SIAM Journal on Numerical Analysis最新文献_第9页

Numerical Analysis for Convergence of a Sample-Wise Backpropagation Method for Training Stochastic Neural Networks 用于训练随机神经网络的采样-明智反向传播方法收敛性的数值分析

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-03-01 DOI: 10.1137/22m1523765

Richard Archibald, Feng Bao, Yanzhao Cao, Hui Sun

{"title":"Numerical Analysis for Convergence of a Sample-Wise Backpropagation Method for Training Stochastic Neural Networks","authors":"Richard Archibald, Feng Bao, Yanzhao Cao, Hui Sun","doi":"10.1137/22m1523765","DOIUrl":"https://doi.org/10.1137/22m1523765","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 593-621, April 2024. Abstract. The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807–2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001008","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 Numerical Framework for Nonlinear Peridynamics on Two-Dimensional Manifolds Based on Implicit P-(EC)[math] Schemes 基于隐式 P-(EC)[math]方案的二维平面上非线性周流体力学数值框架

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-03-01 DOI: 10.1137/22m1498942

Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi

{"title":"A Numerical Framework for Nonlinear Peridynamics on Two-Dimensional Manifolds Based on Implicit P-(EC)[math] Schemes","authors":"Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi","doi":"10.1137/22m1498942","DOIUrl":"https://doi.org/10.1137/22m1498942","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024. Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)[math] formulation of the implicit [math]-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140015424","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

Homogenization of Nondivergence-Form Elliptic Equations with Discontinuous Coefficients and Finite Element Approximation of the Homogenized Problem 具有不连续系数的非发散形式椭圆方程的均质化和均质化问题的有限元逼近

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-03-01 DOI: 10.1137/23m1580279

Timo Sprekeler

{"title":"Homogenization of Nondivergence-Form Elliptic Equations with Discontinuous Coefficients and Finite Element Approximation of the Homogenized Problem","authors":"Timo Sprekeler","doi":"10.1137/23m1580279","DOIUrl":"https://doi.org/10.1137/23m1580279","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 646-666, April 2024. Abstract. We study the homogenization of the equation [math] posed in a bounded convex domain [math] subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic, and symmetric diffusion matrix [math] is merely assumed to be essentially bounded and (if [math]) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax–Milgram-type problem, we obtain [math]-bounds for periodic problems in double-divergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of Hölder continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140015460","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

Virtual Element Methods Without Extrinsic Stabilization 无外在稳定的虚拟元素方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-20 DOI: 10.1137/22m1504196

Chunyu Chen, Xuehai Huang, Huayi Wei

引用次数: 0

A Universal Median Quasi-Monte Carlo Integration 通用中值准蒙特卡罗积分法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-16 DOI: 10.1137/22m1525077

Takashi Goda, Kosuke Suzuki, Makoto Matsumoto

{"title":"A Universal Median Quasi-Monte Carlo Integration","authors":"Takashi Goda, Kosuke Suzuki, Makoto Matsumoto","doi":"10.1137/22m1525077","DOIUrl":"https://doi.org/10.1137/22m1525077","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 533-566, February 2024. Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750213","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

High Order Splitting Methods for SDEs Satisfying a Commutativity Condition 满足换元条件的 SDE 的高阶分裂方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-15 DOI: 10.1137/23m155147x

James M. Foster, Gonçalo dos Reis, Calum Strange

{"title":"High Order Splitting Methods for SDEs Satisfying a Commutativity Condition","authors":"James M. Foster, Gonçalo dos Reis, Calum Strange","doi":"10.1137/23m155147x","DOIUrl":"https://doi.org/10.1137/23m155147x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 500-532, February 2024. Abstract. In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal, and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely, Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs—which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox–Ingersoll–Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh–Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of [math] and outperform schemes previously proposed in the literature.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139739343","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

Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation 用于波方程分布式优化控制的时空有限元方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1137/22m1532962

Richard Löscher, Olaf Steinbach

{"title":"Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation","authors":"Richard Löscher, Olaf Steinbach","doi":"10.1137/22m1532962","DOIUrl":"https://doi.org/10.1137/22m1532962","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024. Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math], this does not define a solution isomorphism. Hence, we use an appropriate state space [math] such that the wave operator becomes an isomorphism from [math] onto [math]. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error [math] between the computed space-time finite element solution [math] and the target function [math] with respect to the regularization parameter [math], and the space-time finite element mesh size [math], depending on the regularity of the desired state [math]. These estimates lead to the optimal choice [math] in order to define the regularization parameter [math] for a given space-time finite element mesh size [math] or to determine the required mesh size [math] when [math] is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139700968","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 Uncertainty Quantification of Eigenvalues and Eigenspaces with Higher Multiplicity 论高倍性特征值和特征空间的不确定性量化

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1137/22m1529324

Jürgen Dölz, David Ebert

引用次数: 0

Convergence Analysis for Bregman Iterations in Minimizing a Class of Landau Free Energy Functionals 最小化一类朗道自由能函数的布雷格曼迭代收敛分析

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1137/22m1517664

Chenglong Bao, Chang Chen, Kai Jiang, Lingyun Qiu

引用次数: 0

Frequency-Explicit A Posteriori Error Estimates for Discontinuous Galerkin Discretizations of Maxwell’s Equations 麦克斯韦方程非连续伽勒金离散化的频率显式后验误差估计值

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-02-06 DOI: 10.1137/22m1516348

Théophile Chaumont-Frelet, Patrick Vega

引用次数: 0