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Long Time Stability and Numerical Stability of Implicit Schemes for Stochastic Heat Equations
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-18 DOI: 10.1137/24m1636691
Xiaochen Yang, Yaozhong Hu
{"title":"Long Time Stability and Numerical Stability of Implicit Schemes for Stochastic Heat Equations","authors":"Xiaochen Yang, Yaozhong Hu","doi":"10.1137/24m1636691","DOIUrl":"https://doi.org/10.1137/24m1636691","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 396-421, February 2025. <br/> Abstract. This paper studies the long time stability of both the solution of a stochastic heat equation on a bounded domain driven by a correlated noise and its approximations. It is popular for researchers to prove the intermittency of the solution, which means that the moments of solution to a stochastic heat equation usually grow to infinity exponentially fast and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation, which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on a bounded domain to write a stochastic heat equation as a system of infinite many stochastic differential equations. We also present numerical experiments which are consistent with our theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"49 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143435501","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
Parameterized Wasserstein Hamiltonian Flow
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-14 DOI: 10.1137/23m159281x
Hao Wu, Shu Liu, Xiaojing Ye, Haomin Zhou
{"title":"Parameterized Wasserstein Hamiltonian Flow","authors":"Hao Wu, Shu Liu, Xiaojing Ye, Haomin Zhou","doi":"10.1137/23m159281x","DOIUrl":"https://doi.org/10.1137/23m159281x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 360-395, February 2025. <br/> Abstract. In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use the parameterized function as a push-forward map to characterize the solution of WHF, and convert the PDE to a finite-dimensional ODE system, which is a Hamiltonian system in the phase space of the parameter manifold. We establish theoretical error bounds for the continuous time approximation scheme in the Wasserstein metric. For the numerical implementation, neural networks are used as push-forward maps. We design an effective symplectic scheme to solve the derived Hamiltonian ODE system so that the method preserves some important quantities such as Hamiltonian. The computation is done by a fully deterministic symplectic integrator without any neural network training. Thus, our method does not involve direct optimization over network parameters and hence can avoid errors introduced by the stochastic gradient descent or similar methods, which are usually hard to quantify and measure in practice. The proposed algorithm is a sampling-based approach that scales well to higher dimensional problems. In addition, the method also provides an alternative connection between the Lagrangian and Eulerian perspectives of the original WHF through the parameterized ODE dynamics.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143417624","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
ContHutch++: Stochastic Trace Estimation For Implicit Integral Operators
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-13 DOI: 10.1137/23m1614365
Jennifer Zvonek, Andrew J. Horning, Alex Townsend
{"title":"ContHutch++: Stochastic Trace Estimation For Implicit Integral Operators","authors":"Jennifer Zvonek, Andrew J. Horning, Alex Townsend","doi":"10.1137/23m1614365","DOIUrl":"https://doi.org/10.1137/23m1614365","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 334-359, February 2025. <br/> Abstract. Hutchinson’s estimator is a randomized algorithm that computes an [math]-approximation to the trace of any positive semidefinite matrix using [math] matrix-vector products. An improvement of Hutchinson’s estimator, known as [math], only requires [math] matrix-vector products. In this paper, we propose a generalization of [math], which we call [math], that uses operator-function products to efficiently estimate the trace of any trace-class integral operator. Our ContHutch++ estimates avoid spectral artifacts introduced by discretization and are accompanied by rigorous high-probability error bounds. We use ContHutch++ to derive a new high-order accurate algorithm for quantum density-of-states and also show how it can estimate electromagnetic fields induced by incoherent sources.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143417622","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
Mixed Finite Element Methods for Linear Cosserat Equations
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-07 DOI: 10.1137/24m1648387
W. M. Boon, O. Duran, J. M. Nordbotten
{"title":"Mixed Finite Element Methods for Linear Cosserat Equations","authors":"W. M. Boon, O. Duran, J. M. Nordbotten","doi":"10.1137/24m1648387","DOIUrl":"https://doi.org/10.1137/24m1648387","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 306-333, February 2025. <br/> Abstract. We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge–Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"62 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143258690","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
Second Order Exponential Splittings in the Presence of Unbounded and Time-Dependent Operators: Construction and Convergence
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-03 DOI: 10.1137/23m1607660
K. Kropielnicka, J. C. Del Valle
{"title":"Second Order Exponential Splittings in the Presence of Unbounded and Time-Dependent Operators: Construction and Convergence","authors":"K. Kropielnicka, J. C. Del Valle","doi":"10.1137/23m1607660","DOIUrl":"https://doi.org/10.1137/23m1607660","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 288-305, February 2025. <br/> Abstract. For linear differential equations of the form [math], [math], with a possibly unbounded operator [math], we construct and deduce error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel’s formula is reformulated: we show that their choice defines the structure of the splitting. Furthermore, the reformulation allows us to consider quadratures based on the Birkhoff interpolation to obtain splittings featuring not only exponentials of [math] or [math] but also time-derivatives of [math] and commutators of [math] and [math]. In this approach, the construction and error analysis of the splittings are carried out simultaneously. We discuss the accuracy of the members of the families. Numerical experiments are presented to complement the theoretical consideration.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143084051","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
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. <br/> 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. <br/> 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. <br/> 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
{"title":"Criticality Measure-Based Error Estimates for Infinite Dimensional Optimization","authors":"Danlin Li, Johannes Milz","doi":"10.1137/24m1647023","DOIUrl":"https://doi.org/10.1137/24m1647023","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 193-213, February 2025. <br/> Abstract. Motivated by optimization with differential equations, we consider optimization problems with Hilbert spaces as decision spaces. As a consequence of their infinite dimensionality, the numerical solution necessitates finite dimensional approximations and discretizations. We develop an approximation framework and demonstrate criticality measure-based error estimates. We consider criticality measures inspired by those used within optimization methods, such as semismooth Newton and (conditional) gradient methods. Furthermore, we show that our error estimates are optimal. Our findings augment existing distance-based error estimates but do not rely on strong convexity or second-order sufficient optimality conditions. Moreover, our error estimates can be used for code verification and validation. We illustrate our theoretical convergence rates on linear, semilinear, and bilinear PDE-constrained optimization problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"111 3S 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143026661","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
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
{"title":"Convergent Finite Difference Schemes for Stochastic Transport Equations","authors":"Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang","doi":"10.1137/23m159946x","DOIUrl":"https://doi.org/10.1137/23m159946x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 149-192, February 2025. <br/> Abstract. We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish [math] stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The [math] estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"52 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020597","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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