Alternative Tilings for the Fast Multipole Method on the Plane

TL;DR

This paper explores alternative tilings like hexagons and triangles for the 2D fast multipole method, demonstrating their theoretical validity and analyzing their computational efficiency and accuracy.

Contribution

It introduces and analyzes hexagon and triangle tilings as new spatial data structures for the FMM, extending beyond traditional square tilings.

Findings

Both tilings satisfy FMM separation properties

Theoretical error bounds are established for both structures

Empirical analysis confirms competitive runtime and accuracy

Abstract

The fast multipole method (FMM) performs fast approximate kernel summation to a specified tolerance $\epsilon$ by using a hierarchical division of the domain, which groups source and receiver points into regions that satisfy local separation and the well-separated pair decomposition properties. While square tilings and quadtrees are commonly used in 2D, we investigate alternative tilings and associated spatial data structures: regular hexagons (septree) and triangles (triangle-quadtree). We show that both structures satisfy separation properties for the FMM and prove their theoretical error bounds and computational costs. Empirical runtime and error analysis of our implementations are provided.