A Directional Equispaced interpolation-based Fast Multipole Method for oscillatory kernels

TL;DR

This paper presents a new directional Fast Multipole Method (FMM) for oscillatory kernels that leverages FFT-accelerated polynomial interpolation on equispaced grids, significantly improving performance in high-frequency regimes.

Contribution

The paper introduces defmm, a novel directional FMM that uses FFT-based polynomial interpolation and symmetry exploitation, optimized for non-uniform particle distributions and high-frequency oscillatory kernels.

Findings

Significant performance gains over existing FMM libraries.

Effective FFT acceleration of polynomial interpolation on equispaced grids.

Algorithmic design tailored for non-uniform particle distributions.

Abstract

Fast Multipole Methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from Boundary Integral Equations (BIEs) in acoustic or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm - directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach, and show how symmetries can be exploited in the Fourier domain. We also describe the algorithmic design of defmm, well-suited for the BIE non-uniform particle distributions, and present performance optimizations on one CPU core. Finally, we exhibit important performance gains on all test cases for defmm over a state-of-the-art FMM library for oscillatory kernels.