On the best approximation of the hierarchical matrix product

TL;DR

This paper introduces a new algorithmic framework for multiplying hierarchical matrices that efficiently finds the best low-rank approximation of their product, achieving near-linear computational cost and improved accuracy.

Contribution

It proposes a novel framework that enhances hierarchical matrix multiplication by optimizing low-rank approximations, reducing computational cost and improving accuracy over existing methods.

Findings

The new method computes the best-approximation of hierarchical matrix products.

It achieves near-linear computational complexity under certain assumptions.

Numerical experiments confirm improved efficiency and accuracy.

Abstract

The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices, resulting in an almost linear cost. However, the computational efficiency of the algorithm is based on a recursive scheme which makes the error analysis quite involved. In this article, we propose a new algorithmic framework for the multiplication of hierarchical matrices. It improves currently known implementations by reducing the multiplication of hierarchical matrices towards finding a suitable low-rank approximation of sums of matrix-products. We propose several compression schemes to address this task. As a consequence, we are able to compute the best-approximation of hierarchical matrix products. A cost analysis shows that, under reasonable assumptions on the low-rank approximation method, the cost of the framework is almost linear with respect to the size of the matrix. Numerical experiments show that the new approach produces indeed the best-approximation of the product of hierarchical matrices for a given tolerance. They also show that the new multiplication can accomplish this task in less computation time than the established multiplication algorithm without error control.