Hierarchical matrix arithmetic with accumulated updates

TL;DR

This paper introduces a new hierarchical matrix algorithm that reduces the number of low-rank updates, significantly decreasing preconditioner setup time for large-scale PDE and integral equation problems.

Contribution

A novel algorithm that minimizes low-rank updates in hierarchical matrix computations, improving efficiency for 3D problem preconditioners.

Findings

Setup time reduced by 50% or more

Fewer low-rank updates needed in the algorithm

Enhanced efficiency for large-scale 3D problems

Abstract

Hierarchical matrices can be used to construct efficient preconditioners for partial differential and integral equations by taking advantage of low-rank structures in triangular factorizations and inverses of the corresponding stiffness matrices. The setup phase of these preconditioners relies heavily on low-rank updates that are responsible for a large part of the algorithm's total run-time, particularly for matrices resulting from three-dimensional problems. This article presents a new algorithm that significantly reduces the number of low-rank updates and can reduce the setup time by 50 percent or more.