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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Vydáno v:SIAM journal on matrix analysis and applications Ročník 31; číslo 5; s. 2899 - 2920
Hlavní autoři: Xia, Jianlin, Gu, Ming
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Témata:
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
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
ISSN:0895-4798, 1095-7162
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Shrnutí: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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ISSN:0895-4798
1095-7162
DOI:10.1137/090750500