Journal of Scientific Computing最新文献

Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations. 自动微分是训练神经网络求解微分方程的关键。

IF 3.3 2区数学

Journal of Scientific Computing Pub Date : 2025-08-01 Epub Date: 2025-06-24 DOI: 10.1007/s10915-025-02965-3

Chuqi Chen, Yahong Yang, Yang Xiang, Wenrui Hao

{"title":"Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations.","authors":"Chuqi Chen, Yahong Yang, Yang Xiang, Wenrui Hao","doi":"10.1007/s10915-025-02965-3","DOIUrl":"10.1007/s10915-025-02965-3","url":null,"abstract":"Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One advantage of the neural network methods for PDEs lies in its automatic differentiation (AD), which necessitates only the sample points themselves, unlike traditional finite difference (FD) approximations that require nearby local points to compute derivatives. In this paper, we quantitatively demonstrate the advantage of AD in training neural networks. The concept of truncated entropy is introduced to characterize the training property. Specifically, through comprehensive experimental and theoretical analyses conducted on random feature models and two-layer neural networks, we discover that the defined truncated entropy serves as a reliable metric for quantifying the residual loss of random feature models and the training speed of neural networks for both AD and FD methods. Our experimental and theoretical analyses demonstrate that, from a training perspective, AD outperforms FD in solving PDEs.","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"104 2","pages":""},"PeriodicalIF":3.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12407148/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145001825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Homotopy Relaxation Training Algorithms for Infinite-Width Two-Layer ReLU Neural Networks. 无限宽双层ReLU神经网络的同伦松弛训练算法。

IF 2.8 2区数学

Journal of Scientific Computing Pub Date : 2025-02-01 Epub Date: 2025-01-03 DOI: 10.1007/s10915-024-02761-5

Yahong Yang, Qipin Chen, Wenrui Hao

引用次数: 0

A Sparse Hierarchical hp-Finite Element Method on Disks and Annuli. 圆盘和环空的稀疏层次hp有限元方法。

IF 2.8 2区数学

Journal of Scientific Computing Pub Date : 2025-01-01 Epub Date: 2025-06-23 DOI: 10.1007/s10915-025-02964-4

Ioannis P A Papadopoulos, Sheehan Olver

{"title":"A Sparse Hierarchical hp-Finite Element Method on Disks and Annuli.","authors":"Ioannis P A Papadopoulos, Sheehan Olver","doi":"10.1007/s10915-025-02964-4","DOIUrl":"https://doi.org/10.1007/s10915-025-02964-4","url":null,"abstract":"We develop a sparse hierarchical hp-finite element method (hp-FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse hp-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. Rotationally anisotropic coefficients that are approximated by low-degree polynomials in Cartesian coordinates also result in sparse linear systems. e consider examples such as a high-frequency Helmholtz equation with radial discontinuities and rotationally anisotropic coefficients, singular source terms, țhe time-dependent Schrödinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"104 2","pages":"51"},"PeriodicalIF":2.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12185591/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144499024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Unified Discontinuous Galerkin Analysis of a Thermo/Poro-viscoelasticity Model. 热/孔粘弹性模型的统一不连续伽辽金分析。

IF 3.3 2区数学

Journal of Scientific Computing Pub Date : 2025-01-01 Epub Date: 2025-09-02 DOI: 10.1007/s10915-025-03016-7

Stefano Bonetti, Mattia Corti

引用次数: 0

A Posteriori Error Analysis for a Coupled Stokes-Poroelastic System with Multiple Compartments. 多隔室stokes -孔隙弹性耦合系统的后验误差分析。

IF 2.8 2区数学

Journal of Scientific Computing Pub Date : 2025-01-01 Epub Date: 2025-03-08 DOI: 10.1007/s10915-025-02814-3

Ivan Fumagalli, Nicola Parolini, Marco Verani

{"title":"A Posteriori Error Analysis for a Coupled Stokes-Poroelastic System with Multiple Compartments.","authors":"Ivan Fumagalli, Nicola Parolini, Marco Verani","doi":"10.1007/s10915-025-02814-3","DOIUrl":"10.1007/s10915-025-02814-3","url":null,"abstract":"The computational effort entailed in the discretization of fluid-poromechanics systems is typically highly demanding. This is particularly true for models of multiphysics flows in the brain, due to the geometrical complexity of the cerebral anatomy-requiring a very fine computational mesh for finite element discretization-and to the high number of variables involved. Indeed, this kind of problems can be modeled by a coupled system encompassing the Stokes equations for the cerebrospinal fluid in the brain ventricles and Multiple-network Poro-Elasticity (MPE) equations describing the brain tissue, the interstitial fluid, and the blood vascular networks at different space scales. The present work aims to rigorously derive a posteriori error estimates for the coupled Stokes-MPE problem, as a first step towards the design of adaptive refinement strategies or reduced order models to decrease the computational demand of the problem. Through numerical experiments, we verify the reliability and optimal efficiency of the proposed a posteriori estimator and identify the role of the different solution variables in its composition.","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"103 1","pages":"22"},"PeriodicalIF":2.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11890245/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143598197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Surrogate Modeling of Resonant Behavior in Scattering Problems Through Adaptive Rational Approximation and Sketching. 基于自适应理性逼近和素描的散射问题共振行为代理建模。

IF 3.3 2区数学

Journal of Scientific Computing Pub Date : 2025-01-01 Epub Date: 2025-08-09 DOI: 10.1007/s10915-025-03020-x

