Adaptive multiplication of rank-structured matrices in linear complexity

TL;DR

This paper introduces a new adaptive algorithm for multiplying hierarchical -matrices in linear time, which constructs bases on-the-fly, improving efficiency over previous methods that required precomputed bases or special structures.

Contribution

The paper presents a general, adaptive -matrix multiplication algorithm with linear complexity that does not require precomputed bases, unlike prior approaches.

Findings

Algorithm achieves linear complexity in practice.

Numerical experiments show significant speed improvements.

The method maintains accuracy while adapting bases dynamically.

Abstract

Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms. The matrix multiplication, a key component of many more advanced numerical algorithms, has so far proven tricky: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance. In this article, a new and fairly general algorithm for multiplying $\mathcal{H}^2$-matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application. The complexity and accuracy are analysed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.