Adaptive Cross Approximation with a Geometrical Pivot Choice: ACA-GP Method

TL;DR

This paper introduces ACA-GP, a novel hybrid method combining algebraic and geometrical pivot selection to improve low-rank approximations in hierarchical matrices used in boundary integral methods.

Contribution

It proposes the ACA-GP method that integrates geometrical pivot choices into the classical ACA, enhancing approximation efficiency and accuracy.

Findings

ACA-GP outperforms classical ACA in tests

Improved low-rank approximations for boundary integral operators

Reduced computational costs with ACA-GP

Abstract

The Adaptive Cross Approximation (ACA) method is widely used to approximate admissible blocks of hierarchical matrices, or H-matrices, from discretized operators in the boundary integral method. These matrices are fully populated, making their storage and manipulation resource-intensive. ACA constructs a low-rank approximation by evaluating only a few rows and columns of the original operator, significantly reducing computational costs. A key aspect of ACA's effectiveness is the selection of pivots, which are entries common to the evaluated row and column of the original matrix. This paper proposes combining the classical, purely algebraic ACA method with a geometrical pivot selection based on the central subsets and extreme property subsets. The method is named ACA-GP, GP stands for Geometrical Pivots. The superiority of the ACA-GP compared to the classical ACA is demonstrated using a classical Green operator for two clouds of interacting points.