Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast Ewald Summation using Prolate Spheroidal Wave Functions

Erik Boström, Anna-Karin Tornberg, Ludvig af Klinteberg

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 Ewald 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.

Fast Ewald Summation using Prolate Spheroidal Wave Functions

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 Ewald 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.
Paper Structure (21 sections, 6 theorems, 88 equations, 2 figures, 2 tables)

This paper contains 21 sections, 6 theorems, 88 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Basic definitions and notation
  4. A fundamental lemma for Laplacian kernel splitting
  5. Numerical mollifiers for the Laplacian kernel
  6. Window functions, periodization and Fourier coefficients
  7. Prolate Spheroidal Wave Functions
  8. Ewald summation
  9. Ewald summation for triply periodic problems
  10. Computing the real-space sum
  11. Direct computation of the Fourier-space sum
  12. Fast Ewald summation
  13. Fast Ewald summation for the Laplace kernel using the first PSWF of order zero
  14. General Ewald decomposition for numerical mollifiers with support $[-1,1]$
  15. A PSWF mollifier for the Ewald split
  16. ...and 6 more sections

Key Result

Lemma 1

Let $f\in L^1(\mathbb{R})$ be even and non-negative, with Fourier transform $\hat{f}$. Define the radial function $g\colon\mathbb{R}^3\to\mathbb{R}$ by with $g(\bm{0})$ defined by continuity. Then

Figures (2)

  • Figure 1: Comparison of the Gaussian and the first PSWF of order zero in real space (left) and Fourier space (right). For equal effective support, the PSWF mollifier requires roughly half as many Fourier modes to achieve the same accuracy.
  • Figure 2: Split function $\Phi_{r_c}^{c_s}(x)$ (left) and residual kernel $R(x)$ (right) for $r_c = 0.5$ and $c_s \in \{1,6,11,\ldots,49\}$. By construction, the split function satisfies $\Phi_{r_c}^{c_s}(r_c)=1$, and the residual kernel is compactly supported on $[-r_c,r_c]$.

Theorems & Definitions (21)

  • Lemma 1
  • Remark 1: Radial extension
  • Remark 2
  • Example 1: Gaussian mollifier
  • Remark 3
  • Theorem 1: osipovProlateSpheroidalWave2013
  • Lemma 2
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 11 more