SIAM Journal on Numerical Analysis最新文献_第4页

Stable and Accurate Least Squares Radial Basis Function Approximations on Bounded Domains 有界域上稳定和精确的最小二乘径向基函数逼近

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-04 DOI: 10.1137/23m1593243

Ben Adcock, Daan Huybrechs, Cecile Piret

{"title":"Stable and Accurate Least Squares Radial Basis Function Approximations on Bounded Domains","authors":"Ben Adcock, Daan Huybrechs, Cecile Piret","doi":"10.1137/23m1593243","DOIUrl":"https://doi.org/10.1137/23m1593243","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2698-2718, December 2024. Abstract. The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape parameter with the degrees of freedom is optimal, resulting in constant overlap between neighboring RBF’s regardless of their number, and we propose an explicit suitable choice of the proportionality constant. We show numerically that highly accurate approximations to smooth functions can also be obtained on bounded domains in several dimensions, using a linear scaling with the degrees of freedom per dimension. We extend the least squares approach to a collocation-based method for the solution of elliptic boundary value problems and illustrate that the combination of centers outside the domain, oversampling, and optimal scaling can result in accuracy close to machine precision in spite of having to solve very ill-conditioned linear systems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142763139","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

A Second-Order, Global-in-Time Energy Stable Implicit-Explicit Runge–Kutta Scheme for the Phase Field Crystal Equation 相场晶体方程的二阶全局能量稳定隐显龙格-库塔格式

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-03 DOI: 10.1137/24m1637623

Hong Zhang, Haifeng Wang, Xueqing Teng

{"title":"A Second-Order, Global-in-Time Energy Stable Implicit-Explicit Runge–Kutta Scheme for the Phase Field Crystal Equation","authors":"Hong Zhang, Haifeng Wang, Xueqing Teng","doi":"10.1137/24m1637623","DOIUrl":"https://doi.org/10.1137/24m1637623","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2667-2697, December 2024. Abstract. We develop a two-stage, second-order, global-in-time energy stable implicit-explicit Runge–Kutta (IMEX RK(2, 2)) scheme for the phase field crystal equation with an [math] time step constraint, and without the global Lipschitz assumption. A linear stabilization term is introduced to the system with Fourier pseudo-spectral spatial discretization, and a novel compact reformulation is devised by rewriting the IMEX RK(2, 2) scheme as an approximation to the variation-of-constants formula. Under the assumption that all stage solutions are a priori bounded in the [math] norm, we first demonstrate that the original energy obtained by this second-order scheme is nonincreasing for any time step with a sufficiently large stabilization parameter. To justify the a priori [math] bound assumption, we establish a uniform-in-time [math] estimate for all stage solutions, subject to an [math] time step constraint. This results in a uniform-in-time bound for all stage solutions through discrete Sobolev embedding from [math] to [math]. Consequently, we achieve an [math] stabilization parameter, ensuring global-in-time energy stability. Additionally, we provide an optimal rate convergence analysis and error estimate for the IMEX RK(2, 2) scheme in the [math] norm. The global-in-time energy stability represents a novel achievement for a two-stage, second-order accurate scheme for a gradient flow without the global Lipschitz assumption. Numerical experiments substantiate the second-order accuracy and energy stability of the proposed scheme.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142760547","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

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