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Long-term behaviour of symmetric partitioned linear multistep methods. I: global error and conservation of invariants 对称分段线性多步方法的长期行为。I：全局误差和不变量的守恒
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-24 DOI: 10.1093/imanum/draf062
Begoña Cano, Ángel Durán, Melquiades Rodríguez
{"title":"Long-term behaviour of symmetric partitioned linear multistep methods. I: global error and conservation of invariants","authors":"Begoña Cano, Ángel Durán, Melquiades Rodríguez","doi":"10.1093/imanum/draf062","DOIUrl":"https://doi.org/10.1093/imanum/draf062","url":null,"abstract":"In this paper an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and conservation of invariants. In particular, symmetric partitioned linear multistep methods with no common roots in their first characteristic polynomials, except unity, appear as efficient methods to approximate nonseparable Hamiltonian systems since they can be explicit and show good long term behaviour at the same time. As a case study, a thorough analysis is given for small oscillations of the double pendulum problem, which is illustrated by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702019","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
Convergence and near-optimal sampling for multivariate function approximations in irregular domains via Vandermonde with Arnoldi 基于Vandermonde和Arnoldi的不规则域多元函数逼近的收敛性和近最优抽样
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-22 DOI: 10.1093/imanum/draf055
Wenqi Zhu, Yuji Nakatsukasa
{"title":"Convergence and near-optimal sampling for multivariate function approximations in irregular domains via Vandermonde with Arnoldi","authors":"Wenqi Zhu, Yuji Nakatsukasa","doi":"10.1093/imanum/draf055","DOIUrl":"https://doi.org/10.1093/imanum/draf055","url":null,"abstract":"Vandermonde matrices are usually exponentially ill-conditioned and often result in unstable approximations. In this paper we introduce and analyse the multivariate Vandermonde with Arnoldi (V+A) method, which is based on least-squares approximation together with a Stieltjes orthogonalization process, for approximating continuous, multivariate functions on $d$-dimensional irregular domains. The V+A method addresses the ill-conditioning of the Vandermonde approximation by creating a set of discrete orthogonal bases with respect to a discrete measure. The V+A method is simple and general, relying only on the domain’s sample points. This paper analyses the sample complexity of the least-squares approximation that uses the V+A method. We show that, for a large class of domains, this approximation gives a well-conditioned and near-optimal $N$-dimensional least-squares approximation using $M={cal O}(N^{2})$ equispaced sample points or $M={cal O}(N^{2}log N)$ random sample points, independently of $d$. We provide a comprehensive analysis of the error estimates and the rate of convergence of the least-squares approximation that uses the V+A method. Based on the multivariate V+A techniques we propose a new variant of the weighted V+A least-squares algorithm that uses only $M={cal O}(Nlog N)$ sample points to achieve a near-optimal approximation. Our initial numerical results validate that the V+A least-squares approximation method provides well-conditioned and near-optimal approximations for multivariate functions on (irregular) domains. Additionally, the (weighted) least-squares approximation that uses the V+A method performs competitively with state-of-the-art orthogonalization techniques and can serve as a practical tool for selecting near-optimal distributions of sample points in irregular domains.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144684797","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
Numerical analysis of lowest-order finite volume methods for a class of Stokes variational inequality problem 一类Stokes变分不等式问题的最低阶有限体积法数值分析
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-21 DOI: 10.1093/imanum/draf056
Feifei Jing, Takahito Kashiwabara, Wenjing Yan
{"title":"Numerical analysis of lowest-order finite volume methods for a class of Stokes variational inequality problem","authors":"Feifei Jing, Takahito Kashiwabara, Wenjing Yan","doi":"10.1093/imanum/draf056","DOIUrl":"https://doi.org/10.1093/imanum/draf056","url":null,"abstract":"Three types of lowest-order finite volume element methods, i.e., the conforming, nonconforming and discontinuous schemes, are introduced and analysed for a variational inequality governed by the stationary Stokes equations. The variational inequality arises due to a nonlinear and nondifferentiable relationship in the slip boundary condition of friction type. This relationship cannot be well combined into a finite volume scheme by a standard procedure based on integration by parts on dual control volumes. Thereby we propose to enforce it pointwisely at cell centres in a dual mesh, which leads to some numerical integration formula for the boundary nonlinear term in the variational inequality. The resulting finite volume schemes can be seen to be equivalent to some finite element methods. We further show their solvability and stability, as well as the a priori error estimates with optimal approximation behaviours. Numerical results are reported to demonstrate the theoretical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144677490","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 conditional gradient homotopy method with applications to semidefinite programming 一个条件梯度同伦方法在半定规划中的应用
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-20 DOI: 10.1093/imanum/draf059
Pavel Dvurechensky, Gabriele Iommazzo, Shimrit Shtern, Mathias Staudigl
{"title":"A conditional gradient homotopy method with applications to semidefinite programming","authors":"Pavel Dvurechensky, Gabriele Iommazzo, Shimrit Shtern, Mathias Staudigl","doi":"10.1093/imanum/draf059","DOIUrl":"https://doi.org/10.1093/imanum/draf059","url":null,"abstract":"We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art semidefinite programming solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"96 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144669720","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
Approximation in Hilbert spaces of the Gaussian and related analytic kernels 希尔伯特空间中高斯核及相关解析核的近似
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-15 DOI: 10.1093/imanum/draf050
Toni Karvonen, Yuya Suzuki
