Explicit approximate inverse preconditioning techniques

Abstract

The 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.