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

## TL;DR

This paper introduces an accuracy-controlled ${
m H}^2$-matrix-matrix product algorithm that dynamically adjusts cluster bases during computation, maintaining linear complexity and preserving the matrix structure, suitable for large-scale electromagnetic problems.

## Contribution

It presents a novel ${
m H}^2$-matrix-matrix product algorithm with explicit accuracy control via dynamic cluster basis changes, ensuring linear complexity and structure preservation.

## Key findings

- Achieves accurate ${
m H}^2$-matrix products with linear complexity
- Demonstrates efficiency in large-scale electromagnetic simulations
- Maintains matrix structure during computation

## Abstract

${\cal H}^2$-matrix constitutes a general mathematical framework for efficient computation of both partial-differential-equation and integral-equation-based operators. Existing linear-complexity ${\cal H}^2$ matrix-matrix product (MMP) algorithm lacks explicit accuracy control, while controlling accuracy without compromising linear complexity is challenging. In this paper, we develop an accuracy controlled ${\cal H}^2$ matrix-matrix product 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 ${\cal H}^2$ matrix-matrix product 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 ${\cal H}^2$-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 ${\cal H}^2$-matrices. 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.