Factorized sparse approximate inverse preconditioners can be sensitive to the ordering of the unknowns. This sensitivity can be exploited to decrease the amount of fill-in in the approximate inverse factorization, and/or to increase the amount of parallelism available in the construction of the preconditioner. The question is, how does the ordering of the unknowns affect the rate of convergence of the preconditioned iteration? To get some insight, we experimented with different reorderings and partitioning strategies on a number of test matrices from several application areas.
In particular, we considered symmetric orderings for sparse direct solvers, and nonsymmetric orderings aimed at producing matrices with a zero-free diagonal. We also considered reorderings induced by some basic graph partitioning heuristics. While our main focus was on the factorized approximate inverse strategies based on conjugation, we also experimented with methods based on Frobenius norm minimization and on bordering.
In this talk we report on the results of our experiments, which show that the robustness and performance of sparse approximate inverse preconditioners can be significantly enhanced by reordering the coefficient matrix. We make recommendations on which reorderings should be used, and which are likely to give poor results. We contrast the situation for factorized approximate inverses with that for ILU-type techniques, and we offer some simple graph-theoretical arguments to help explain the results of our experiments.