Preconditioners for hierarchical matrices based on their extended sparse form

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.