A two-level sparse approximate inverse preconditioner for unsymmetric matrices

Sparse approximate inverse (SPAI) preconditioners are effective in accelerating iterative solutions of a large class of unsymmetric linear systems and their inherent parallelism has been widely explored. The effectiveness of SPAI relies on the assumption of the unknown true inverse admitting a sparse approximation. Furthermore, for the usual right SPAI, one must restrict the number of non-zeros in each column to control the overall construction cost and this restriction can reduce the effectiveness of such preconditioners. To extend the applicability of SPAI, this paper proposes to use two-level preconditioning: possible dense columns of the true inverse, skipped by right SPAI (column-wise), will be better approximated by left SPAI (row-wise). Essentially, we approximate the true inverse by sparse matrices via a Gauss–Jordan like decomposition. Numerical experiments on a class of benchmark test matrices show that our new idea of two-level preconditioning can lead to a major enhancement to the standard SPAI method.