SIAM Journal on Numerical Analysis最新文献_第2页

Primal Hybrid Finite Element Method for the Helmholtz Equation 亥姆霍兹方程的原始混合有限元法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-07 DOI: 10.1137/24m1654038

A. Bendali

{"title":"Primal Hybrid Finite Element Method for the Helmholtz Equation","authors":"A. Bendali","doi":"10.1137/24m1654038","DOIUrl":"https://doi.org/10.1137/24m1654038","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 54-80, February 2025. Abstract. This study addresses some previously unexplored issues concerning the stability and error bounds of the primal hybrid finite element method. This method relaxes the strong interelement continuity conditions on the unknown [math] of a boundary-value problem, set in terms of a second-order elliptic partial differential equation, by means of a Lagrange multiplier [math] defined on the mesh skeleton. We show how the decomposition of the space of shape functions for the approximation of [math] allows us to derive conditions, simple to verify, ensuring the inf-sup condition of Brezzi and Babuška, crucial for the stability of the discrete problem, both in two and three dimensions. An adaptation of the analysis of the mixed finite element approximation of the Helmholtz equation [math] in [G. J. Fix and R. A. Nicolaides, SIAM J. Numer. Anal., 17 (1980), pp. 779–786] enables us to give stability conditions and error bounds, explicit in the mesh size [math] and [math]. Some numerical experiments show that the primal hybrid finite element method is more robust than the usual continuous finite element method with regard to dispersion anomalies, known as the pollution effect, particularly on well-smoothed meshes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142935722","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

Sharp Preasymptotic Error Bounds for the Helmholtz [math]-FEM Helmholtz [math]-FEM的尖锐前渐近误差界

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-06 DOI: 10.1137/23m1546178

J. Galkowski, E. A. Spence

{"title":"Sharp Preasymptotic Error Bounds for the Helmholtz [math]-FEM","authors":"J. Galkowski, E. A. Spence","doi":"10.1137/23m1546178","DOIUrl":"https://doi.org/10.1137/23m1546178","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 1-22, February 2025. Abstract. In the analysis of the [math]-version of the finite-element method (FEM), with fixed polynomial degree [math], applied to the Helmholtz equation with wavenumber [math], the asymptotic regime is when [math] is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here [math] is the [math] norm of the Helmholtz solution operator, with [math] for nontrapping problems. In the preasymptotic regime, one expects that if [math] is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition either realized exactly using the Dirichlet-to-Neumann map on the boundary of a ball or approximated either by a radial perfectly matched layer (PML) or an impedance boundary condition. Previously, such bounds for [math] were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalization of the “elliptic-projection” argument (the argument used to obtain the result for [math]), which can be applied to a wide variety of abstract Helmholtz-type problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142934606","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

Recovery Based Linear Finite Element Methods for Hamilton–Jacobi–Bellman Equation with Cordes Coefficients 带Cordes系数的Hamilton-Jacobi-Bellman方程的恢复线性有限元方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-06 DOI: 10.1137/23m1579297

Tianyang Chu, Hailong Guo, Zhimin Zhang

引用次数: 0

Erratum: Multidimensional Sum-Up Rounding for Elliptic Control Systems 更正：椭圆控制系统的多维总结舍入

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-17 DOI: 10.1137/24m1674169

Paul Manns, Christian Kirches

引用次数: 0

Swarm-Based Gradient Descent Meets Simulated Annealing 基于群的梯度下降与模拟退火

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-17 DOI: 10.1137/24m1657808

Zhiyan Ding, Martin Guerra, Qin Li, Eitan Tadmor

{"title":"Swarm-Based Gradient Descent Meets Simulated Annealing","authors":"Zhiyan Ding, Martin Guerra, Qin Li, Eitan Tadmor","doi":"10.1137/24m1657808","DOIUrl":"https://doi.org/10.1137/24m1657808","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2745-2781, December 2024. Abstract. We introduce a novel method, called swarm-based simulated annealing (SSA), for nonconvex optimization which is at the interface between the swarm-based gradient-descent (SBGD) [J. Lu et al., arXiv:2211.17157; E. Tadmor and A. Zenginoglu, Acta Appl. Math., 190 (2024)] and simulated annealing (SA) [V. Cerny, J. Optim. Theory Appl., 45 (1985), pp. 41–51; S. Kirkpatrick et al., Science, 220 (1983), pp. 671–680; S. Geman and C.-R. Hwang, SIAM J. Control Optim., 24 (1986), pp. 1031–1043]. Similarly to SBGD, we introduce a swarm of agents, each identified with a position, [math] and mass [math], to explore the ambient space. Similarly to SA, the agents proceed in the gradient descent direction, and are subject to Brownian motion. The annealing rate, however, is dictated by a decreasing function of their mass. As a consequence, instead of the SA protocol for time-decreasing temperature, here the swarm decides how to “cool down” agents, depending on their own accumulated mass. The dynamics of masses is coupled with the dynamics of positions: agents at higher ground transfer (part of) their mass to those at lower ground. Consequently, the resulting SSA optimizer is dynamically divided between heavier, cooler agents viewed as “leaders” and lighter, warmer agents viewed as “explorers.” Mean-field convergence analysis and benchmark optimizations demonstrate the effectiveness of the SSA method as a multidimensional global optimizer.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841384","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

Corrigendum: A New Lagrange Multiplier Approach for Constructing Structure-Preserving Schemes, II. Bound Preserving 勘误：构造保结构方案的一种新的拉格朗日乘子方法，2。绑定保存

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-17 DOI: 10.1137/24m1670895

Qing Cheng, Jie Shen

引用次数: 0

Multiple Relaxation Exponential Runge–Kutta Methods for the Nonlinear Schrödinger Equation 非线性Schrödinger方程的多重松弛指数龙格-库塔方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-13 DOI: 10.1137/23m1606034

Dongfang Li, Xiaoxi Li

引用次数: 0

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