Direct Solution of General ${\mathcal{H}}^2$ -Matrices With Controlled Accuracy and Concurrent Change of Cluster Bases for Electromagnetic Analysis

Abstract

The dense matrix resulting from an integral equation (IE)-based solution of Maxwell’s equations can be compactly represented by an ${\mathcal H}^{2}$ -matrix with controlled accuracy. In this paper, we develop a new direct solution for general ${\mathcal H}^{2}$ -matrices. The new solution possesses not only explicitly controlled accuracy but also a full change of cluster basis to efficiently account for the updates to the original matrix during the direct solution procedure. The change of cluster basis is performed concurrently with the direct solution, without increasing its computational complexity. Comparisons with the state-of-the-art direct solutions have demonstrated a much reduced CPU run time of the new solution, in addition to a significantly improved accuracy, which is also controllable. Rapid and accurate direct solutions of millions of unknowns have been obtained on a single CPU core.