Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A [asymptotically =] R^T R up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method com...
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| Published in: | SIAM journal on matrix analysis and applications Vol. 31; no. 5; pp. 2899 - 2920 |
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| Language: | English |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2010
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| ISSN: | 0895-4798, 1095-7162 |
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| Abstract | Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A [asymptotically =] R^T R up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method compresses off-diagonal blocks with rank-revealing orthogonal decompositions. In the meantime, positive semidefinite terms are automatically and implicitly added to Schur complements in the factorization so that the approximation R^T R is guaranteed to exist and be positive definite. The approximate factorization can be used as a structured preconditioner which does not break down. No extra stabilization step is needed. When A has an off-diagonal low-rank property, or when the off-diagonal blocks of A have small numerical ranks, the preconditioner is data sparse and is especially efficient. Furthermore, the method has a good potential to give satisfactory preconditioning bounds even if this low-rank property is not obvious. Numerical experiments are used to demonstrate the performance of the method. The method can be used to provide effective structured preconditioners for large sparse problems when combined with some sparse matrix techniques. The hierarchical compression scheme in this work is also useful in the development of more HSS algorithms. [PUBLICATION ABSTRACT] |
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| AbstractList | Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A approximately R theta R up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method compresses off-diagonal blocks with rank-revealing orthogonal decompositions. In the meantime, positive semidefinite terms are automatically and implicitly added to Schur complements in the factorization so that the approximation R theta R is guaranteed to exist and be positive definite. The approximate factorization can be used as a structured preconditioner which does not break down. No extra stabilization step is needed. When A has an off-diagonal low-rank property, or when the off-diagonal blocks of A have small numerical ranks, the preconditioner is data sparse and is especially efficient. Furthermore, the method has a good potential to give satisfactory preconditioning bounds even if this low-rank property is not obvious. Numerical experiments are used to demonstrate the performance of the method. The method can be used to provide effective structured preconditioners for large sparse problems when combined with some sparse matrix techniques. The hierarchical compression scheme in this work is also useful in the development of more HSS algorithms. Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A [asymptotically =] R^T R up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method compresses off-diagonal blocks with rank-revealing orthogonal decompositions. In the meantime, positive semidefinite terms are automatically and implicitly added to Schur complements in the factorization so that the approximation R^T R is guaranteed to exist and be positive definite. The approximate factorization can be used as a structured preconditioner which does not break down. No extra stabilization step is needed. When A has an off-diagonal low-rank property, or when the off-diagonal blocks of A have small numerical ranks, the preconditioner is data sparse and is especially efficient. Furthermore, the method has a good potential to give satisfactory preconditioning bounds even if this low-rank property is not obvious. Numerical experiments are used to demonstrate the performance of the method. The method can be used to provide effective structured preconditioners for large sparse problems when combined with some sparse matrix techniques. The hierarchical compression scheme in this work is also useful in the development of more HSS algorithms. [PUBLICATION ABSTRACT] |
| Author | Xia, Jianlin Gu, Ming |
| Author_xml | – sequence: 1 givenname: Jianlin surname: Xia fullname: Xia, Jianlin – sequence: 2 givenname: Ming surname: Gu fullname: Gu, Ming |
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| Cites_doi | 10.1016/j.laa.2005.02.037 10.1090/S0025-5718-04-01716-8 10.1007/s006070050020 10.1002/nla.1680010207 10.1137/S0895479803436652 10.1090/S0025-5718-1980-0559197-0 10.1007/PL00021408 10.1016/0024-3795(87)90103-0 10.1002/nla.329 10.1145/356044.356047 10.1016/j.jcp.2004.10.033 10.1137/S1064827599356900 10.1007/BF01436084 10.1007/s00211-005-0595-4 10.1016/S0024-3795(96)00301-1 10.1007/s10092-005-0107-z 10.1137/S0895479891223781 10.1137/050636012 10.1007/s006070070031 10.1002/nla.525 10.1002/(SICI)1099-1506(199811/12)5:6<483::AID-NLA156>3.3.CO;2-Z 10.1016/j.laa.2006.10.021 10.1137/S0895479802405884 10.1007/s00607-004-0080-4 10.1137/080720693 10.1016/j.laa.2007.08.038 10.1137/0710032 10.1007/s00791-008-0098-9 10.1137/0917055 10.1002/nla.461 10.1007/s00211-002-0445-6 10.1137/040615845 10.1007/s00607-005-0146-y 10.1016/S0955-7997(02)00152-2 10.1137/0720040 10.1007/s10915-008-9240-6 10.1137/07068179X 10.1002/nme.1620200511 10.1007/s00607-002-1450-4 10.1137/09074543X |
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| Keywords | Triangular matrix Compression Stabilization Schur compensation Numerical data Cholesky method Compensation 65F30 Development Sparse matrix hierarchically semiseparable structure Schur complement Approximation off-diagonal compression robust preconditioner Rank Algorithm Factorization Numerical analysis Symmetric matrix 15A12 Linear algebra Positive definite matrix Potential method Performance Preconditioning 65F05 |
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| References | Meijerink J. A. (R42) 1977; 31 R41 R40 R21 R20 R23 R45 R22 R44 R25 R47 R24 R46 R27 R26 R29 Hackbusch W. (R34) 2000; 64 R1 R2 R4 R5 R6 R7 R8 R9 R30 R10 R31 R12 R11 R33 R14 R13 R35 R38 R15 R37 R18 R17 R39 R19 |
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| Snippet | Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A [asymptotically =] R^T R up to any desired accuracy,... Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A approximately R theta R up to any desired accuracy,... |
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| SubjectTerms | Algebra Algorithms Approximation Breaking down Cholesky factorization Complement Compressing Exact sciences and technology Factorization Linear and multilinear algebra, matrix theory Mathematical analysis Mathematical models Mathematics Matrix Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Sciences and techniques of general use Studies |
| Title | Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices |
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