Refining an approximate inverse

Direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades. The advances in graph theoretic algorithms have not received enough attention from the iterative methods community. For example, we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner. For very sparse preconditioners, however, a reformulation of the algorithm with outer products can exploit even more zeros to good advantage. We also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings. The article ends by proposing a block version of the algorithm for further gains in efficiency and robustness.