On numerical modeling performance of generalized preconditioned methods

Abstract

During the last decades, research efforts have been focused on the derivation of effective preconditioned iterative methods. The preconditioned iterative methods are mainly categorized into implicit preconditioned methods and explicit preconditioned methods. In this manuscript we review implicit preconditioned methods, based on incomplete and approximate factorization, and explicit preconditioned methods, based on sparse approximate inverses and explicit approximate inverses. Modified Moore-Penrose conditions are presented and theoretical estimates for the sensitivity of the explicit approximate inverse matrix of the explicit preconditioned method are derived. Finally, the performance of the preconditioned iterative methods is illustrated by solving characteristic 2D elliptic problem and numerical results are given indicating a qualitative agreement with the theoretical estimates.