Fast Ewald summation for electrostatic potentials with arbitrary periodicity

Abstract

A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the free-space Poisson problem in three, two or one dimensions, depending on the number of non-periodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only marginally increase the total run time. Our comparisons also show that the computational cost of the SE method for the free-space case is typically about four times more expensive as compared to the triply periodic case.The Gaussian window function previously used in the SE method, is here compared to an approximation of the Kaiser-Bessel window function, recently introduced in [2]. With a carefully tuned shape parameter that is selected based on an error estimate for this new window function, runtimes for the SE method can be further reduced.

Aliasing error of the exp$(β\sqrt{1-z^2})$ kernel in the nonuniform fast Fourier transform

“…It has a wide range of applications, including magnetic resonance imaging [17,10,21], computed tomography [12], optical coherence tomography [36], synthetic aperture radar [1], spectral interpolation between grids [20,Sec. 6] [14], and electrostatics in molecular dynamics [24,31]; for reviews see [20,16,3]. Given nonuniform points x j , j = 1, .…”

“…being simpler than either KB or PSWF, yet empirically having the same optimal rate and very similar errors. The ES kernel has since been used to accelerate Spectral Ewald codes for periodic electrostatic sums [31]. Its Fourier transform φES,β (ξ) is not known analytically, yet is easily evaluated by quadrature [3,Sec.…”

“…The main goal of this paper is to prove the following aliasing error convergence theorem applying to (7), which places this simple kernel (and hence algorithms which use it [3,31]) on a rigorous footing. The rate will depend on the fixed upsampling factor σ > 1.…”

Fast Ewald summation for Stokes flow with arbitrary periodicity

“…The method was extended to the free-space case for all three kernels by [23]. The current paper serves to complete this development, much in the same way as was recently done for electrostatics by [24], by adding the missing pieces (singly periodic case for all three kernels, and doubly periodic case for the stresslet and rotlet), and unifying all periodic cases within a single framework. This opens up the possibility to perform efficient simulations of three-dimensional Stokes flow with arbitrary periodicity (D = 3, 2, 1, 0), using boundary integral and potential methods.…”

“…Thus, the FMM will typically be faster for highly nonuniform point distributions (especially in free space), while the SE method may be faster for uniform distributions, as shown e.g. by [23,24]. Yet another alternative method is found in [49], in which the long-range interaction is represented by auxiliary sources.…”