AN APPROACH TO MAKING SPAI AND PSAI PRECONDITIONING EFFECTIVE FOR LARGE IRREGULAR SPARSE LINEAR SYSTEMS

作者：Jia, Zhongxiao^*; Zhang, Qian

来源：SIAM Journal on Scientific Computing, 2013, 35(4): A1903-A1927.

DOI：10.1137/120900800

• preconditioning
• sparse approximate inverse
• irregular sparse
• regular sparse
• Sherman-Morrison-Woodbury formula
• F-norm minimization
• Krylov solver

We investigate the adaptive sparse approximate inverse (SPAI) and adaptive power sparse approximate inverse (PSAI) preconditioning procedures and shed light on two important features of them: (i) For the large linear system Ax = b with A irregular sparse, i. e., with A having s relatively dense columns, SPAI may be very costly to implement, and the resulting sparse approximate inverses may be ineffective for preconditioning. PSAI can be effective for preconditioning but may require excessive storage and be unacceptably time consuming. (ii) The situation is improved drastically when A is regular sparse, that is, all its columns are sparse. In this case, both SPAI and PSAI are efficient. Moreover, SPAI and, especially, PSAI are more likely to construct effective preconditioners. Motivated by these features, we propose an approach to making SPAI and PSAI more practical for Ax = b with A irregular sparse. We first split A into a regular sparse (A) over tilde and a matrix of low rank s. Then exploiting the Sherman-Morrison-Woodbury formula, we transform Ax = b into s+1 new linear systems with the same coefficient matrix (A) over tilde, use SPAI and PSAI to compute sparse approximate inverses of (A) over tilde efficiently, and apply Krylov iterative methods to solve the preconditioned linear systems. Theoretically, we consider the nonsingularity and conditioning of (A) over tilde obtained from some important classes of matrices. We show how to recover an approximate solution of Ax = b from those of the s+1 new systems and how to design reliable stopping criteria for the s+1 systems to guarantee that the approximate solution of Ax = b satisfies a desired accuracy. Given the fact that irregular sparse linear systems are common in applications, this approach widely extends the practicability of SPAI and PSAI. Numerical results demonstrate the considerable superiority of our approach to the direct application of SPAI and PSAI to Ax = b.

• 出版日期2013
• 单位清华大学