HODLR3D: Hierarchical matrices for $N$-body problems in three dimensions

TL;DR

HODLR3D introduces a hierarchical matrix structure for 3D $N$-body problems, exploiting low-rank off-diagonal blocks to enable efficient computations with near-linear complexity.

Contribution

This paper develops HODLR3D, a novel hierarchical matrix format for 3D $N$-body problems, demonstrating its low-rank properties and computational advantages over existing methods.

Findings

All off-diagonal blocks are rank deficient for Laplace kernel in 3D.

HODLR3D achieves almost linear scaling in storage and computation.

Numerical experiments confirm the efficiency and scalability of HODLR3D.

Abstract

This article introduces HODLR3D, a class of hierarchical matrices arising out of $N$-body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the $N$-body problems in three dimensions are numerically low-rank. For the Laplace kernel in $3$D, which is widely encountered, we prove that all the off-diagonal matrix sub-blocks are rank deficient in finite precision. We also obtain the growth of the rank as a function of the size of these matrix sub-blocks. For other kernels in three dimensions, we numerically illustrate a similar scaling in rank for the different off-diagonal sub-blocks. We leverage this hierarchical low-rank structure to construct HODLR3D representation, with which we accelerate matrix-vector products. The storage and computational complexity of the HODLR3D matrix-vector product scales almost linearly with system size. We demonstrate the computational performance of HODLR3D representation through various numerical experiments. Further, we explore the performance of the HODLR3D representation on distributed memory systems. HODLR3D, described in this article, is based on a weak admissibility condition. Among the hierarchical matrices with different weak admissibility conditions in $3$D, only in HODLR3D did the rank of the admissible off-diagonal blocks not scale with any power of the system size. Thus, the storage and the computational complexity of the HODLR3D matrix-vector product remain tractable for $N$-body problems with large system sizes.