Foundations of Computational Mathematics最新文献_第4页

A Sheaf-Theoretic Construction of Shape Space 形状空间的 Sheaf 理论构造

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-05-16 DOI: 10.1007/s10208-024-09650-1

Shreya Arya, Justin Curry, Sayan Mukherjee

引用次数: 0

Discrete Weber Inequalities and Related Maxwell Compactness for Hybrid Spaces over Polyhedral Partitions of Domains with General Topology 具有一般拓扑学的多面体分区域上混合空间的离散韦伯不等式及相关麦克斯韦紧凑性

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-04-16 DOI: 10.1007/s10208-024-09648-9

Simon Lemaire, Silvano Pitassi

引用次数: 0

Sum-of-Squares Relaxations for Information Theory and Variational Inference 信息论和变量推理的平方和松弛

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-04-05 DOI: 10.1007/s10208-024-09651-0

{"title":"Sum-of-Squares Relaxations for Information Theory and Variational Inference","authors":"","doi":"10.1007/s10208-024-09651-0","DOIUrl":"https://doi.org/10.1007/s10208-024-09651-0","url":null,"abstract":"<h3>Abstract</h3> We consider extensions of the Shannon relative entropy, referred to as f-divergences. Three classical related computational problems are typically associated with these divergences: (a) estimation from moments, (b) computing normalizing integrals, and (c) variational inference in probabilistic models. These problems are related to one another through convex duality, and for all of them, there are many applications throughout data science, and we aim for computationally tractable approximation algorithms that preserve properties of the original problem such as potential convexity or monotonicity. In order to achieve this, we derive a sequence of convex relaxations for computing these divergences from non-centered covariance matrices associated with a given feature vector: starting from the typically non-tractable optimal lower-bound, we consider an additional relaxation based on “sums-of-squares”, which is is now computable in polynomial time as a semidefinite program. We also provide computationally more efficient relaxations based on spectral information divergences from quantum information theory. For all of the tasks above, beyond proposing new relaxations, we derive tractable convex optimization algorithms, and we present illustrations on multivariate trigonometric polynomials and functions on the Boolean hypercube.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140533954","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

Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs 求解高维和多尺度椭圆 PDE 的稀疏谱方法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-04-02 DOI: 10.1007/s10208-024-09649-8

{"title":"Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs","authors":"","doi":"10.1007/s10208-024-09649-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09649-8","url":null,"abstract":"<h3>Abstract</h3> In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, “[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues” [Boyd in Chebyshev and Fourier spectral methods, second rev ed. Dover Publications, Mineola, NY, 2001, pg. 194]. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, sparse Fourier transform paired with randomized rank-1 lattice methods. The resulting sparse spectral method rapidly and automatically determines a set of Fourier basis functions whose span is guaranteed to contain an accurate approximation of the solution of a given elliptic PDE. This much smaller, near-optimal Fourier basis is then used to efficiently solve the given PDE in a runtime which only depends on the PDE’s data compressibility and ellipticity properties, while breaking the curse of dimensionality and relieving linear dependence on any multiscale structure in the original problem. Theoretical performance of the method is established herein with convergence analysis in the Sobolev norm for a general class of non-constant diffusion equations, as well as pointers to technical extensions of the convergence analysis to more general advection–diffusion–reaction equations. Numerical experiments demonstrate good empirical performance on several multiscale and high-dimensional example problems, further showcasing the promise of the proposed methods in practice.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140343465","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 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":null,"pages":null},"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":null,"pages":null},"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