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

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 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 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.