A Preconditioner Based on Low-Rank Approximation of Schur Complements

TL;DR

This paper presents a novel preconditioner for large matrices based on hierarchical low-rank approximations of Schur complements, enabling efficient inversion without full matrix assembly, tested on finite element problems.

Contribution

Introduces a preconditioner using hierarchical low-rank Schur complement approximations, inspired by nested dissection, with a fast $LDM^t$ factorization approach that does not require full matrix assembly.

Findings

Effective in accelerating convergence for finite element discretizations.

Enables fast inversion and application of the preconditioner.

Applicable to elliptic and hyperbolic PDE problems.

Abstract

We introduce a preconditioner based on a hierarchical low-rank compression scheme of Schur complements. The construction is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi-Separable matrices. We build the preconditioner as an approximate $LDM^t$ factorization of a given matrix $A$, and no knowledge of $A$ in assembled form is required by the construction. The $LDM^t$ factorization is amenable to fast inversion, and the action of the inverse can be determined fast as well. We investigate the behavior of the preconditioner in the context of DG finite element approximations of elliptic and hyperbolic problems.