Foundations of Computational Mathematics最新文献_第7页

Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions 具有阻抗边界条件的麦克斯韦方程组的波数显式hp-FEM分析

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-11-14 DOI: 10.1007/s10208-023-09626-7

J. M. Melenk, S. A. Sauter

引用次数: 0

Communication Lower Bounds for Nested Bilinear Algorithms via Rank Expansion of Kronecker Products 基于Kronecker乘积秩展开的嵌套双线性算法的通信下界

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-11-06 DOI: 10.1007/s10208-023-09633-8

Caleb Ju, Yifan Zhang, Edgar Solomonik

{"title":"Communication Lower Bounds for Nested Bilinear Algorithms via Rank Expansion of Kronecker Products","authors":"Caleb Ju, Yifan Zhang, Edgar Solomonik","doi":"10.1007/s10208-023-09633-8","DOIUrl":"https://doi.org/10.1007/s10208-023-09633-8","url":null,"abstract":"We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen’s algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds by use of the rank expansion, or the minimum rank of any fixed size subset of columns of a matrix, for each of the three matrices encoding a bilinear algorithm. This framework provides lower bounds for a class of dependency directed acyclic graphs (DAGs) corresponding to the execution of a given bilinear algorithm, in contrast to other approaches that yield bounds for specific DAGs. However, our lower bounds only apply to executions that do not compute the same DAG node multiple times. Two bilinear algorithms can be nested by taking Kronecker products between their encoding matrices. Our main result is a lower bound on the rank expansion of a matrix constructed by a Kronecker product derived from lower bounds on the rank expansion of the Kronecker product’s operands. We apply the rank expansion lower bounds to obtain novel communication lower bounds for nested Toom-Cook convolution, Strassen’s algorithm, and fast algorithms for contraction of partially symmetric tensors.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"30 2","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509666","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

Error Analysis for 2D Stochastic Navier–Stokes Equations in Bounded Domains with Dirichlet Data Dirichlet数据有界域中二维随机Navier-Stokes方程的误差分析

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-26 DOI: 10.1007/s10208-023-09621-y

Dominic Breit, Andreas Prohl

引用次数: 0

Approximation of Deterministic Mean Field Games with Control-Affine Dynamics 具有控制仿射动力学的确定性平均场对策的逼近

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09629-4

Justina Gianatti, Francisco J. Silva

{"title":"Approximation of Deterministic Mean Field Games with Control-Affine Dynamics","authors":"Justina Gianatti, Francisco J. Silva","doi":"10.1007/s10208-023-09629-4","DOIUrl":"https://doi.org/10.1007/s10208-023-09629-4","url":null,"abstract":"We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean field games with control on the acceleration (see Achdou et al. in NoDEA Nonlinear Differ Equ Appl 27(3):33, 2020; Cannarsa and Mendico in Minimax Theory Appl 5(2):221-250, 2020; Cardaliaguet and Mendico in Nonlinear Anal 203: 112185, 2021). We focus our attention on the approximation of such mean field games by analogous problems in discrete time and finite state space which fall in the framework of (Gomes in J Math Pures Appl (9) 93(3):308-328, 2010). For these approximations, we show the existence and, under an additional monotonicity assumption, uniqueness of solutions. In our main result, we establish the convergence of equilibria of the discrete mean field games problems towards equilibria of the continuous one. Finally, we provide some numerical results for two MFG problems. In the first one, the dynamics of a typical player is nonlinear with respect to the state and, in the second one, a typical player controls its acceleration.As per journal style, reference citation should be expanded form in abstract. So kindly check and confirm the reference citation present in the abstract is correct.Please change \"Gomes in\" below by \"Gomes et al. in \"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"30 11","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509661","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: 3

A Construction of $$C^r$$ Conforming Finite Element Spaces in Any Dimension 任意维$$C^r$$共形有限元空间的构造

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09627-6

Jun Hu, Ting Lin, Qingyu Wu

引用次数: 1

Convex Analysis on Hadamard Spaces and Scaling Problems Hadamard空间的凸分析与标度问题

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09628-5

Hiroshi Hirai

引用次数: 4

A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow 平均曲率流参数有限元逼近分析的一种新方法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09622-x

Genming Bai, Buyang Li

{"title":"A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow","authors":"Genming Bai, Buyang Li","doi":"10.1007/s10208-023-09622-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09622-x","url":null,"abstract":"Parametric finite element methods have achieved great success in approximating the evolution of surfaces under various different geometric flows, including mean curvature flow, Willmore flow, surface diffusion, and so on. However, the convergence of Dziuk’s parametric finite element method, as well as many other widely used parametric finite element methods for these geometric flows, remains open. In this article, we introduce a new approach and a corresponding new framework for the analysis of parametric finite element approximations to surface evolution under geometric flows, by estimating the projected distance from the numerically computed surface to the exact surface, rather than estimating the distance between particle trajectories of the two surfaces as in the currently available numerical analyses. The new framework can recover some hidden geometric structures in geometric flows, such as the full (H^1) parabolicity in mean curvature flow, which is used to prove the convergence of Dziuk’s parametric finite element method with finite elements of degree (k ge 3) for surfaces in the three-dimensional space. The new framework introduced in this article also provides a foundational mathematical tool for analyzing other geometric flows and other parametric finite element methods with artificial tangential motions to improve the mesh quality.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"30 9","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509663","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

Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms 生成函数条形码对哈密顿微分的逼近

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09631-w

Pazit Haim-Kislev, Ofir Karin

引用次数: 0

Gaussian Beam Ansatz for Finite Difference Wave Equations 有限差分波动方程的高斯光束Ansatz

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09632-9

Umberto Biccari, Enrique Zuazua

{"title":"Gaussian Beam Ansatz for Finite Difference Wave Equations","authors":"Umberto Biccari, Enrique Zuazua","doi":"10.1007/s10208-023-09632-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09632-9","url":null,"abstract":"This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"42 24","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516805","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

How Much Can One Learn a Partial Differential Equation from Its Solution? 从一个偏微分方程的解中可以学到多少？

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2023-10-17 DOI: 10.1007/s10208-023-09620-z

Yuchen He, Hongkai Zhao, Yimin Zhong

引用次数: 0