Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation

Abstract

We introduce an accurate and efficient method for a class of nonlocal potential evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipolar potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel and Taylor expansion of the density. Starting from the convolution formulation, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. Hence, the potential is separated into a regular integral and a near-field singular correction integral, where the first integral is computed with the Fourier pseudospectral method and the latter singular one can be well resolved utilizing a low-order Taylor expansion of the density. Both evaluations can be accelerated by fast Fourier transforms (FFT). The new method is accurate (14-16 digits), efficient (O(N log N ) complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelable.

A GPU accelerated and error-controlled solver for the unbounded Poisson equation in three dimensions

“…Taylor expansion of the density in the near-zone allows to treat the correction by analytical integration, where involved derivatives are computed by FFT as well. The method is efficient and mathematically proven to yield full control over the maximum computation error [1]. We now give a brief description of the method, where we also emphasize novel aspects relevant to the implementation.…”

“…The purpose of this paper is to provide the interested reader with a MATLAB implementation of an efficient and mathematically analyzed method [1] for solving the free-space/unbounded Poisson equation. More precisely, the method presented in this paper solves −∆u(x) = ρ(x), x ∈ R 3 , lim |x|→∞ |u(x)| = 0,…”

“…via the well-known representation of the solution to (1) as the convolution of the density ρ with the free-space Green's function U (x) = 1…”

“…The problem (1) is fundamental in many fields of physics, e.g. quantum chemistry [2][3][4][5][6][7], particle physics [8,9] or astrophysics [10].…”

Optimal control of the self-bound dipolar droplet formation process

“…Consequently, the only remaining difficulty is the computation of the convolution in (18), which however is a ubiquitous problem in computational physics and scientific computing in general. In fact, the solution of (18) A variety of efficient numerical methods has been developed to solve the Poisson equation in unbounded domains [50][51][52][53][54][55][56][57]. While most of them are based on formulation (18) it is also possible to employ (19) directly.…”