Preconditioners for hierarchical matrices based on their extended sparse form

TL;DR

This paper introduces new preconditioners for dense matrices from boundary integral equations, utilizing their extended sparse form, with the most effective being a reverse-Schur based approach, improving solution efficiency.

Contribution

The paper develops novel preconditioners for -matrices based on their extended sparse form, enhancing iterative solver performance for boundary integral equation systems.

Findings

Reverse-Schur preconditioning is highly effective.

Preconditioners significantly reduce iteration counts.

Numerical experiments confirm improved efficiency.

Abstract

In this paper we consider linear systems with dense-matrices which arise from numerical solution of boundary integral equations. Such matrices can be well-approximated with $\mathcal{H}^2$-matrices. We propose several new preconditioners for such matrices that are based on the equivalent \emph{sparse extended form} of $\mathcal{H}^2$-matrices. In the numerical experiments we show that the most efficient approach is based on the so-called reverse-Schur preconditioning technique.