SIAM Journal on Numerical Analysis最新文献_第3页

Orthogonal Polynomial Approximation and Extended Dynamic Mode Decomposition in Chaos 混沌中的正交多项式逼近与扩展动态模态分解

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-20 DOI: 10.1137/23m1597873

Caroline Wormell

{"title":"Orthogonal Polynomial Approximation and Extended Dynamic Mode Decomposition in Chaos","authors":"Caroline Wormell","doi":"10.1137/23m1597873","DOIUrl":"https://doi.org/10.1137/23m1597873","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 122-148, February 2025. Abstract. Extended dynamic mode decomposition (EDMD) is a data-driven tool for forecasting and model reduction of dynamics, which has been extensively taken up in the physical sciences. While the method is conceptually simple, in deterministic chaos it is unclear what its properties are or even what it converges to. In particular, it is not clear how EDMD’s least-squares approximation treats the classes of differentiable functions on which chaotic systems act. We develop for the first time a general, rigorous theory of EDMD on the simplest examples of chaotic maps: analytic expanding maps of the circle. To do this, we prove a new, basic approximation result in the theory of orthogonal polynomials on the unit circle (OPUC) and apply methods from transfer operator theory. We show that in the infinite-data limit, the least-squares projection error is exponentially small for trigonometric polynomial observable dictionaries. As a result, we show that forecasts and Koopman spectral data produced using EDMD in this setting converge to the physically meaningful limits, exponentially fast with respect to the size of the dictionary. This demonstrates that with only a relatively small polynomial dictionary, EDMD can be very effective, even when the sampling measure is not uniform. Furthermore, our OPUC result suggests that data-based least-squares projection may be a very effective approximation strategy more generally.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"107 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990649","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 Energy-Stable Parametric Finite Element Method for the Planar Willmore Flow 平面Willmore流的能量稳定参数有限元法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-13 DOI: 10.1137/24m1633893

Weizhu Bao, Yifei Li

引用次数: 0

VEM-Nitsche Fully Discrete Polytopal Scheme for Frictionless Contact-Mechanics 无摩擦接触力学的VEM-Nitsche全离散多边形格式

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-09 DOI: 10.1137/24m1660218

Mohamed Laaziri, Roland Masson

{"title":"VEM-Nitsche Fully Discrete Polytopal Scheme for Frictionless Contact-Mechanics","authors":"Mohamed Laaziri, Roland Masson","doi":"10.1137/24m1660218","DOIUrl":"https://doi.org/10.1137/24m1660218","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 81-102, February 2025. Abstract. This work targets the discretization of contact-mechanics accounting for small strains, linear elastic constitutive laws, and fractures or faults represented as a network of co-dimension one planar interfaces. This type of model coupled with Darcy flow plays an important role typically for the simulation of fault reactivation by fluid injection in geological storage or the hydraulic fracture stimulation in enhanced geothermal systems. To simplify the presentation, a frictionless contact behavior at matrix fracture interfaces is considered, although the scheme developed in this work readily extends to more complex contact models such as the Mohr–Coulomb friction. To account for the geometrical complexity of subsurface, our discretization is based on the first order virtual element method (VEM), which generalizes the [math] finite element method to polytopal meshes. Following previous works in the finite element framework, the contact conditions are enforced in a weak sense using Nitsche’s formulation based on additional consistent penalization terms. We perform, in a fully discrete framework, the well-posedness and convergence analysis showing an optimal first order error estimate with minimal regularity assumptions. Numerical experiments confirm our theoretical findings and exhibit the good behavior of the nonlinear semismooth Newton solver.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142936909","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

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