Fast Ewald Summation using Prolate Spheroidal Wave Functions

TL;DR

This paper introduces a novel Ewald summation method using prolate spheroidal wave functions, achieving higher accuracy with fewer Fourier modes in molecular simulations.

Contribution

It develops a PSWF-based Ewald summation technique with rigorous error estimates and explicit parameter choices, outperforming Gaussian and B-spline methods.

Findings

PSWF-based Ewald summation requires fewer Fourier modes.

Achieves target accuracy with smaller window supports.

Provides explicit error bounds and parameter selection.

Abstract

Fast Ewald summation efficiently evaluates Coulomb interactions and is widely used in molecular dynamics simulations. It is based on a split into a short-range and a long-range part, where evaluation of the latter is accelerated using the fast Fourier transform (FFT). The accuracy and computational cost depend critically on the mollifier in the kernel split and the window function used in the spreading and interpolation steps that enable the use of the FFT. The first prolate spheroidal wavefunction (PSWF) has optimal concentration in real and Fourier space simultaneously, and is used when defining both a mollifier and a window function. We provide a complete description of the method and derive rigorous error estimates. In addition, we obtain closed-form approximations of the Fourier truncation and aliasing errors, yielding explicit parameter choices for the achieved error to closely match the prescribed tolerance. Numerical experiments confirm the analysis: PSWF-based Ewald summation achieves a given accuracy with significantly fewer Fourier modes and smaller window supports than Gaussian- and B-spline-based approaches, providing a superior alternative to existing Ewald methods for particle simulations.