Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast convolution solver based on far-field smooth approximation

Xin Liu, Yong Zhang

TL;DR

The paper develops a fast, spectrally accurate solver for convolution-type nonlocal potentials on bounded domains by introducing a far-field smooth kernel approximation. It splits the potential into a regular part solved by trapezoidal rule with FFT and a singular part handled in Fourier space, then merges both into a single discrete convolution with an axis-symmetric tensor that remains efficient even under strong anisotropy. The method provides rigorous error estimates and demonstrates spectral accuracy across Coulomb, Poisson, Biharmonic, Yukawa, and dipole-dipole kernels, with memory and time complexity nearly optimal and independent of anisotropy strength. This yields a practical, easy-to-implement framework suitable for high-accuracy simulations in physics and engineering where nonlocal interactions arise. Overall, the approach offers a robust, scalable alternative to existing kernel-truncation or Gaussian-sum based methods, broadening the toolkit for fast nonlocal potential computations.

Abstract

The convolution potential arises in a wide variety of application areas, and its efficient and accurate evaluation encounters three challenges: singularity, nonlocality and anisotropy. We introduce a fast algorithm based on a far-field smooth approximation of the kernel, where the bounded domain Fourier transform, one of the most essential difficulties, is well approximated by the whole space Fourier transform which usually admits explicit formula. The convolution is split into a regular and singular integral, and they are well resolved by trapezoidal rule and Fourier spectral method respectively. The scheme is simplified to a discrete convolution and is implemented efficiently with Fast Fourier Transform (FFT). Importantly, the tensor generation procedure is quite simple, highly efficient and independent of the anisotropy strength. It is easy to implement and achieves spectral accuracy with nearly optimal efficiency and minimum memory requirement. Rigorous error estimates and extensive numerical investigations, together with a comprehensive comparison, showcase its superiorities for different kernels.

Fast convolution solver based on far-field smooth approximation

TL;DR

The paper develops a fast, spectrally accurate solver for convolution-type nonlocal potentials on bounded domains by introducing a far-field smooth kernel approximation. It splits the potential into a regular part solved by trapezoidal rule with FFT and a singular part handled in Fourier space, then merges both into a single discrete convolution with an axis-symmetric tensor that remains efficient even under strong anisotropy. The method provides rigorous error estimates and demonstrates spectral accuracy across Coulomb, Poisson, Biharmonic, Yukawa, and dipole-dipole kernels, with memory and time complexity nearly optimal and independent of anisotropy strength. This yields a practical, easy-to-implement framework suitable for high-accuracy simulations in physics and engineering where nonlocal interactions arise. Overall, the approach offers a robust, scalable alternative to existing kernel-truncation or Gaussian-sum based methods, broadening the toolkit for fast nonlocal potential computations.

Abstract

The convolution potential arises in a wide variety of application areas, and its efficient and accurate evaluation encounters three challenges: singularity, nonlocality and anisotropy. We introduce a fast algorithm based on a far-field smooth approximation of the kernel, where the bounded domain Fourier transform, one of the most essential difficulties, is well approximated by the whole space Fourier transform which usually admits explicit formula. The convolution is split into a regular and singular integral, and they are well resolved by trapezoidal rule and Fourier spectral method respectively. The scheme is simplified to a discrete convolution and is implemented efficiently with Fast Fourier Transform (FFT). Importantly, the tensor generation procedure is quite simple, highly efficient and independent of the anisotropy strength. It is easy to implement and achieves spectral accuracy with nearly optimal efficiency and minimum memory requirement. Rigorous error estimates and extensive numerical investigations, together with a comprehensive comparison, showcase its superiorities for different kernels.
Paper Structure (16 sections, 2 theorems, 36 equations, 3 figures, 5 tables)

This paper contains 16 sections, 2 theorems, 36 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Numerical method
  3. Evaluation of the regular integral $\Phi^{\rm{R}}({\textbf{x}})$
  4. Evaluation of the singular integral $\Phi^{\rm S}({\textbf{x}})$
  5. Discrete tensor structure
  6. Anisotropic density
  7. Error estimates
  8. Far-field smooth approximation
  9. Proof
  10. Numerical results
  11. The Coulomb potentials in 2D/3D
  12. The Poisson potential in 2D
  13. Application to some common potentials
  14. Conclusions
  15. Error estimates
  16. ...and 1 more sections

Key Result

Theorem 1

\newlabelFourierApproximation0 For smooth density $\rho(\mathbf{x})$ that is compactly supported in $\bm{\mathrm{R}}_L^{\bm{\gamma}}$, the following estimates hold true for any positive integer $m >0$.

Figures (3)

  • Figure 1: Cigar-shaped (left) and pancaked-shaped (right) densities.
  • Figure 2: Slice plot of $U^{\varepsilon}$ (left) and the error $(U-U^{\varepsilon})$ (right) for Coulomb kernel $U = 1/r$.
  • Figure 3: Timing results of the pre-computation part versus increasing anisotropy strength $\gamma_f$.

Theorems & Definitions (10)

  • Remark 2.1: Parameter choice of $\varepsilon$ for isotropic case
  • Remark 2.2: Parameter choice of $\varepsilon$ for anisotropic case
  • Remark 2.3: Comparison of the tensor generation procedure
  • Theorem 1
  • Theorem 2: Far-field approximation
  • Remark 4.1
  • Example 1
  • Example 2
  • Example 3
  • Example 4