arXiv - MATH - Numerical Analysis最新文献_第5页

High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well-balancedness and positivity 浅水方程自适应移动网格上的高阶精确结构保持有限体积方案：均衡性和正向性

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-15 DOI: arxiv-2409.09600

Zhihao Zhang, Huazhong Tang, Kailiang Wu

{"title":"High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well-balancedness and positivity","authors":"Zhihao Zhang, Huazhong Tang, Kailiang Wu","doi":"arxiv-2409.09600","DOIUrl":"https://doi.org/arxiv-2409.09600","url":null,"abstract":"This paper develops high-order accurate, well-balanced (WB), and\u0000positivity-preserving (PP) finite volume schemes for shallow water equations on\u0000adaptive moving structured meshes. The mesh movement poses new challenges in\u0000maintaining the WB property, which not only depends on the balance between flux\u0000gradients and source terms but is also affected by the mesh movement. To\u0000address these complexities, the WB property in curvilinear coordinates is\u0000decomposed into flux source balance and mesh movement balance. The flux source\u0000balance is achieved by suitable decomposition of the source terms, the\u0000numerical fluxes based on hydrostatic reconstruction, and appropriate\u0000discretization of the geometric conservation laws (GCLs). Concurrently, the\u0000mesh movement balance is maintained by integrating additional schemes to update\u0000the bottom topography during mesh adjustments. The proposed schemes are\u0000rigorously proven to maintain the WB property by using the discrete GCLs and\u0000these two balances. We provide rigorous analyses of the PP property under a\u0000sufficient condition enforced by a PP limiter. Due to the involvement of mesh\u0000metrics and movement, the analyses are nontrivial, while some standard\u0000techniques, such as splitting high-order schemes into convex combinations of\u0000formally first-order PP schemes, are not directly applicable. Various numerical\u0000examples validate the high-order accuracy, high efficiency, WB, and PP\u0000properties of the proposed schemes.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"215O 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Robust Training of Neural Networks at Arbitrary Precision and Sparsity 以任意精度和稀疏度进行神经网络的鲁棒性训练

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-14 DOI: arxiv-2409.09245

Chengxi Ye, Grace Chu, Yanfeng Liu, Yichi Zhang, Lukasz Lew, Andrew Howard

引用次数: 0

Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods 使用精确或近似交互邻域的非局部问题有限元方法的误差估计

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-14 DOI: arxiv-2409.09270

Qiang Du, Hehu Xie, Xiaobo Yin, Jiwei Zhang

引用次数: 0

Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor 基于张量的同步和块状三焦张量的低空白度

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-14 DOI: arxiv-2409.09313

Daniel Miao, Gilad Lerman, Joe Kileel

引用次数: 0

Two-grid convergence theory for symmetric positive semidefinite linear systems 对称正半有限线性系统的双网格收敛理论

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-14 DOI: arxiv-2409.09442

Xuefeng Xu

引用次数: 0

Factorization method for inverse elastic cavity scattering 反弹性空腔散射的因式分解法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-14 DOI: arxiv-2409.09434

Shuxin Li, Junliang Lv, Yi Wang

{"title":"Factorization method for inverse elastic cavity scattering","authors":"Shuxin Li, Junliang Lv, Yi Wang","doi":"arxiv-2409.09434","DOIUrl":"https://doi.org/arxiv-2409.09434","url":null,"abstract":"This paper is concerned with the inverse elastic scattering problem to\u0000determine the shape and location of an elastic cavity. By establishing a\u0000one-to-one correspondence between the Herglotz wave function and its kernel, we\u0000introduce the far-field operator which is crucial in the factorization method.\u0000We present a theoretical factorization of the far-field operator and rigorously\u0000prove the properties of its associated operators involved in the factorization.\u0000Unlike the Dirichlet problem where the boundary integral operator of the\u0000single-layer potential involved in the factorization of the far-field operator\u0000is weakly singular, the boundary integral operator of the conormal derivative\u0000of the double-layer potential involved in the factorization of the far-field\u0000operator with Neumann boundary conditions is hypersingular, which forces us to\u0000prove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we\u0000present theoretical analyses of the factorization method for various\u0000illumination and measurement cases, including compression-wave illumination and\u0000compression-wave measurement, shear-wave illumination and shear-wave\u0000measurement, and full-wave illumination and full-wave measurement. In addition,\u0000we also consider the limited aperture problem and provide a rigorous\u0000theoretical analysis of the factorization method in this case. Numerous\u0000numerical experiments are carried out to demonstrate the effectiveness of the\u0000proposed method, and to analyze the influence of various factors, such as\u0000polarization direction, frequency, wavenumber, and multi-scale scatterers on\u0000the reconstructed results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Randomized sketched TT-GMRES for linear systems with tensor structure 张量结构线性系统的随机草图 TT-GMRES

