A residual based sparse approximate inverse preconditioning procedure for large sparse linear systems

We propose a residual based sparse approximate inverse (RSAI) preconditioning procedure, for the large sparse linear system A x =b . Different from the SParse Approximate Inverse (SPAI) algorithm proposed by Grote and Huckle (SIAM Journal on Scientific Computing, 18 (1997), pp. 838–853.), RSAI uses only the dominant other than all the information on the current residual and augments sparsity patterns adaptively during loops. In order to control the sparsity of M , we develop two practical algorithms RSAI(f i x ) and RSAI(t o l ). RSAI(f i x ) retains the prescribed number of large nonzero entries and adjusts their positions in each column of M among all available ones, in which the number of large entries is increased by a fixed number at each loop. In contrast, the existing indices of M by SPAI are untouched in subsequent loops and a few most profitable indices are added to each column of M from the new candidates in the next loop. RSAI(t o l ) is a tolerance based dropping algorithm and retains all large entries by dynamically dropping small ones below some tolerances, and it better suits for the problem where the numbers of large entries in the columns of A ^?1 differ greatly. When the two preconditioners M have almost the same or comparable numbers of nonzero entries, the numerical experiments on real‐world problems demonstrate that RSAI(f i x ) is highly competitive with SPAI and can outperform the latter for some problems. We also make comparisons of RSAI(f i x ), RSAI(t o l ), and power sparse approximate inverse(t o l ) proposed Jia and Zhu (Numerical Linear Algebra with Applications, 16 (2009), pp.259–299.) and incomplete LU factorization type methods and draw some general conclusions.