Modified block-approximate factorization strategies

Abstract

Summary. Two variants of modified incomplete block-matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations. Both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations (PDEs), even in the case of quasi-singular problems.

A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

“…Other, more robust methods, include the dynamic perturbation method in [9], where a criterion based on the difference (LI -LA)e was used, the methods in [33] and the relaxation method in [ 111. See also [25] for techniques aimed at better robustness.…”

Eigenvalue bounds for block preconditioned matrices

“…Furthermore, interest in this method has increased recently because of its practically ideal suitability to implementation on parallel computers. As to the incomplete block factorization methods, they have been intensively studied during the last decade (see, e.g., [1,[3][4][5][6][8][9][10][11][12]) since they proved to be efficient as preconditioners when solving different problems mainly by the conjugate gradient method.This paper is specific, in particular, in the fact that a number of bounds for the block Jacobi scaling and incomplete block factorization preconditionings are derived under the same assumptions, which permits one to compare the efficiency of the two types of preconditionings. In our opinion, an attempt to introduce a quantitative characterization of block matrix partitionings that can be used when analyzing block preconditioning methods is also of interest.…”