On well-separated sets and fast multipole methods

TL;DR

This paper analyzes well-separated sets in fast multipole methods, providing a new error estimate for the multipole acceptance criterion and exploring asymmetric variants for improved computational efficiency.

Contribution

It introduces a generalized error analysis for the multipole acceptance criterion and examines asymmetric multipole divisions for better implementation on modern architectures.

Findings

Derived a relative error estimate for the multipole acceptance criterion.

Showed that asymmetric multipole divisions can lead to more balanced and efficient computations.

Enhanced understanding of cluster interactions in fast multipole methods.

Abstract

The notion of well-separated sets is crucial in fast multipole methods as the main idea is to approximate the interaction between such sets via cluster expansions. We revisit the one-parameter multipole acceptance criterion in a general setting and derive a relative error estimate. This analysis benefits asymmetric versions of the method, where the division of the multipole boxes is more liberal than in conventional codes. Such variants offer a particularly elegant implementation with a balanced multipole tree, a feature which might be very favorable on modern computer architectures.