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On Polynomial Interpolation in the Monomial Basis 关于单项式基上的多项式插值
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-03-05 DOI: 10.1137/23m1623215
Zewen Shen, Kirill Serkh
{"title":"On Polynomial Interpolation in the Monomial Basis","authors":"Zewen Shen, Kirill Serkh","doi":"10.1137/23m1623215","DOIUrl":"https://doi.org/10.1137/23m1623215","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 469-494, April 2025. <br/> Abstract. In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation over general regions in the complex plane using the monomial basis. Our analysis also yields a new upper bound for the condition number of an arbitrary Vandermonde matrix, which generalizes several previous results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546179","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
Discretization of Total Variation in Optimization with Integrality Constraints 具有完整性约束的优化中总变分的离散化
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-25 DOI: 10.1137/24m164608x
Annika Schiemann, Paul Manns
{"title":"Discretization of Total Variation in Optimization with Integrality Constraints","authors":"Annika Schiemann, Paul Manns","doi":"10.1137/24m164608x","DOIUrl":"https://doi.org/10.1137/24m164608x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 437-460, February 2025. <br/> Abstract. We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart–Thomas functions, which is known from the literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [IMA J. Numer. Anal., 43 (2022), pp. 692–736] no longer holds. Even weaker [math]-convergence results no longer hold because the recovery sequences generally need to attain noninteger values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart–Thomas ansatz, which enables us to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143495222","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: Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle 勘误：通过均分原理生成网格的域分解方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-25 DOI: 10.1137/24m1693453
Martin J. Gander, Ronald D. Haynes, Felix Kwok
{"title":"Corrigendum: Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle","authors":"Martin J. Gander, Ronald D. Haynes, Felix Kwok","doi":"10.1137/24m1693453","DOIUrl":"https://doi.org/10.1137/24m1693453","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 461-467, February 2025. <br/> Abstract. Various nonlinear Schwarz domain decomposition methods were proposed to solve the one-dimensional equidistribution principle in [M. J. Gander and R. D. Haynes, SIAM J. Numer. Anal., 50 (2012), pp. 2111-2135]. A corrected proof of convergence for the linearized Schwarz algorithm presented in section 3.2, under additional hypotheses, is presented here. An alternative linearized Schwarz algorithm for equidistributed grid generation is also provided.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"83 1 Pt 2 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143495221","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
Rational Methods for Abstract, Linear, Nonhomogeneous Problems without Order Reduction 无阶约抽象、线性、非齐次问题的有理方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-02-24 DOI: 10.1137/24m165942x
Carlos Arranz-Simón, César Palencia
{"title":"Rational Methods for Abstract, Linear, Nonhomogeneous Problems without Order Reduction","authors":"Carlos Arranz-Simón, César Palencia","doi":"10.1137/24m165942x","DOIUrl":"https://doi.org/10.1137/24m165942x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 422-436, February 2025. <br/> Abstract. Starting from an A-stable rational approximation to [math] of order [math], [math], families of stable methods are proposed to time discretize abstract IVPs of the type [math]. These numerical procedures turn out to be of order [math], thus overcoming the order reduction phenomenon, and only one evaluation of [math] per step is required.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143485943","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
Long Time Stability and Numerical Stability of Implicit Schemes for Stochastic Heat Equations 随机热方程隐式格式的长时间稳定性和数值稳定性
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 参数化瓦瑟斯坦-哈密顿流
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 隐式积分算子的随机迹估计
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 线性Cosserat方程的混合有限元方法
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 无界时变算子存在下的二阶指数分裂：构造与收敛
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 脉动流体力学中Dean-Kawasaki方程的多层蒙特卡罗方法
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
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