^{2d}) is every (f in {L^2({mathbb R}^d)}) uniquely determined (up to a global phase factor) by phaseless samples of the form $$begin{aligned} |V_gf(Lambda )| = left{ |V_gf(lambda )|: lambda in Lambda right} , end{aligned}$$where (V_gf) denotes the STFT of f with respect to g. 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 (Lambda ) is a lattice, i.e (Lambda = A{mathbb Z}^{2d}, A in textrm{GL}(2d,{mathbb R})). 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 $$begin{aligned} Lambda = A left( sqrt{{mathbb Z}} right) ^{2d}, sqrt{{mathbb Z}} = { pm sqrt{n}: n in {mathbb N}_0 }, end{aligned}$$guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians","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}

引用次数: 0

Low-Dimensional Invariant Embeddings for Universal Geometric Learning 通用几何学习的低维不变嵌入

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-02-08 DOI: 10.1007/s10208-024-09641-2

Nadav Dym, Steven J. Gortler

{"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":"This paper studies separating invariants: mappings on D-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 D. 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 (2D+1 ) 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 (D+1) 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.","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}

引用次数: 0

Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling 奇异的 B 序列和 S 序列：代数结构和不变度量采样的阶次条件

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-01-19 DOI: 10.1007/s10208-023-09638-3

Eugen Bronasco

引用次数: 0