Parallel preconditioning technique based on AGM (Arithmetic-Geometric Mean) of eigenvalues without sequential substitution computation

Abstract

We propose a new preconditioning technique: PAGME (preconditioning based on arithmetic-geometric mean) of eigenvalues suited to parallel computation. PAGME does not need sequential computation of forward and backward substitution. In general, these substitutions prevent from efficient parallelization. Therefore the speed of computation deteriorates greatly on parallel computers. In PAGME, however, computation of upper and lower triangular matrix and vector multiplication which can be computed in parallel mode replaces computation of forward and backward substitution included in incomplete factorization of matrix. Therefore, efficient computation on parallel computers can be realized based on the theoretical aspect. Through some numerical experiments, we verify efficiency and effectiveness of the preconditioned CG method with PAGME for some problems.