{"title":"Approximation in Hilbert spaces of the Gaussian and related analytic kernels","authors":"Toni Karvonen, Yuya Suzuki","doi":"10.1093/imanum/draf050","DOIUrl":"https://doi.org/10.1093/imanum/draf050","url":null,"abstract":"We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = exp (-tfrac{1}{2}varepsilon ^{2}(x-y)^{2})$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(varepsilon /2)^{n} (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates that we use to bound the Hilbert space norm of a weighted polynomial fooling function.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144639793","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
Inf-sup stable discretization of the quasi-static Biot’s equations in poroelasticity 孔隙弹性准静态Biot方程的非稳定离散化
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-07 DOI: 10.1093/imanum/draf032
Christian Kreuzer, Pietro Zanotti
{"title":"Inf-sup stable discretization of the quasi-static Biot’s equations in poroelasticity","authors":"Christian Kreuzer, Pietro Zanotti","doi":"10.1093/imanum/draf032","DOIUrl":"https://doi.org/10.1093/imanum/draf032","url":null,"abstract":"We propose a new full discretization of the Biot’s equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters and the time horizon. We further construct an interpolant showing how the error decays for smooth solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"189 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144577905","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
Finite element approximation of penalized elastoplastic torsion problem with nonconstant source term 非常源项惩罚弹塑性扭转问题的有限元逼近
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-03 DOI: 10.1093/imanum/draf052
Franz Chouly, Tom Gustafsson, Patrick Hild
{"title":"Finite element approximation of penalized elastoplastic torsion problem with nonconstant source term","authors":"Franz Chouly, Tom Gustafsson, Patrick Hild","doi":"10.1093/imanum/draf052","DOIUrl":"https://doi.org/10.1093/imanum/draf052","url":null,"abstract":"This study is concerned with the finite element approximation of the elastoplastic torsion problem. We focus on the case of a nonconstant source term, which cannot be easily recast into an obstacle problem as can be done in the case of a constant source term. We present a simple formulation that penalizes the constraint directly on the gradient norm of the solution. We study its well-posedness, derive error estimates and present numerical results to illustrate the theory.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144565936","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 Riemannian inexact Newton method for solving the orthogonal INDSCAL problem in multidimensional scaling 求解多维标度中正交INDSCAL问题的黎曼非精确牛顿法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-07-03 DOI: 10.1093/imanum/draf047
Xue-lin Zhou, Chao-qian Li, Jiao-fen Li, Xue-feng Duan
{"title":"A Riemannian inexact Newton method for solving the orthogonal INDSCAL problem in multidimensional scaling","authors":"Xue-lin Zhou, Chao-qian Li, Jiao-fen Li, Xue-feng Duan","doi":"10.1093/imanum/draf047","DOIUrl":"https://doi.org/10.1093/imanum/draf047","url":null,"abstract":"The well-known individual differences scaling (INDSCAL) model is intended for simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. In this work the problem of fitting the orthogonal INDSCAL model to the data is reformulated and studied as a matrix optimization problem on the product manifold of orthonormal and diagonal matrices. A Riemannian inexact Newton method is proposed to address the underlying problem, with the global and quadratic convergence of the proposed method established under some mild assumptions. Furthermore, the positive definiteness condition of the Riemannian Hessian of the objective function at a solution is derived. Some numerical experiments are provided to illustrate the efficiency of the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"152 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144566072","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
Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case 非凸随机梯度情况下非调整广义哈密顿蒙特卡罗的反射耦合
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-06-29 DOI: 10.1093/imanum/draf045
Martin Chak, Pierre Monmarché
{"title":"Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case","authors":"Martin Chak, Pierre Monmarché","doi":"10.1093/imanum/draf045","DOIUrl":"https://doi.org/10.1093/imanum/draf045","url":null,"abstract":"Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion, which are commonly used in molecular dynamics simulations. To accommodate the degenerate noise structure corresponding to inertia existing in the chain, a characteristically discrete-in-time coupling and contraction proof is devised. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance and total variation are also given, together with numerical bias estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513176","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 theta methods for free stochastic differential equations 自由随机微分方程的随机θ方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-06-29 DOI: 10.1093/imanum/draf044
Yuanling Niu, Jiaxin Wei, Zhi Yin, Dan Zeng
{"title":"Stochastic theta methods for free stochastic differential equations","authors":"Yuanling Niu, Jiaxin Wei, Zhi Yin, Dan Zeng","doi":"10.1093/imanum/draf044","DOIUrl":"https://doi.org/10.1093/imanum/draf044","url":null,"abstract":"We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Assuming that the drift coefficient of the free stochastic differential equations is operator Lipschitz and the diffusion coefficients are locally operator Lipschitz we prove the strong convergence of the numerical methods. Moreover, we investigate the exponential stability in mean square of the equations and the numerical methods. In particular, the free stochastic theta methods with $theta in [1/2, 1]$ can inherit the exponential stability of original equations for any given step size. Our methods offer better stability than the free Euler–Maruyama method. Numerical results are reported to confirm these theoretical findings and show the efficiency of our methods compared with the free Euler–Maruyama method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513174","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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