On Least-Squares Approximate Inverse-Based Preconditioners

Abstract

We discuss approximate inverse preconditioners based on Frobenius-norm minimization. We introduce a novel adaptive algorithm based on truncated Neumann matrix expansions for selecting the sparsity pattern of the preconditioner. The construction of the approximate inverse is based on a dual dropping strategy, namely a threshold to drop small entries and a maximum number of nonzero entries per column. We introduce a post-processing stabilization technique to deflate some of the smallest eigenvalues in the spectrum of the preconditioned matrix which can potentially disturb the convergence. Results of preliminary experiments are reported on a set of linear systems arising from different application fields to illustrate the potential of the proposed algorithm for preconditioning effectively iterative Krylov solvers.