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Matrix-Free Inexact Preconditioning Techniques for Isogeometric Tensor-Product Discretizations. 等几何张量积离散化的无矩阵不精确预处理技术。
IF 2.6 2区 数学
Journal of Scientific Computing Pub Date : 2026-01-01 Epub Date: 2026-04-27 DOI: 10.1007/s10915-026-03253-4
Michał Ł Mika, René R Hiemstra, Dominik Schillinger
{"title":"Matrix-Free Inexact Preconditioning Techniques for Isogeometric Tensor-Product Discretizations.","authors":"Michał Ł Mika, René R Hiemstra, Dominik Schillinger","doi":"10.1007/s10915-026-03253-4","DOIUrl":"10.1007/s10915-026-03253-4","url":null,"abstract":"<p><p>We propose a matrix-free inexact preconditioning strategy for elliptic partial differential equations discretized by the isogeometric Galerkin method on tensor-product spline spaces. We base our preconditioner on an approximation of the discrete linear operator by a sum of Kronecker product matrices. The action of its inverse on a vector of coefficients is approximated by an inner preconditioned conjugate gradient solve. The forward problem is solved by the inexact preconditioned conjugate gradient method. The complexity of the Kronecker matrix-vector products in the inner iteration is lower than the complexity of the matrix-vector products in the forward problem, leading to a reduced number of iterations and significant performance gains. We show the robustness, efficiency and effectiveness of our approach in test problems involving the Poisson equation and linear elasticity, and illustrate the performance gain with respect to preconditioning techniques based on fast diagonalization. The proposed method is implemented in our open-source Julia framework for spline based discretization methods.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"107 3","pages":"81"},"PeriodicalIF":2.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13121230/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147787376","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
Fast Numerical Solvers for Parameter Identification Problems in Mathematical Biology. 数学生物学参数辨识问题的快速数值求解。
IF 2.6 2区 数学
Journal of Scientific Computing Pub Date : 2026-01-01 Epub Date: 2026-03-16 DOI: 10.1007/s10915-025-03170-y
Karolína Benková, John W Pearson, Mariya Ptashnyk
{"title":"Fast Numerical Solvers for Parameter Identification Problems in Mathematical Biology.","authors":"Karolína Benková, John W Pearson, Mariya Ptashnyk","doi":"10.1007/s10915-025-03170-y","DOIUrl":"10.1007/s10915-025-03170-y","url":null,"abstract":"<p><p>In this paper, we consider effective discretization strategies and iterative solvers for nonlinear PDE-constrained optimization models of pattern evolution within biological processes. Upon a Sequential Quadratic Programming linearization of the optimization problem, we devise appropriate time-stepping schemes and discrete approximations of the cost functionals such that the discretization and optimization operations are commutative, a highly desirable property of a discretization of such problems. We formulate the large-scale, coupled linear systems in such a way that efficient preconditioned iterative methods can be applied within a Krylov subspace solver. Numerical experiments demonstrate the viability and efficiency of our approach.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"107 1","pages":"24"},"PeriodicalIF":2.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12992359/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147482075","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
High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge-Kutta Methods. 基于显式龙格-库塔方法的高阶BUG动态低秩积分器。
IF 2.6 2区 数学
Journal of Scientific Computing Pub Date : 2026-01-01 Epub Date: 2026-05-13 DOI: 10.1007/s10915-026-03317-5
Fabio Nobile, Sébastien Riffaud
{"title":"High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge-Kutta Methods.","authors":"Fabio Nobile, Sébastien Riffaud","doi":"10.1007/s10915-026-03317-5","DOIUrl":"https://doi.org/10.1007/s10915-026-03317-5","url":null,"abstract":"<p><p>In this work, we introduce high-order Basis-Update & Galerkin (BUG) integrators based on explicit Runge-Kutta methods for large-scale matrix differential equations. These dynamical low-rank integrators extend the BUG integrator [1] to arbitrary explicit Runge-Kutta schemes by performing a BUG step at each stage of the method. The resulting Runge-Kutta BUG (RK-BUG) integrators are robust with respect to small singular values, fully forward in time, and high-order accurate, while enabling conservation and rank adaptivity. We prove that RK-BUG integrators retain the order of convergence of the underlying Runge-Kutta method until the error reaches a plateau corresponding to the low-rank truncation error, which vanishes as the rank becomes full. This theoretical analysis is supported by several numerical experiments. The results demonstrate the high-order convergence of the RK-BUG integrator and its superior accuracy compared to other existing dynamical low-rank integrators.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"107 3","pages":"102"},"PeriodicalIF":2.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13677646/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149225247","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
Convergence Analysis of Iterative Deep Learning Algorithms for Fully Nonlinear BSPDEs in Non-Markovian Utility Maximization. 非马尔可夫效用最大化下全非线性BSPDEs迭代深度学习算法的收敛性分析
IF 2.6 2区 数学
Journal of Scientific Computing Pub Date : 2026-01-01 Epub Date: 2026-07-28 DOI: 10.1007/s10915-026-03416-3
Jingtang Ma, Haofei Wu, H Harry Zheng
{"title":"Convergence Analysis of Iterative Deep Learning Algorithms for Fully Nonlinear BSPDEs in Non-Markovian Utility Maximization.","authors":"Jingtang Ma, Haofei Wu, H Harry Zheng","doi":"10.1007/s10915-026-03416-3","DOIUrl":"10.1007/s10915-026-03416-3","url":null,"abstract":"<p><p>In utility maximization with non-Markovian setting, the value function satisfies a stochastic Hamilton-Jacobi-Bellman (HJB) equation, a fully nonlinear backward stochastic partial differential equation (BSPDE). We propose iterative deep learning algorithms for such BSPDEs and analyze their convergence. We derive the error estimate of the time discretization scheme for BSPDEs, that of the policy iteration scheme for discretized BSPDEs, and that of the iterative deep learning scheme for discretized BSPDEs. We also test the algorithms and show their convergence and accuracy with numerical examples, including Markovian Heston volatility model and non-Markovian rough volatility model.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"108 3","pages":"95"},"PeriodicalIF":2.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13415659/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148622373","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
