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Convergence analysis for minimum action methods coupled with a finite difference method 最小作用法与有限差分法的收敛性分析
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-06-21 DOI: 10.1093/imanum/drae038
Jialin Hong, Diancong Jin, Derui Sheng
{"title":"Convergence analysis for minimum action methods coupled with a finite difference method","authors":"Jialin Hong, Diancong Jin, Derui Sheng","doi":"10.1093/imanum/drae038","DOIUrl":"https://doi.org/10.1093/imanum/drae038","url":null,"abstract":"The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $theta $-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141452971","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
Precise error bounds for numerical approximations of fractional HJB equations 分数 HJB 方程数值近似的精确误差范围
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-06-12 DOI: 10.1093/imanum/drae030
Indranil Chowdhury, Espen R Jakobsen
{"title":"Precise error bounds for numerical approximations of fractional HJB equations","authors":"Indranil Chowdhury, Espen R Jakobsen","doi":"10.1093/imanum/drae030","DOIUrl":"https://doi.org/10.1093/imanum/drae030","url":null,"abstract":"We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $sigma in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $mathcal{O}big (h^{frac{1}{2}}big )$.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"2014 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315615","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
CIP-stabilized virtual elements for diffusion-convection-reaction problems 扩散-对流-反应问题的 CIP 稳定虚拟元素
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-31 DOI: 10.1093/imanum/drae020
L Beirão da Veiga, C Lovadina, M Trezzi
{"title":"CIP-stabilized virtual elements for diffusion-convection-reaction problems","authors":"L Beirão da Veiga, C Lovadina, M Trezzi","doi":"10.1093/imanum/drae020","DOIUrl":"https://doi.org/10.1093/imanum/drae020","url":null,"abstract":"The Virtual Element Method (VEM) for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the VEM, the stability and the error analysis requires particular care—especially in treating the advective term. Some numerical tests are presented to support the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141185236","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
Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems 适用于粗糙哈密顿系统的交映法随机修正方程
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-24 DOI: 10.1093/imanum/drae019
Chuchu Chen, Jialin Hong, Chuying Huang
{"title":"Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems","authors":"Chuchu Chen, Jialin Hong, Chuying Huang","doi":"10.1093/imanum/drae019","DOIUrl":"https://doi.org/10.1093/imanum/drae019","url":null,"abstract":"We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096648","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
Discrete anisotropic curve shortening flow in higher codimension 高标度离散各向异性曲线缩短流
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-24 DOI: 10.1093/imanum/drae015
Klaus Deckelnick, Robert Nürnberg
{"title":"Discrete anisotropic curve shortening flow in higher codimension","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.1093/imanum/drae015","DOIUrl":"https://doi.org/10.1093/imanum/drae015","url":null,"abstract":"We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{mathbb{R}}}^{d}$, $dgeq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096758","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
High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation 用于平流扩散方程保守公式的高阶拉格朗日-加勒金方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-18 DOI: 10.1093/imanum/drae018
Rodolfo Bermejo, Manuel Colera
{"title":"High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation","authors":"Rodolfo Bermejo, Manuel Colera","doi":"10.1093/imanum/drae018","DOIUrl":"https://doi.org/10.1093/imanum/drae018","url":null,"abstract":"We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $mu $ is large, the error is $O(h^{k+1}+varDelta t^{q})$, whereas in the advective regime, $mu ll 1$, the convergence is $O(min (h^{k},frac{h^{k+1} }{varDelta t})+varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $mu ^{-1}$ dependence.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"127 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954632","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 estimator for mixed finite element discretizations of the curl-curl problem 卷曲问题混合有限元离散的均衡估计器
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-17 DOI: 10.1093/imanum/drae007
T Chaumont-Frelet
{"title":"An equilibrated estimator for mixed finite element discretizations of the curl-curl problem","authors":"T Chaumont-Frelet","doi":"10.1093/imanum/drae007","DOIUrl":"https://doi.org/10.1093/imanum/drae007","url":null,"abstract":"We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"126 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954005","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
Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations 针对非稳态不可压缩磁流体动力学方程的两种 RPC-SAV 方案的最佳收敛性分析
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-15 DOI: 10.1093/imanum/drae016
Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili Tang
{"title":"Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations","authors":"Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili Tang","doi":"10.1093/imanum/drae016","DOIUrl":"https://doi.org/10.1093/imanum/drae016","url":null,"abstract":"In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $beta =0$ and RPC-SAV-CN when $beta in (0,frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-alpha _{1}varDelta tnu varDelta (widetilde {textbf {u}}^{n+1}-{textbf {u}}^{n})$ and $alpha _{2}varDelta tsigma ^{-1}mbox {curl}mbox {curl} (textbf {H}^{n+1}-textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140949377","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
Efficient function approximation in enriched approximation spaces 丰富近似空间中的高效函数近似
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-12 DOI: 10.1093/imanum/drae017
Astrid Herremans, Daan Huybrechs
{"title":"Efficient function approximation in enriched approximation spaces","authors":"Astrid Herremans, Daan Huybrechs","doi":"10.1093/imanum/drae017","DOIUrl":"https://doi.org/10.1093/imanum/drae017","url":null,"abstract":"An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140915068","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
Variational data assimilation with finite-element discretization for second-order parabolic interface equation 二阶抛物面界面方程的有限元离散化变量数据同化
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-05-11 DOI: 10.1093/imanum/drae010
Xuejian Li, Xiaoming He, Wei Gong, Craig C Douglas
{"title":"Variational data assimilation with finite-element discretization for second-order parabolic interface equation","authors":"Xuejian Li, Xiaoming He, Wei Gong, Craig C Douglas","doi":"10.1093/imanum/drae010","DOIUrl":"https://doi.org/10.1093/imanum/drae010","url":null,"abstract":"In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140910594","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
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