Foundations of Computational Mathematics最新文献

Conley Index for Multivalued Maps on Finite Topological Spaces

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-12-09 DOI: 10.1007/s10208-024-09685-4

Jonathan Barmak, Marian Mrozek, Thomas Wanner

引用次数: 0

Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-12-09 DOI: 10.1007/s10208-024-09682-7

James Demmel, Ioana Dumitriu, Ryan Schneider

{"title":"Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization","authors":"James Demmel, Ioana Dumitriu, Ryan Schneider","doi":"10.1007/s10208-024-09682-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09682-7","url":null,"abstract":"We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any (n times n) matrix pencil (A, B). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel and Dumitriu (Technical Report 2010). We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni and Srivastava (Foundations of Computational Mathematics 2023). In particular, we show that perturbing and scaling (A, B) regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of (A, B) in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible S, T and diagonal D such that (||A - SDT^{-1}||_2 le varepsilon ) and (||B - ST^{-1}||_2 le varepsilon ) in at most (O left( log ^2 left( frac{n}{varepsilon } right) T_{text {MM}}(n) right) ) operations, where (T_{text {MM}}(n)) is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact-arithmetic matrix pencil diagonalization.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"47 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142796989","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

Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in $$mathbb {R}^n$$

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-12-04 DOI: 10.1007/s10208-024-09659-6

Kendrick Shepherd, Deepesh Toshniwal

{"title":"Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in $$mathbb {R}^n$$","authors":"Kendrick Shepherd, Deepesh Toshniwal","doi":"10.1007/s10208-024-09659-6","DOIUrl":"https://doi.org/10.1007/s10208-024-09659-6","url":null,"abstract":"Given a domain (Omega subset mathbb {R}^n), the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on (Omega ), and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When (Omega ) is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such (Omega ), we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"82 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776456","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

Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex 德拉姆复数顶点补丁中 p-稳健局部重构的有约束和无约束稳定离散最小化

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-25 DOI: 10.1007/s10208-024-09674-7

Théophile Chaumont-Frelet, Martin Vohralík

{"title":"Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex","authors":"Théophile Chaumont-Frelet, Martin Vohralík","doi":"10.1007/s10208-024-09674-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09674-7","url":null,"abstract":"We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the (H^1), ({varvec{H}}(textbf{curl})), or ({varvec{H}}({text {div}})) spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of p. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best–global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in (H^1) and constrained minimization in ({varvec{H}}({text {div}})) have been previously treated in the literature. Along with improvement of the results in the (H^1) and ({varvec{H}}({text {div}})) cases, our key contribution is the treatment of the ({varvec{H}}(textbf{curl})) framework. This enables us to cover the whole de Rham diagram in three space dimensions in a single setting.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"113 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142713198","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

Tribute to Nick Higham 向尼克-海勒姆致敬

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-22 DOI: 10.1007/s10208-024-09680-9

Alan Edelman

引用次数: 0

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints 近端伽勒金：用于点式约束的结构保留有限元方法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-20 DOI: 10.1007/s10208-024-09681-8

Brendan Keith, Thomas M. Surowiec

{"title":"Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints","authors":"Brendan Keith, Thomas M. Surowiec","doi":"10.1007/s10208-024-09681-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09681-8","url":null,"abstract":"The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"99 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678312","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

Classification of Finite Groups: Recent Developements and Open Problems 有限群的分类：最新发展和未决问题

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-12 DOI: 10.1007/s10208-024-09688-1

Bettina Eick

引用次数: 0

Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation 使用二元公式对动态优化运输进行离散化的定量收敛

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-11 DOI: 10.1007/s10208-024-09686-3

Sadashige Ishida, Hugo Lavenant

引用次数: 0

Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory 利用算子值自由概率论计算非交换内等级

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-11 DOI: 10.1007/s10208-024-09684-5

Johannes Hoffmann, Tobias Mai, Roland Speicher

引用次数: 0

Gabor Phase Retrieval via Semidefinite Programming 通过半定量编程实现 Gabor 相位检索

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-07 DOI: 10.1007/s10208-024-09683-6

Philippe Jaming, Martin Rathmair

{"title":"Gabor Phase Retrieval via Semidefinite Programming","authors":"Philippe Jaming, Martin Rathmair","doi":"10.1007/s10208-024-09683-6","DOIUrl":"https://doi.org/10.1007/s10208-024-09683-6","url":null,"abstract":"We consider the problem of reconstructing a function (fin L^2({mathbb R})) given phase-less samples of its Gabor transform, which is defined by $$begin{aligned} {mathcal {G}}f(x,y) :=2^{frac{1}{4}} int _{mathbb R}f(t) e^{-pi (t-x)^2} e^{-2pi i y t},text{ d }t,quad (x,y)in {mathbb R}^2. end{aligned}$$More precisely, given sampling positions (Omega subseteq {mathbb R}^2) the task is to reconstruct f (up to global phase) from measurements ({|{mathcal {G}}f(omega )|: ,omega in Omega }). This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"61 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142597481","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