Davide Pradovera, Ralf Hiptmair, Ilaria Perugia

{"title":"Surrogate Modeling of Resonant Behavior in Scattering Problems Through Adaptive Rational Approximation and Sketching.","authors":"Davide Pradovera, Ralf Hiptmair, Ilaria Perugia","doi":"10.1007/s10915-025-03020-x","DOIUrl":"10.1007/s10915-025-03020-x","url":null,"abstract":"This paper describes novel algorithms for the identification of (almost-)resonant behavior in scattering problems. Our methods, relying on rational approximation, aim at building surrogate models of what we call \"field amplification\", defined as the norm of the solution operator of the scattering problem, which we express through boundary-integral equations. To provide our techniques with theoretical foundations, we first derive results linking the field amplification to the spectral properties of the operator that defines the scattering problem. Such results are then used to justify the use of rational approximation in the surrogate-modeling task. Some of our proposed methods apply rational approximation in a \"standard\" way, building a rational approximant for either the solution operator directly or, in the interest of computational efficiency, for a randomly \"sketched\" version of it. Our other \"hybrid\" approaches are more innovative, combining rational-approximation-assisted root-finding with approximation using radial basis functions. Three key features of our methods are that (i) they are agnostic of the strategy used to discretize the scattering problem, (ii) they do not require any computations involving non-real wavenumbers, and (iii) they can adjust to different settings through the use of adaptive sampling strategies. We carry out some numerical experiments involving 2D scatterers to compare our approaches. In our tests, two of our approaches (one standard, one hybrid) emerge as the best performers, with one or the other being preferable, depending on whether emphasis is placed on accuracy or efficiency.","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"104 3","pages":"104"},"PeriodicalIF":3.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12335407/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144823108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems. 求解拟线性椭圆方程和系统的有效的张量积拟牛顿法。

IF 2.8 2区数学

Journal of Scientific Computing Pub Date : 2025-01-01 Epub Date: 2025-04-30 DOI: 10.1007/s10915-025-02897-y

Wenrui Hao, Sun Lee, Xiangxiong Zhang

{"title":"An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems.","authors":"Wenrui Hao, Sun Lee, Xiangxiong Zhang","doi":"10.1007/s10915-025-02897-y","DOIUrl":"10.1007/s10915-025-02897-y","url":null,"abstract":"In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method reduces the computational overhead typical of traditional Newton methods while handling the large, sparse matrices generated from discretized PDEs. We also provide a convergence analysis demonstrating local convergence to the exact solution under optimal choices for the regularization parameter, ensuring stability and efficiency in each iteration. Numerical experiments in two- and three-dimensional domains validate the proposed method's robustness and computational gains with tensor product implementation. This approach offers a promising pathway for accelerating quasi-linear elliptic equations and systems solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities.","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"103 3","pages":"89"},"PeriodicalIF":2.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12043795/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144057248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery 基于非线性层次矩阵因数分解的张量环逼近法实现多维图像复原

IF 2.5 2区数学

Journal of Scientific Computing Pub Date : 2024-09-16 DOI: 10.1007/s10915-024-02670-7

Wei-Hao Wu, Ting-Zhu Huang, Xi-Le Zhao, Hao Zhang, Zhi-Long Han

{"title":"Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery","authors":"Wei-Hao Wu, Ting-Zhu Huang, Xi-Le Zhao, Hao Zhang, Zhi-Long Han","doi":"10.1007/s10915-024-02670-7","DOIUrl":"https://doi.org/10.1007/s10915-024-02670-7","url":null,"abstract":"Recently, tensor ring (TR) approximation has received increasing attention in multi-dimensional image processing. In TR approximation, the key backbone is the shallow matrix factorizations, which approximate the circular unfolding of the multi-dimensional image. However, the shallow matrix factorization limits the standard TR approximation’s ability to represent images with complex details and textures. To address this limitation, we propose a nonlinear hierarchical matrix factorization-based tensor ring (NHTR) approximation. Specifically, instead of the shallow matrix factorization, we introduce the nonlinear hierarchical matrix factorization in NHTR approximation to approximate circularly (lceil frac{N}{2}rceil )-modes unfoldings of an N-th order tensor. Benefiting from the powerful expressive capability of the nonlinear hierarchical matrix factorization, the proposed NHTR approximation can faithfully capture fine details of the clean image compared to classical tensor ring approximation. Empowered with the proposed NHTR, we build a multi-dimensional image recovery model and establish a theoretical error bound between the recovered image and the clean image based on the proposed model. To solve the highly nonlinear and hierarchical optimization problem, we develop an efficient alternating minimization-based algorithm. Experiments on multispectral images and color videos conclusively demonstrate the superior performance of our method over the compared state-of-the-art methods in multi-dimensional image recovery.","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"51 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256274","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

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation 分数科特韦格-德-弗里斯方程的完全离散有限差分方案

IF 2.5 2区数学

Journal of Scientific Computing Pub Date : 2024-09-14 DOI: 10.1007/s10915-024-02672-5

Mukul Dwivedi, Tanmay Sarkar

引用次数: 0

Curvature-Dependent Elastic Bending Total Variation Model for Image Inpainting with the SAV Algorithm 利用 SAV 算法绘制图像的曲率相关弹性弯曲总变化模型

IF 2.5 2区数学

Journal of Scientific Computing Pub Date : 2024-09-12 DOI: 10.1007/s10915-024-02666-3

Caixia Nan, Zhonghua Qiao, Qian Zhang

引用次数: 0