Simple non-extensive sparsification of the hierarchical matrices

Abstract

In this paper, we consider the matrices approximated in ${\mathscr{H}}^{2}$ format. The direct solution, as well as the preconditioning of systems with such matrices, is a challenging problem. We propose a non-extensive sparse factorization of the ${\mathscr{H}}^{2}$ matrix that allows to substitute direct ${\mathscr{H}}^{2}$ solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed of parameters of the ${\mathscr{H}}^{2}$ matrix. In the numerical experiments, we show the consistency of this approach in comparison with the other approximate block low-rank hierarchical solvers, such as HODLR [3], H2Lib [5], and IFMM [11].