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:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Univ. of Minnesota, Twin Cities, MN (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (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:
- 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. https://doi.org/10.1137/16M110486X
Li, Ruipeng, and Saad, Yousef. Tue .
"Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners". United States. https://doi.org/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},
number = 3,
volume = 38,
place = {United States},
year = {Tue Aug 01 00:00:00 EDT 2017},
month = {Tue Aug 01 00:00:00 EDT 2017}
}
Web of Science
Works referenced in this record:
An $$\mathcal O (N \log N)$$ O ( N log N ) Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices: With Application to Radial Basis Function Interpolation
journal, April 2013
- Ambikasaran, Sivaram; Darve, Eric
- Journal of Scientific Computing, Vol. 57, Issue 3
An Approximate Minimum Degree Ordering Algorithm
journal, October 1996
- Amestoy, Patrick R.; Davis, Timothy A.; Duff, Iain S.
- SIAM Journal on Matrix Analysis and Applications, Vol. 17, Issue 4
Algorithm 837
journal, September 2004
- Amestoy, Patrick R.; Davis, Timothy A.; Duff, Iain S.
- ACM Transactions on Mathematical Software, Vol. 30, Issue 3, p. 381-388
A fast block low-rank dense solver with applications to finite-element matrices
journal, January 2016
- Aminfar, AmirHossein; Ambikasaran, Sivaram; Darve, Eric
- Journal of Computational Physics, Vol. 304
Overlapping Domain Decomposition Algorithms for General Sparse Matrices
journal, May 1996
- Cai, Xiao-Chuan; Saad, Yousef
- Numerical Linear Algebra with Applications, Vol. 3, Issue 3
A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems
journal, January 1999
- Cai, Xiao-Chuan; Sarkis, Marcus
- SIAM Journal on Scientific Computing, Vol. 21, Issue 2
Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication
journal, July 1999
- Catalyurek, U. V.; Aykanat, C.
- IEEE Transactions on Parallel and Distributed Systems, Vol. 10, Issue 7
A Fast Solver for HSS Representations via Sparse Matrices
journal, January 2007
- Chandrasekaran, S.; Dewilde, P.; Gu, M.
- SIAM Journal on Matrix Analysis and Applications, Vol. 29, Issue 1
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations
journal, January 2006
- Chandrasekaran, S.; Gu, M.; Pals, T.
- SIAM Journal on Matrix Analysis and Applications, Vol. 28, Issue 3
Approximate Inverse Preconditioners via Sparse-Sparse Iterations
journal, May 1998
- Chow, Edmond; Saad, Yousef
- SIAM Journal on Scientific Computing, Vol. 19, Issue 3
Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation
journal, February 2011
- Engquist, Björn; Ying, Lexing
- Communications on Pure and Applied Mathematics, Vol. 64, Issue 5
An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling
journal, January 2016
- Ghysels, Pieter; Li, Xiaoye S.; Rouet, François-Henry
- SIAM Journal on Scientific Computing, Vol. 38, Issue 5
Construction and Arithmetics of H -Matrices
journal, August 2003
- Grasedyck, Lars; Hackbusch, Wolfgang
- Computing, Vol. 70, Issue 4
A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices
journal, April 1999
- Hackbusch, W.
- Computing, Vol. 62, Issue 2
Data-sparse Approximation by Adaptive ℋ2-Matrices
journal, September 2002
- Hackbusch, W.; Börm, S.
- Computing, Vol. 69, Issue 1, p. 1-35
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
journal, January 2011
- Halko, N.; Martinsson, P. G.; Tropp, J. A.
- SIAM Review, Vol. 53, Issue 2
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998
- Karypis, George; Kumar, Vipin
- SIAM Journal on Scientific Computing, Vol. 20, Issue 1
A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering
journal, January 1998
- Karypis, George; Kumar, Vipin
- Journal of Parallel and Distributed Computing, Vol. 48, Issue 1
An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
journal, October 1950
- Lanczos, C.
