Block-adaptive Cross Approximation of Discrete Integral Operators

TL;DR

This paper introduces a block-adaptive extension of the ACA method, enabling more efficient matrix approximation by adjusting accuracy per block based on error estimates, thus improving iterative solution efficiency.

Contribution

The paper develops a novel block-adaptive ACA method that integrates error estimation to adaptively refine matrix approximations during iterative solutions.

Findings

Enhanced efficiency in matrix approximation for integral operators.

Automatic error-driven adaptation improves solution accuracy.

Interlaced assembly and solution reduce computational costs.

Abstract

In this article we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the block-wise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution.