Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices

Abstract. Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A RT 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 RT 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 prop...

• Type: Journal
• Year: 2010
• Where: SIAMMAX
• Authors: Jianlin Xia, Ming Gu
• Keywords: Factorization, Low-rank Property, Positive Definite, SIAMMAX 2010