- Journal of Research of the National Bureau of Standards, Vol. 45, Issue 4
ℋ-matrices for Convection-diffusion Problems with Constant Convection
journal, June 2003
- Borne, Sabine Le
- Computing, Vol. 70, Issue 3
H-matrix Preconditioners in Convection-Dominated Problems
journal, January 2006
- Le Borne, Sabine; Grasedyck, Lars
- SIAM Journal on Matrix Analysis and Applications, Vol. 27, Issue 4
Divide and Conquer Low-Rank Preconditioners for Symmetric Matrices
journal, January 2013
- Li, Ruipeng; Saad, Yousef
- SIAM Journal on Scientific Computing, Vol. 35, Issue 4
GPU-accelerated preconditioned iterative linear solvers
journal, October 2012
- Li, Ruipeng; Saad, Yousef
- The Journal of Supercomputing, Vol. 63, Issue 2
Schur complement-based domain decomposition preconditioners with low-rank corrections: SCHUR COMPLEMENT-BASED LOW-RANK PRECONDITIONING
journal, April 2016
- Li, Ruipeng; Xi, Yuanzhe; Saad, Yousef
- Numerical Linear Algebra with Applications, Vol. 23, Issue 4
pARMS: a parallel version of the algebraic recursive multilevel solver
journal, January 2003
- Li, Zhongze; Saad, Yousef; Sosonkina, Masha
- Numerical Linear Algebra with Applications, Vol. 10, Issue 5-6
The Lanczos algorithm with selective orthogonalization
journal, January 1979
- Parlett, B. N.; Scott, D. S.
- Mathematics of Computation, Vol. 33, Issue 145
Partitioning Sparse Matrices with Eigenvectors of Graphs
journal, July 1990
- Pothen, Alex; Simon, Horst D.; Liou, Kang-Pu
- SIAM Journal on Matrix Analysis and Applications, Vol. 11, Issue 3
A Distributed-Memory Package for Dense Hierarchically Semi-Separable Matrix Computations Using Randomization
journal, June 2016
- Rouet, François-Henry; Li, Xiaoye S.; Ghysels, Pieter
- ACM Transactions on Mathematical Software, Vol. 42, Issue 4
A Flexible Inner-Outer Preconditioned GMRES Algorithm
journal, March 1993
- Saad, Youcef
- SIAM Journal on Scientific Computing, Vol. 14, Issue 2
ARMS: an algebraic recursive multilevel solver for general sparse linear systems
journal, January 2002
- Saad, Y.; Suchomel, B.
- Numerical Linear Algebra with Applications, Vol. 9, Issue 5
The Lanczos algorithm with partial reorthogonalization
journal, January 1984
- Simon, Horst D.
- Mathematics of Computation, Vol. 42, Issue 165
An Algebraic Multilevel Preconditioner with Low-Rank Corrections for Sparse Symmetric Matrices
journal, January 2016
- Xi, Yuanzhe; Li, Ruipeng; Saad, Yousef
- SIAM Journal on Matrix Analysis and Applications, Vol. 37, Issue 1
A Rational Function Preconditioner For Indefinite Sparse Linear Systems
journal, January 2017
- Xi, Yuanzhe; Saad, Yousef
- SIAM Journal on Scientific Computing, Vol. 39, Issue 3
Efficient Structured Multifrontal Factorization for General Large Sparse Matrices
journal, January 2013
- Xia, Jianlin
- SIAM Journal on Scientific Computing, Vol. 35, Issue 2
Fast algorithms for hierarchically semiseparable matrices
journal, November 2010
- Xia, Jianlin; Chandrasekaran, Shivkumar; Gu, Ming
- Numerical Linear Algebra with Applications, Vol. 17, Issue 6
Superfast Multifrontal Method for Large Structured Linear Systems of Equations
journal, January 2010
- Xia, Jianlin; Chandrasekaran, Shivkumar; Gu, Ming
- SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 3
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
journal, January 2010
- Xia, Jianlin; Gu, Ming
- SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 5
Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration
journal, December 2006
- Zhou, Yunkai; Saad, Yousef; Tiago, Murilo L.
- Physical Review E, Vol. 74, Issue 6
Works referencing / citing this record:
An ensemble Kalman filter approach based on operator splitting for solving nonlinear Hammerstein type ill-posed operator equations
journal, October 2018
- Yang, Xiao-Mei; Deng, Zhi-Liang
- Modern Physics Letters B, Vol. 32, Issue 28