NESTING APPROXIMATE INVERSES FOR IMPROVED PRECONDITIONING AND ALGEBRAIC MULTIGRID SMOOTHING\ast

Approximate inverses are a very powerful tool for both preconditioning and algebraic multigrid smoothing. One of their main features is their high degree of parallelism that makes them extremely effective for high performance computing. One of their main drawbacks, however, is that the set-up cost grows very quickly with the density, so that it is practically impossible to increase their accuracy by just allowing more entries. In this work, we consider approximate inverses in factored form and show that nesting more factors can greatly enhance their effectiveness at a reasonable computational cost. We additionally provide strategies and theoretical insights aimed at mitigating the computational burden associated with the triple matrix product required in the initial stage of nesting. The effectiveness of the proposed approach is demonstrated through numerical experiments arising from a broad spectrum of real-world applications.