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

• Authors: Jianlin Xia;Ming Gu
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2010

Given a symmetric positive definite matrix $A$, we compute a structured approximate Cholesky factorization $A\approx\mathbf{R}^{T}\mathbf{R}$ up to any desired accuracy, where $\mathbf{R}$ 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 $\mathbf{R}^{T}\mathbf{R}$ 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.