The Effect of Quadrature on the Convergence of Policy Iteration for Hamilton-Jacobi-Bellman Equations. 正交对Hamilton-Jacobi-Bellman方程策略迭代收敛性的影响。
IF 2.6 2区 数学
Journal of Scientific Computing Pub Date : 2026-01-01 Epub Date: 2026-10-08 DOI: 10.1007/s10915-026-03480-9
T Hall, I Smears, E Süli, H Wells
{"title":"The Effect of Quadrature on the Convergence of Policy Iteration for Hamilton-Jacobi-Bellman Equations.","authors":"T Hall, I Smears, E Süli, H Wells","doi":"10.1007/s10915-026-03480-9","DOIUrl":"https://doi.org/10.1007/s10915-026-03480-9","url":null,"abstract":"<p><p>Modern finite element libraries allow users to express partial differential equations directly in variational form, with the added convenience of automatic quadrature selection. In the context of Hamilton-Jacobi-Bellman (HJB) equations, automatic quadrature selection can result in nonmatching quadratures between different terms that may lead to loss of convergence of the policy iteration, which is otherwise expected from theory to converge superlinearly. The simple remedy of enforcing matching quadrature recovers the expected superlinear convergence.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"109 3","pages":"79"},"PeriodicalIF":2.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13649925/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149171221","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
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 2.6 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":"<p><p>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.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"104 2","pages":""},"PeriodicalIF":2.6,"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 2.6 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
{"title":"Homotopy Relaxation Training Algorithms for Infinite-Width Two-Layer ReLU Neural Networks.","authors":"Yahong Yang, Qipin Chen, Wenrui Hao","doi":"10.1007/s10915-024-02761-5","DOIUrl":"10.1007/s10915-024-02761-5","url":null,"abstract":"<p><p>In this paper, we present a novel training approach called the Homotopy Relaxation Training Algorithm (HRTA), aimed at accelerating the training process in contrast to traditional methods. Our algorithm incorporates two key mechanisms: one involves building a homotopy activation function that seamlessly connects the linear activation function with the <math><mi>R</mi> <mi>e</mi> <mi>L</mi> <mi>U</mi></math> activation function; the other technique entails relaxing the homotopy parameter to enhance the training refinement process. We have conducted an in-depth analysis of this novel method within the context of the neural tangent kernel (NTK), revealing significantly improved convergence rates. Our experimental results, especially when considering networks with larger widths, validate the theoretical conclusions. This proposed HRTA exhibits the potential for other activation functions and deep neural networks.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"102 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12074661/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144038484","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
Mimetic Metrics for the DGSEM. DGSEM的模拟度量。
IF 3.3 2区 数学
Journal of Scientific Computing Pub Date : 2025-01-01 Epub Date: 2025-10-14 DOI: 10.1007/s10915-025-03082-x
Daniel Bach, Andrés Rueda-Ramírez, David A Kopriva, Gregor J Gassner
{"title":"Mimetic Metrics for the DGSEM.","authors":"Daniel Bach, Andrés Rueda-Ramírez, David A Kopriva, Gregor J Gassner","doi":"10.1007/s10915-025-03082-x","DOIUrl":"https://doi.org/10.1007/s10915-025-03082-x","url":null,"abstract":"<p><p>Free-stream preservation is an essential property for numerical solvers on curvilinear grids. Key to this property is that the metric terms of the curvilinear mapping satisfy discrete metric identities, i.e., have zero divergence. Divergence-free metric terms are furthermore essential for entropy stability on curvilinear grids. We present a new way to compute the metric terms for discontinuous Galerkin spectral element methods (DGSEMs) that guarantees they are divergence-free. The proposed mimetic approach uses projections that fit within the de Rham Cohomology.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"105 2","pages":"57"},"PeriodicalIF":3.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12521316/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145309771","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
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 <i>hp</i>-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":"<p><p>We develop a sparse hierarchical <i>hp</i>-finite element method (<i>hp</i>-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 <i>hp</i>-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.</p>","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
{"title":"Unified Discontinuous Galerkin Analysis of a Thermo/Poro-viscoelasticity Model.","authors":"Stefano Bonetti, Mattia Corti","doi":"10.1007/s10915-025-03016-7","DOIUrl":"10.1007/s10915-025-03016-7","url":null,"abstract":"<p><p>We present and analyze a discontinuous Galerkin method for the numerical modeling of a Kelvin-Voigt thermo/poro-viscoelastic problem. We present the derivation of the model and we develop a stability analysis in the continuous setting that holds both for the full inertial and quasi-static problems and that is robust with respect to most of the physical parameters of the problem. For spatial discretization, we propose an arbitrary-order weighted symmetric interior penalty scheme that supports general polytopal grids and is robust with respect to strong heterogeneities in the model coefficients. For the semi-discrete problem, we prove the extension of the stability result demonstrated in the continuous setting and we provide an a-priori error estimate. A wide set of numerical simulations is presented to assess the convergence and robustness properties of the proposed method. Moreover, we test the scheme with literature and physically sound test cases for <i>proof-of-concept</i> applications in the geophysical context.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"105 1","pages":"11"},"PeriodicalIF":3.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12405049/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145001808","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
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