Fully algebraic domain decomposition preconditioners with adaptive spectral bounds

TL;DR

This paper introduces a new family of algebraic domain decomposition preconditioners called AWG, which use spectral bounds to improve the conditioning of symmetric positive definite systems without requiring matrix assembly.

Contribution

It presents the full development, spectral analysis, and numerical validation of AWG preconditioners, a novel algebraic approach that avoids partial matrix assembly.

Findings

Spectral bounds are established for the AWG preconditioners.

Enriching the coarse space with spectral modes improves the condition number.

Numerical experiments confirm the effectiveness of the new preconditioners.

Abstract

In this article a new family of preconditioners is introduced for symmetric positive definite linear systems. The new preconditioners, called the AWG preconditioners (for Algebraic-Woodbury-GenEO) are constructed algebraically. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable non-assembled form is known by construction. The extra cost is a second (and to this day rather expensive) coarse solve in the preconditioner. One of the AWG preconditioners has already been presented in the short preprint [38]. This article is the first full presentation of the larger family of AWG preconditioners. It includes proofs of the spectral bounds as well as numerical illustrations.