Natively Periodic Fast Multipole Method: Approximating the Optimal Green Function

TL;DR

This paper introduces an approximation for the optimal Green function in the Fast Multipole Method that enables native periodic boundary conditions with high accuracy, avoiding complex lattice sum computations.

Contribution

It derives a practical approximation for the optimal Green function, improving the implementation of periodic FMM without lattice sums or hybrid methods.

Findings

Approximation accurate to better than 1e-3 in LMAX norm

Approximation accurate to better than 1e-4 in L2 norm

Enables efficient, native periodic FMM implementations

Abstract

The Fast Multipole Method (FMM) obeys periodic boundary conditions "natively" if it uses a periodic Green function for computing the multipole expansion in the interaction zone of each FMM oct-tree node. One can define the "optimal" Green function for such a method that results in the numerical solution that converges to the equivalent Particle-Mesh solution in the limit of sufficiently high order of multipoles. A discrete functional equation for the optimal Green function can be derived, but is not practically useful as methods for its solution are not known. Instead, this paper presents an approximation for the optimal Green function that is accurate to better than 1e-3 in LMAX norm and 1e-4 in L2 norm for practically useful multipole counts. Such an approximately optimal Green function offers a practical way for implementing FMM with periodic boundary conditions "natively", without the need to compute lattice sums or to rely on hybrid FMM-PM approaches.