Accuracy Controlled Structure-Preserving ${\mathcal{ H}}^{2}$ -Matrix-Matrix Product in Linear Complexity With Change of Cluster Bases

Abstract

${\mathcal{ H}}^{2}$ -matrix constitutes a general mathematical framework for efficient computation of both partial-differential-equation (PDE) and integral-equation (IE)-based operators. Existing linear-complexity ${\mathcal{ H}}^{2}$ matrix-matrix product (MMP) algorithm lacks explicit accuracy control, while controlling accuracy without compromising linear complexity is challenging. In this article, we develop an accuracy controlled ${\mathcal{ H}}^{2}$ MMP algorithm by instantaneously changing the cluster bases during the matrix product computation based on prescribed accuracy. Meanwhile, we retain the computational complexity of the overall algorithm to be linear. Different from the existing ${\mathcal{ H}}^{2}$ MMP algorithm where formatted multiplications are performed using the original cluster bases, in the proposed algorithm, all additions and multiplications are either exact or computed based on prescribed accuracy. Furthermore, the original ${\mathcal{ H}}^{2}$ -matrix structure is preserved in the matrix product. While achieving optimal complexity for constant-rank matrices, the computational complexity of the proposed algorithm is also minimized for variable-rank ${\mathcal{ H}}^{2}$ -matrices. For example, it has a complexity of $O(NlogN)$ for computing electrically large volume IEs, where $N$ is matrix size. The proposed work serves as a fundamental arithmetic in the development of fast solvers for large-scale electromagnetic analysis. Applications to both large-scale capacitance extraction and electromagnetic scattering problems involving millions of unknowns on a single core have demonstrated the accuracy and efficiency of the proposed algorithm.