HODLR2D: A new class of Hierarchical matrices

TL;DR

HODLR2D introduces a hierarchical low-rank matrix representation for 2D N-body problems, enabling fast matrix-vector multiplication with near-linear scaling, and is applicable to large dense linear systems.

Contribution

The paper presents HODLR2D, a novel hierarchical matrix class that achieves efficient matrix-vector products with proven logarithmic rank bounds, improving scalability for large 2D problems.

Findings

Matrix-vector product scales as O(pN log(N))

Maximum rank p is O(log(N) log(log(N)))

HODLR2D demonstrates good parallel scalability

Abstract

This article introduces HODLR2D, a new hierarchical low-rank representation for a class of dense matrices arising out of $N$ body problems in two dimensions. Using this new hierarchical framework, we propose a new fast matrix-vector product that scales almost linearly. We apply this fast matrix-vector product to accelerate the iterative solution of large dense linear systems arising out of radial basis function interpolation and discretized integral equation. The space and computational complexity of HODLR2D matrix-vector products scales as $\mathcal{O}(pN \log(N))$, where $p$ is the maximum rank of the compressed matrix subblocks. We also prove that $p \in \mathcal{O}(\log(N)\log(\log(N)))$, which ensures that the storage and computational complexity of HODLR2D matrix-vector products remain tractable for large $N$. Additionally, we also present the parallel scalability of HODLR2D as part of this article.