Toward a new fully algebraic preconditioner for symmetric positive definite problems

TL;DR

This paper introduces a novel algebraic domain decomposition preconditioner for symmetric positive definite matrices that does not require Neumann matrices, utilizing the GenEO coarse space and Woodbury identity for efficient solutions.

Contribution

It presents the first algebraic preconditioner that avoids Neumann matrices, combining GenEO coarse space and Woodbury identity for improved efficiency.

Findings

Preliminary numerical results demonstrate effectiveness.

Preconditioner simplifies implementation by avoiding Neumann matrices.

Potential for enhanced scalability in solving large linear systems.

Abstract

A new domain decomposition preconditioner is introduced for efficiently solving linear systems Ax = b with a symmetric positive definite matrix A. The particularity of the new preconditioner is that it is not necessary to have access to the so-called Neumann matrices (i.e.: the matrices that result from assembling the variational problem underlying A restricted to each subdomain). All the components in the preconditioner can be computed with the knowledge only of A (and this is the meaning given here to the word algebraic). The new preconditioner relies on the GenEO coarse space for a matrix that is a low-rank modification of A and on the Woodbury matrix identity. The idea underlying the new preconditioner is introduced here for the first time with a first version of the preconditioner. Some numerical illustrations are presented. A more extensive presentation including some improved variants of the new preconditioner can be found in [7] (https://hal.archives-ouvertes.fr/hal-03258644).