SIAM Journal on Numerical Analysis最新文献

On the Existence of Minimizers in Shallow Residual ReLU Neural Network Optimization Landscapes 论浅残差 ReLU 神经网络优化景观中最小值的存在性

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-26 DOI: 10.1137/23m1556241

Steffen Dereich, Arnulf Jentzen, Sebastian Kassing

引用次数: 0

A Domain Decomposition Method for Stochastic Evolution Equations 随机演化方程的领域分解法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-20 DOI: 10.1137/24m1629845

Evelyn Buckwar, Ana Djurdjevac, Monika Eisenmann

{"title":"A Domain Decomposition Method for Stochastic Evolution Equations","authors":"Evelyn Buckwar, Ana Djurdjevac, Monika Eisenmann","doi":"10.1137/24m1629845","DOIUrl":"https://doi.org/10.1137/24m1629845","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024. Abstract. In recent years, stochastic partial differential equations (SPDEs) have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"81 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems II: Neumann–Neumann Algorithms 抛物线最优控制问题的新时域分解方法 II：诺伊曼-诺伊曼算法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-19 DOI: 10.1137/24m1634424

Martin J. Gander, Liu-Di Lu

引用次数: 0

The Mean-Field Ensemble Kalman Filter: Near-Gaussian Setting 平均场集合卡尔曼滤波器：近高斯背景

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-15 DOI: 10.1137/24m1628207

J. A. Carrillo, F. Hoffmann, A. M. Stuart, U. Vaes

{"title":"The Mean-Field Ensemble Kalman Filter: Near-Gaussian Setting","authors":"J. A. Carrillo, F. Hoffmann, A. M. Stuart, U. Vaes","doi":"10.1137/24m1628207","DOIUrl":"https://doi.org/10.1137/24m1628207","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2549-2587, December 2024. Abstract. The ensemble Kalman filter is widely used in applications because, for high-dimensional filtering problems, it has a robustness that is not shared, for example, by the particle filter; in particular, it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue, we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. We prove two types of results: The first type comprises a stability estimate controlling the error made by the ensemble Kalman filter in terms of the difference between the true filtering distribution and a nearby Gaussian, and the second type uses this stability result to show that, in a neighborhood of Gaussian problems, the ensemble Kalman filter makes a small error in comparison with the true filtering distribution. Our analysis is developed for the mean-field ensemble Kalman filter. We rewrite the update equations for this filter and for the true filtering distribution in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters, and we prove various stability estimates for the maps defining the evolution of the two filters in this metric. Using these stability estimates, we prove results of the first and second types in the weighted total variation metric. We also provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean-field description of the unscented Kalman filter.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited 时延系统的 Lanczos Tau 框架：帕代逼近与重新定位

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-13 DOI: 10.1137/24m164611x

Evert Provoost, Wim Michiels

引用次数: 0

Spherical Designs for Approximations on Spherical Caps 球形帽上的近似球形设计

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-11 DOI: 10.1137/23m1555417

Chao Li, Xiaojun Chen

{"title":"Spherical Designs for Approximations on Spherical Caps","authors":"Chao Li, Xiaojun Chen","doi":"10.1137/23m1555417","DOIUrl":"https://doi.org/10.1137/23m1555417","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2506-2528, December 2024. Abstract. A spherical [math]-design is a set of points on the unit sphere, which provides an equal weight quadrature rule integrating exactly all spherical polynomials of degree at most [math] and has a sharp error bound for approximations on the sphere. This paper introduces a set of points called a spherical cap [math]-subdesign on a spherical cap [math] with center [math] and radius [math] induced by the spherical [math]-design. We show that the spherical cap [math]-subdesign provides an equal weight quadrature rule integrating exactly all zonal polynomials of degree at most [math] and all functions expanded by orthonormal functions on the spherical cap which are defined by shifted Legendre polynomials of degree at most [math]. We apply the spherical cap [math]-subdesign and the orthonormal basis functions on the spherical cap to non-polynomial approximation of continuous functions on the spherical cap and present theoretical approximation error bounds. We also apply spherical cap [math]-subdesigns to sparse signal recovery on the upper hemisphere, which is a spherical cap with [math]. Our theoretical and numerical results show that spherical cap [math]-subdesigns can provide a good approximation on spherical caps.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"153 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142598300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

