Explicit approximate inverse preconditioning techniques

Abstract

SummaryThe numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU -type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient -type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.

On the performance of parallel normalized explicit preconditioned conjugate gradient type methods

“…the additional mumber of diagonals retained next to the main diagonal in the lower and upper part of the inverse, [11,12,13,14,15,16].…”

“…An important achievement over the last decades is the appearance and use of preconditioned iterative methods, for solving a linear system Au=s, [2,4,9,10,11,17,18,21,22]. The preconditioned form of the linear system is MAu=Ms, where M is a suitable preconditioner, satisfying the following conditions: (i) MA should have a "clustered" spectrum, (ii) M can be efficiently computed in parallel and (iii) finally "M × vector" should be fast to compute in parallel, [9,11,12,17].…”

“…The preconditioned form of the linear system is MAu=Ms, where M is a suitable preconditioner, satisfying the following conditions: (i) MA should have a "clustered" spectrum, (ii) M can be efficiently computed in parallel and (iii) finally "M × vector" should be fast to compute in parallel, [9,11,12,17].…”

“…Additionally sparse approximate inverses by minimizing the Frobenious norm of the error have been presented and can be implemented on parallel systems [17,21]. In recent years explicit preconditioned methods, based on approximate inverse matrix algorithms, have been used for solving efficiently sparse linear systems, [11]. The effectiveness of the explicit approximate inverse preconditioning method is related to the fact that the derived classes of approximate inverses exhibit a similar "fuzzy" structure as the coefficient matrix and are close approximants to the coefficient matrix, [11,13].…”

“…It is known that adaptive approximate factorization and approximate inverse matrix algorithms are in general tediously complicated. However as the demand for solving boundary value problems grows, the need to use efficient sparse finite element (FE) linear equations solvers based on approximate (5) (2) factorization procedures and approximate inverse algorithms becomes one of great importance, [11].…”