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Title: Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners

Abstract

This study presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits domain decomposition (DD) and low-rank corrections. The DD approach decouples the matrix and, once inverted, a low-rank approximation is applied by exploiting the Sherman--Morrison--Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with pARMS, a DD-based parallel incomplete LU (ILU) preconditioning method, are presented for solving Poisson's equation and linear elasticity problems.

Authors:
 [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Univ. of Minnesota, Twin Cities, MN (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1438714
Report Number(s):
LLNL-JRNL-727122
Journal ID: ISSN 0895-4798
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM Journal on Matrix Analysis and Applications
Additional Journal Information:
Journal Volume: 38; Journal Issue: 3; Journal ID: ISSN 0895-4798
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Sherman-Morrison-Woodbury formula; low-rank approximation; distributed sparse linear systems; parallel preconditioner; incomplete LU factorization; Krylov subspace method; domain decomposition

Citation Formats

Li, Ruipeng, and Saad, Yousef. Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners. United States: N. p., 2017. Web. doi:10.1137/16M110486X.
Li, Ruipeng, & Saad, Yousef. Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners. United States. doi:10.1137/16M110486X.
Li, Ruipeng, and Saad, Yousef. Tue . "Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners". United States. doi:10.1137/16M110486X. https://www.osti.gov/servlets/purl/1438714.
@article{osti_1438714,
title = {Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners},
author = {Li, Ruipeng and Saad, Yousef},
abstractNote = {This study presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits domain decomposition (DD) and low-rank corrections. The DD approach decouples the matrix and, once inverted, a low-rank approximation is applied by exploiting the Sherman--Morrison--Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with pARMS, a DD-based parallel incomplete LU (ILU) preconditioning method, are presented for solving Poisson's equation and linear elasticity problems.},
doi = {10.1137/16M110486X},
journal = {SIAM Journal on Matrix Analysis and Applications},
issn = {0895-4798},
number = 3,
volume = 38,
place = {United States},
year = {2017},
month = {8}
}

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