{"title":"地球物理标记单元内模拟中严重条件不良线性系统的稀疏求解器比较","authors":"Marcel Ferrari","doi":"arxiv-2409.11515","DOIUrl":null,"url":null,"abstract":"Solving sparse linear systems is a critical challenge in many scientific and\nengineering fields, particularly when these systems are severely\nill-conditioned. This work aims to provide a comprehensive comparison of\nvarious solvers designed for such problems, offering valuable insights and\nguidance for domain scientists and researchers. We develop the tools required\nto accurately evaluate the performance and correctness of 16 solvers from 11\nstate-of-the-art numerical libraries, focusing on their effectiveness in\nhandling ill-conditioned matrices. The solvers were tested on linear systems\narising from a coupled hydro-mechanical marker-in-cell geophysical simulation.\nTo address the challenge of computing accurate error bounds on the solution, we\nintroduce the Projected Adam method, a novel algorithm to efficiently compute\nthe condition number of a matrix without relying on eigenvalues or singular\nvalues. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO,\nUMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This\nwork serves as a resource for selecting appropriate solvers, understanding the\nimpact of condition numbers, and improving the robustness of numerical\nsolutions in practical applications.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations\",\"authors\":\"Marcel Ferrari\",\"doi\":\"arxiv-2409.11515\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Solving sparse linear systems is a critical challenge in many scientific and\\nengineering fields, particularly when these systems are severely\\nill-conditioned. This work aims to provide a comprehensive comparison of\\nvarious solvers designed for such problems, offering valuable insights and\\nguidance for domain scientists and researchers. We develop the tools required\\nto accurately evaluate the performance and correctness of 16 solvers from 11\\nstate-of-the-art numerical libraries, focusing on their effectiveness in\\nhandling ill-conditioned matrices. The solvers were tested on linear systems\\narising from a coupled hydro-mechanical marker-in-cell geophysical simulation.\\nTo address the challenge of computing accurate error bounds on the solution, we\\nintroduce the Projected Adam method, a novel algorithm to efficiently compute\\nthe condition number of a matrix without relying on eigenvalues or singular\\nvalues. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO,\\nUMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This\\nwork serves as a resource for selecting appropriate solvers, understanding the\\nimpact of condition numbers, and improving the robustness of numerical\\nsolutions in practical applications.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"55 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.11515\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11515","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations
Solving sparse linear systems is a critical challenge in many scientific and
engineering fields, particularly when these systems are severely
ill-conditioned. This work aims to provide a comprehensive comparison of
various solvers designed for such problems, offering valuable insights and
guidance for domain scientists and researchers. We develop the tools required
to accurately evaluate the performance and correctness of 16 solvers from 11
state-of-the-art numerical libraries, focusing on their effectiveness in
handling ill-conditioned matrices. The solvers were tested on linear systems
arising from a coupled hydro-mechanical marker-in-cell geophysical simulation.
To address the challenge of computing accurate error bounds on the solution, we
introduce the Projected Adam method, a novel algorithm to efficiently compute
the condition number of a matrix without relying on eigenvalues or singular
values. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO,
UMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This
work serves as a resource for selecting appropriate solvers, understanding the
impact of condition numbers, and improving the robustness of numerical
solutions in practical applications.