On efficient Monte Carlo preconditioners and hybrid Monte Carlo methods for linear algebra

Abstract

An enhanced version of a stochastic SParse Approximate Inverse (SPAI) preconditioner for general matrices is presented in this paper. This is a Monte Carlo preconditioner based on Markov Chain Monte Carlo (MCMC) methods to compute a rough approximate matrix inverse first, which can further be optimized by an iterative filter process and a parallel refinement, to enhance the accuracy of the inverse and the preconditioner respectively. The above Monte Carlo preconditioner is further used to solve systems of linear algebraic equations thus delivering hybrid stochastic/deterministic algorithms. The advantage of the proposed approach is that the sparse Monte Carlo matrix inversion has a computational complexity linear of the size of the matrix, it is inherently parallel and thus can be obtained very efficiently for large matrices and can be used also as an efficient preconditioner while solving systems of linear algebraic equations. Computational experiments on the Monte Carlo preconditioners and the hybrid algorithms using BiCGSTAB and GMRES as SLAEs solvers are presented and the results are compared to those of MSPAI (parallel and optimized version of the deterministic SPAI) and combined MSPAI and BiCGSTAB and GMRES approaches to solve SLAEs. The experiment are carried out on classes of matrices from the matrix market and show the efficiency of the proposed approach.