A hierarchical random compression method for kernel matrices

TL;DR

This paper introduces a hierarchical random compression method (HRCM) for kernel matrices that efficiently reduces computational complexity in fast kernel summations by combining hierarchical frameworks with randomized sampling, achieving near-linear scaling.

Contribution

The paper presents a novel HRCM that uses uniform random sampling at each hierarchical level to compress kernel matrices without pre-computing sampling distributions, independent of matrix size.

Findings

Achieves O(N log N) complexity for kernel matrix compression.

Validates efficiency and accuracy with electrostatic and Helmholtz kernels.

Works effectively for matrices with thousands of elements, providing 3-4 digit accuracy.

Abstract

In this paper, we propose a hierarchical random compression method (HRCM) for kernel matrices in fast kernel summations. The HRCM combines the hierarchical framework of the H-matrix and a randomized sampling technique of the column and row spaces for far-field interaction kernel matrices. We show that a uniform column/row sampling (with a given sample size) of a far-field kernel matrix, with- out the need and associated cost to pre-compute a costly sampling distribution, will give a low-rank compression of such low-rank matrices, independent of the matrix sizes and only dependent on the separation of the source and target locations. This far-field random compression technique is then implemented at each level of the hierarchical decomposition for general kernel matrices, resulting in an O(N logN) random compression method. Error and complexity analysis for the HRCM are included. Numerical results for electrostatic and Helmholtz wave kernels have vali- dated the efficiency and accuracy of the proposed method with a cross-over matrix size, in comparison of direct O(N^2) summations, in the order of thousands for a 3-4 digits relative accuracy.