SIAM Journal on Numerical Analysis最新文献

Erratum: Multidimensional Sum-Up Rounding for Elliptic Control Systems

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

Corrigendum: A New Lagrange Multiplier Approach for Constructing Structure-Preserving Schemes, II. Bound Preserving

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

Swarm-Based Gradient Descent Meets Simulated Annealing

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

Multiple Relaxation Exponential Runge–Kutta Methods for the Nonlinear Schrödinger Equation

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

Stable and Accurate Least Squares Radial Basis Function Approximations on Bounded Domains

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-04 DOI: 10.1137/23m1593243

Ben Adcock, Daan Huybrechs, Cecile Piret

{"title":"Stable and Accurate Least Squares Radial Basis Function Approximations on Bounded Domains","authors":"Ben Adcock, Daan Huybrechs, Cecile Piret","doi":"10.1137/23m1593243","DOIUrl":"https://doi.org/10.1137/23m1593243","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2698-2718, December 2024. Abstract. The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape parameter with the degrees of freedom is optimal, resulting in constant overlap between neighboring RBF’s regardless of their number, and we propose an explicit suitable choice of the proportionality constant. We show numerically that highly accurate approximations to smooth functions can also be obtained on bounded domains in several dimensions, using a linear scaling with the degrees of freedom per dimension. We extend the least squares approach to a collocation-based method for the solution of elliptic boundary value problems and illustrate that the combination of centers outside the domain, oversampling, and optimal scaling can result in accuracy close to machine precision in spite of having to solve very ill-conditioned linear systems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142763139","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 Second-Order, Global-in-Time Energy Stable Implicit-Explicit Runge–Kutta Scheme for the Phase Field Crystal Equation

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-12-03 DOI: 10.1137/24m1637623

Hong Zhang, Haifeng Wang, Xueqing Teng

{"title":"A Second-Order, Global-in-Time Energy Stable Implicit-Explicit Runge–Kutta Scheme for the Phase Field Crystal Equation","authors":"Hong Zhang, Haifeng Wang, Xueqing Teng","doi":"10.1137/24m1637623","DOIUrl":"https://doi.org/10.1137/24m1637623","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2667-2697, December 2024. Abstract. We develop a two-stage, second-order, global-in-time energy stable implicit-explicit Runge–Kutta (IMEX RK(2, 2)) scheme for the phase field crystal equation with an [math] time step constraint, and without the global Lipschitz assumption. A linear stabilization term is introduced to the system with Fourier pseudo-spectral spatial discretization, and a novel compact reformulation is devised by rewriting the IMEX RK(2, 2) scheme as an approximation to the variation-of-constants formula. Under the assumption that all stage solutions are a priori bounded in the [math] norm, we first demonstrate that the original energy obtained by this second-order scheme is nonincreasing for any time step with a sufficiently large stabilization parameter. To justify the a priori [math] bound assumption, we establish a uniform-in-time [math] estimate for all stage solutions, subject to an [math] time step constraint. This results in a uniform-in-time bound for all stage solutions through discrete Sobolev embedding from [math] to [math]. Consequently, we achieve an [math] stabilization parameter, ensuring global-in-time energy stability. Additionally, we provide an optimal rate convergence analysis and error estimate for the IMEX RK(2, 2) scheme in the [math] norm. The global-in-time energy stability represents a novel achievement for a two-stage, second-order accurate scheme for a gradient flow without the global Lipschitz assumption. Numerical experiments substantiate the second-order accuracy and energy stability of the proposed scheme.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142760547","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

On the Existence of Minimizers in Shallow Residual ReLU Neural Network Optimization Landscapes 论浅残差 ReLU 神经网络优化景观中最小值的存在性

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-26 DOI: 10.1137/23m1556241

Steffen Dereich, Arnulf Jentzen, Sebastian Kassing

引用次数: 0

A Domain Decomposition Method for Stochastic Evolution Equations 随机演化方程的领域分解法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-20 DOI: 10.1137/24m1629845

Evelyn Buckwar, Ana Djurdjevac, Monika Eisenmann

{"title":"A Domain Decomposition Method for Stochastic Evolution Equations","authors":"Evelyn Buckwar, Ana Djurdjevac, Monika Eisenmann","doi":"10.1137/24m1629845","DOIUrl":"https://doi.org/10.1137/24m1629845","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024. Abstract. In recent years, stochastic partial differential equations (SPDEs) have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"81 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679164","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

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems II: Neumann–Neumann Algorithms 抛物线最优控制问题的新时域分解方法 II：诺伊曼-诺伊曼算法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-19 DOI: 10.1137/24m1634424

Martin J. Gander, Liu-Di Lu

引用次数: 0

The Mean-Field Ensemble Kalman Filter: Near-Gaussian Setting 平均场集合卡尔曼滤波器：近高斯背景

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-11-15 DOI: 10.1137/24m1628207

J. A. Carrillo, F. Hoffmann, A. M. Stuart, U. Vaes

{"title":"The Mean-Field Ensemble Kalman Filter: Near-Gaussian Setting","authors":"J. A. Carrillo, F. Hoffmann, A. M. Stuart, U. Vaes","doi":"10.1137/24m1628207","DOIUrl":"https://doi.org/10.1137/24m1628207","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2549-2587, December 2024. Abstract. The ensemble Kalman filter is widely used in applications because, for high-dimensional filtering problems, it has a robustness that is not shared, for example, by the particle filter; in particular, it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue, we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. We prove two types of results: The first type comprises a stability estimate controlling the error made by the ensemble Kalman filter in terms of the difference between the true filtering distribution and a nearby Gaussian, and the second type uses this stability result to show that, in a neighborhood of Gaussian problems, the ensemble Kalman filter makes a small error in comparison with the true filtering distribution. Our analysis is developed for the mean-field ensemble Kalman filter. We rewrite the update equations for this filter and for the true filtering distribution in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters, and we prove various stability estimates for the maps defining the evolution of the two filters in this metric. Using these stability estimates, we prove results of the first and second types in the weighted total variation metric. We also provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean-field description of the unscented Kalman filter.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642576","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