An Operator Preconditioned Combined Field Integral Equation for Electromagnetic Scattering 电磁散射的算子预处理组合场积分方程

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-07 DOI: 10.1137/23m1581674

Van Chien Le, Kristof Cools

引用次数: 0

An Energy-Based Discontinuous Galerkin Method for the Nonlinear Schrödinger Equation with Wave Operator 带波算子的非线性薛定谔方程的基于能量的非连续伽勒金方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-04 DOI: 10.1137/23m1597496

Kui Ren, Lu Zhang, Yin Zhou

{"title":"An Energy-Based Discontinuous Galerkin Method for the Nonlinear Schrödinger Equation with Wave Operator","authors":"Kui Ren, Lu Zhang, Yin Zhou","doi":"10.1137/23m1597496","DOIUrl":"https://doi.org/10.1137/23m1597496","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2459-2483, December 2024. Abstract. This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schrödinger equation with the wave operator. The focus of the study is on the energy-conserving or energy-dissipating behavior of the method with some simple mesh-independent numerical fluxes we designed. We establish error estimates in the energy norm that require careful selection of a weak formulation for the auxiliary equation involving the time derivative of the displacement variable. A critical part of the convergence analysis is to establish the [math] error bounds for the time derivative of the approximation error in the displacement variable by using the equation that determines its mean value. Using a special weak formulation, we show that one can create a linear system for the time evolution of the unknowns even when dealing with nonlinear properties in the original problem. Numerical experiments were performed to demonstrate the optimal convergence of the scheme in the [math] norm. These experiments involved specific choices of numerical fluxes combined with specific choices of approximation spaces.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142580294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems 失效问题的平衡通量后验误差估算器

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-04 DOI: 10.1137/23m1627195

Annalisa Buffa, Ondine Chanon, Denise Grappein, Rafael Vázquez, Martin Vohralík

{"title":"An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems","authors":"Annalisa Buffa, Ondine Chanon, Denise Grappein, Rafael Vázquez, Martin Vohralík","doi":"10.1137/23m1627195","DOIUrl":"https://doi.org/10.1137/23m1627195","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2439-2458, December 2024. Abstract. An a posteriori error estimator based on an equilibrated flux reconstruction is proposed for defeaturing problems in the context of finite element discretizations. Defeaturing consists in the simplification of a geometry by removing features that are considered not relevant for the approximation of the solution of a given PDE. In this work, the focus is on a Poisson equation with Neumann boundary conditions on the feature boundary. The estimator accounts both for the so-called defeaturing error and for the numerical error committed by approximating the solution on the defeatured domain. Unlike other estimators that were previously proposed for defeaturing problems, the use of the equilibrated flux reconstruction allows us to obtain a sharp bound for the numerical component of the error. Furthermore, it does not require the evaluation of the normal trace of the numerical flux on the feature boundary: this makes the estimator well suited for finite element discretizations, in which the normal trace of the numerical flux is typically discontinuous across elements. The reliability of the estimator is proven and verified on several numerical examples. Its capability to identify the most relevant features is also shown, in anticipation of a future application to an adaptive strategy.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"41 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142580295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

The A Posteriori Error Estimates of the FE Approximation of Defective Eigenvalues for Non-Self-Adjoint Eigenvalue Problems 非自相交特征值问题的缺陷特征值 FE 近似的后验误差估计值

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-04 DOI: 10.1137/23m162065x

Yidu Yang, Shixi Wang, Hai Bi

引用次数: 0