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-14 DOI: arxiv-2409.09471

Alberto Bucci, Davide Palitta, Leonardo Robol

{"title":"Randomized sketched TT-GMRES for linear systems with tensor structure","authors":"Alberto Bucci, Davide Palitta, Leonardo Robol","doi":"arxiv-2409.09471","DOIUrl":"https://doi.org/arxiv-2409.09471","url":null,"abstract":"In the last decade, tensors have shown their potential as valuable tools for\u0000various tasks in numerical linear algebra. While most of the research has been\u0000focusing on how to compress a given tensor in order to maintain information as\u0000well as reducing the storage demand for its allocation, the solution of linear\u0000tensor equations is a less explored venue. Even if many of the routines\u0000available in the literature are based on alternating minimization schemes\u0000(ALS), we pursue a different path and utilize Krylov methods instead. The use\u0000of Krylov methods in the tensor realm is not new. However, these routines often\u0000turn out to be rather expensive in terms of computational cost and ALS\u0000procedures are preferred in practice. We enhance Krylov methods for linear\u0000tensor equations with a panel of diverse randomization-based strategies which\u0000remarkably increase the efficiency of these solvers making them competitive\u0000with state-of-the-art ALS schemes. The up-to-date randomized approaches we\u0000employ range from sketched Krylov methods with incomplete orthogonalization and\u0000structured sketching transformations to streaming algorithms for tensor\u0000rounding. The promising performance of our new solver for linear tensor\u0000equations is demonstrated by many numerical results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Lattice Boltzmann framework for multiphase flows by Eulerian-Eulerian Navier-Stokes equations 通过欧拉-欧拉纳维-斯托克斯方程实现多相流的晶格玻尔兹曼框架

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-13 DOI: arxiv-2409.10399

Matteo Maria Piredda, Pietro Asinari

引用次数: 0

FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition FB-HyDON：通过超网络和有限基域分解对复杂 PDE 进行参数高效的物理信息算子学习

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-13 DOI: arxiv-2409.09207

Milad Ramezankhani, Rishi Yash Parekh, Anirudh Deodhar, Dagnachew Birru

{"title":"FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition","authors":"Milad Ramezankhani, Rishi Yash Parekh, Anirudh Deodhar, Dagnachew Birru","doi":"arxiv-2409.09207","DOIUrl":"https://doi.org/arxiv-2409.09207","url":null,"abstract":"Deep operator networks (DeepONet) and neural operators have gained\u0000significant attention for their ability to map infinite-dimensional function\u0000spaces and perform zero-shot super-resolution. However, these models often\u0000require large datasets for effective training. While physics-informed operators\u0000offer a data-agnostic learning approach, they introduce additional training\u0000complexities and convergence issues, especially in highly nonlinear systems. To\u0000overcome these challenges, we introduce Finite Basis Physics-Informed\u0000HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic\u0000domain decomposition. By leveraging hypernetworks and finite basis functions,\u0000FB-HyDON effectively mitigates the training limitations associated with\u0000existing physics-informed operator learning methods. We validated our approach\u0000on the high-frequency harmonic oscillator, Burgers' equation at different\u0000viscosity levels, and Allen-Cahn equation demonstrating substantial\u0000improvements over other operator learning models.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Neural network Approximations for Reaction-Diffusion Equations -- Homogeneous Neumann Boundary Conditions and Long-time Integrations 反应-扩散方程的神经网络近似 -- 均质新曼边界条件和长时间积分

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-13 DOI: arxiv-2409.08941

Eddel Elí Ojeda Avilés, Jae-Hun Jung, Daniel Olmos Liceaga

{"title":"Neural network Approximations for Reaction-Diffusion Equations -- Homogeneous Neumann Boundary Conditions and Long-time Integrations","authors":"Eddel Elí Ojeda Avilés, Jae-Hun Jung, Daniel Olmos Liceaga","doi":"arxiv-2409.08941","DOIUrl":"https://doi.org/arxiv-2409.08941","url":null,"abstract":"Reaction-Diffusion systems arise in diverse areas of science and engineering.\u0000Due to the peculiar characteristics of such equations, analytic solutions are\u0000usually not available and numerical methods are the main tools for\u0000approximating the solutions. In the last decade, artificial neural networks\u0000have become an active area of development for solving partial differential\u0000equations. However, several challenges remain unresolved with these methods\u0000when applied to reaction-diffusion equations. In this work, we focus on two\u0000main problems. The implementation of homogeneous Neumann boundary conditions\u0000and long-time integrations. For the homogeneous Neumann boundary conditions, we\u0000explore four different neural network methods based on the PINN approach. For\u0000the long time integration in Reaction-Diffusion systems, we propose a domain\u0000splitting method in time and provide detailed comparisons between different\u0000implementations of no-flux boundary conditions. We show that the domain\u0000splitting method is crucial in the neural network approach, for long time\u0000integration in Reaction-Diffusion systems. We demonstrate numerically that\u0000domain splitting is essential for avoiding local minima, and the use of\u0000different boundary conditions further enhances the splitting technique by\u0000improving numerical approximations. To validate the proposed methods, we\u0000provide numerical examples for the Diffusion, the Bistable and the Barkley\u0000equations and provide a detailed discussion and comparisons of the proposed\u0000methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0