Iterative representing set selection for nested cross approximation

TL;DR

This paper introduces a fast, hierarchical algebraic method for matrix approximation that efficiently constructs $\\mathcal{H}^2$-approximations using an iterative selection of representing sets, without needing pre-selected proxy surfaces.

Contribution

It presents a novel iterative approach for selecting representing sets in nested cross approximation, eliminating the need for pre-defined proxy surfaces and ensuring linear complexity.

Findings

Method is effective for electrostatic and boundary integral problems.

Algorithm demonstrates linear complexity and robustness.

Open-source Python implementation available.

Abstract

A new fast algebraic method for obtaining an $\mathcal{H}^2$-approximation of a matrix from its entries is presented. The main idea behind the method is based on the nested representation and the maximum-volume principle to select submatrices in low-rank matrices. A special iterative approach for the computation of so-called representing sets is established. The main advantage of the method is that it uses only the hierarchical partitioning of the matrix and does not require special "proxy surfaces" to be selected in advance. The numerical experiments for the electrostatic problem and for the boundary integral operator confirm the effectiveness and robustness of the approach. The complexity is linear in the matrix size and polynomial in the ranks. The algorithm is implemented as an open-source Python package that is available online.