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

Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-31 DOI: 10.1137/23m1617345

Federico Cornalba, Julian Fischer

{"title":"Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics","authors":"Federico Cornalba, Julian Fischer","doi":"10.1137/23m1617345","DOIUrl":"https://doi.org/10.1137/23m1617345","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 262-287, February 2025. Abstract. Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"19 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071527","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 Priori Analysis of a Tensor ROM for Parameter Dependent Parabolic Problems

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-28 DOI: 10.1137/23m1616844

Alexander V. Mamonov, Maxim A. Olshanskii

{"title":"A Priori Analysis of a Tensor ROM for Parameter Dependent Parabolic Problems","authors":"Alexander V. Mamonov, Maxim A. Olshanskii","doi":"10.1137/23m1616844","DOIUrl":"https://doi.org/10.1137/23m1616844","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 239-261, February 2025. Abstract. A space–time–parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD—a traditional dimension reduction technique—yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD–ROMs) benefits both the practical efficiency of the ROM and its amenability for rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD–ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD–ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties and LRTD accuracy. The estimate depends on the local smoothness rather than on the Kolmogorov [math]-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"24 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143055038","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 Approximation of Discontinuous Solutions of the Semilinear Wave Equation

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-27 DOI: 10.1137/24m1635879

Jiachuan Cao, Buyang Li, Yanping Lin, Fangyan Yao

{"title":"Numerical Approximation of Discontinuous Solutions of the Semilinear Wave Equation","authors":"Jiachuan Cao, Buyang Li, Yanping Lin, Fangyan Yao","doi":"10.1137/24m1635879","DOIUrl":"https://doi.org/10.1137/24m1635879","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 214-238, February 2025. Abstract. A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than preexisting methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that [math] for [math]. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition [math], where [math] and [math] denote the time step size and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"20 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143050861","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

Criticality Measure-Based Error Estimates for Infinite Dimensional Optimization

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-23 DOI: 10.1137/24m1647023

Danlin Li, Johannes Milz

引用次数: 0

Convergent Finite Difference Schemes for Stochastic Transport Equations 随机输运方程的收敛有限差分格式

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2025-01-22 DOI: 10.1137/23m159946x

Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang

引用次数: 0

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