A Residual Based Sparse Approximate Inverse Preconditioning Procedure for Large Sparse Linear Systems

Abstract

The SPAI algorithm, a sparse approximate inverse preconditioning technique for large sparse linear systems, proposed by Grote and Huckle [SIAM J. Sci. Comput., 18 (1997), pp.~838--853.], is based on the F-norm minimization and computes a sparse approximate inverse $M$ of a large sparse matrix $A$ adaptively. However, SPAI may be costly to seek the most profitable indices at each loop and $M$ may be ineffective for preconditioning. In this paper, we propose a residual based sparse approximate inverse preconditioning procedure (RSAI), which, unlike SPAI, is based on only the {\em dominant} rather than all information on the current residual and augments sparsity patterns adaptively during the loops. RSAI is less costly to seek indices and is more effective to capture a good approximate sparsity pattern of $A^{-1}$ than SPAI. To control the sparsity of $M$ and reduce computational cost, we develop a practical RSAI($tol$) algorithm that drops small nonzero entries adaptively during the process. Numerical experiments are reported to demonstrate that RSAI($tol$) is at least competitive with SPAI and can be considerably more efficient and effective than SPAI. They also indicate that RSAI($tol$) is comparable to the PSAI($tol$) algorithm proposed by one of the authors in 2009.