Fast and accurate evaluation of nonlocal coulomb and dipole-dipole interactions via the nonuniform FFT

Shidong Jiang, Leslie Greengard, Weizhu Bao

Research output: Contribution to journalArticle

Abstract

We present a fast and accurate algorithm for the evaluation of nonlocal (longrange) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel ψ(x) and a density function ψ(x) = |ψ(x)|2 for some complexvalued wave function ψψ(x), permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel ρ U(k) has a singularity and/or ρ(k) = 0 at the origin k = 0 in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in k which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in U (k) at the origin is canceled so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function (ρx) is smooth and decays sufficiently quickly as |x| → 8. More precisely, the calculation requires O(N logN) operations, where N is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.

Original languageEnglish (US)
Pages (from-to)B777-B794
JournalSIAM Journal on Scientific Computing
Volume36
Issue number5
DOIs
StatePublished - 2014

Fingerprint

Convolution
Fast Fourier transforms
Probability density function
Dipole
Fourier Integral
Discretization
Density Function
Phase Space
Evaluation
Wave functions
Interaction
Singularity
kernel
Polar Space
Cartesian Grid
Fourier transforms
Fourier Method
Free Space
Fast Fourier transform
Wave Function

Keywords

  • Coulomb interaction
  • Dipole-dipole interaction
  • Interaction energy
  • Nonlocal
  • Nonuniform FFT
  • Poisson equation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

Fast and accurate evaluation of nonlocal coulomb and dipole-dipole interactions via the nonuniform FFT. / Jiang, Shidong; Greengard, Leslie; Bao, Weizhu.

In: SIAM Journal on Scientific Computing, Vol. 36, No. 5, 2014, p. B777-B794.

Research output: Contribution to journalArticle

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