Fourier Based Fast Multipole Method for the Helmholtz Equation

TL;DR

This paper introduces a Fourier-based formulation of the fast multipole method for the Helmholtz equation, improving computational efficiency and simplifying error analysis by replacing spherical harmonics with Fourier basis functions.

Contribution

The paper develops a novel Fourier-based Helmholtz FMM that accelerates computations and simplifies error analysis compared to traditional spherical harmonic approaches.

Findings

Fourier basis functions enable faster interpolation via FFTs.

Derived algorithms for quadrature selection based on error bounds.

Numerical verification confirms improved efficiency and accuracy.

Abstract

The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function.