Fast Ewald summation for electrostatic potentials with arbitrary periodicity

A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three spatial 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 computational cost is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The SE method will always be most efficient for the triply periodic case as the cost of computing FFTs will then be the smallest, whereas the computational cost of the rest of the algorithm is essentially independent of periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime. Our comparisons also show that the computational cost of the SE method in the free-space case is around four times that of the triply periodic case. The Gaussian window function previously used in the SE method, is here compared to a piecewise polynomial approximation of the Kaiser-Bessel window function. 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. Furthermore, we consider different methods for computing the force, and compare the runtime of the SE method with that of the Fast Multipole Method.