Improvement of Hierarchical Matrices with Adaptive Cross Approximation for Large-scale Simulation

Abstract

We propose an improved method for hierarchical-matrices (H-matrices) using adaptive cross approximation (ACA) as the low-rank approximation. The improvement consists of a kind of normalization and a new stopping criterion for the ACA. By using the proposed method, we can avoid the trouble that ranks of approximated matrices increase rapidly as the matrix size increases when the conventional H-matrices with ACA are employed to an integral equation whose kernel function has high-order singularities. In particular, application of the proposed method enables us to perform large-scale simulations such that the conventional H-matrices with ACA fail to construct the low-rank approximation. Applicability of the proposed method is confirmed through numerical experiments on an earthquake cycle simulation.