Kaiser-Bessel basis for particle-mesh interpolation

Fast Ewald summation for electrostatic potentials with arbitrary periodicity

A unified treatment for the fast and spectrally accurate evaluation of electrostatic potentials with periodic boundary conditions in any or none of the three spatial dimensions is presented. Ewald decomposition is used to split the problem into real-space and Fourier-space parts, and the Fast Fourier Transform (FFT)-based Spectral Ewald (SE) method is used to accelerate computation of the latter, yielding the total runtime O(N⁡log(N)) for N sources. A key component is a new FFT-based solution technique for the free-space Poisson problem. The computational cost is further reduced by a new adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The SE method is most efficient in the triply periodic case where the cost of computing FFTs is the lowest, whereas the rest of the algorithm is essentially independent of periodicity. We show that removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime, and in the free-space case, the runtime is around four times that of the triply periodic case. The Gaussian window function previously used in the SE method is compared with a new piecewise polynomial approximation of the Kaiser-Bessel window, which further reduces the runtime. We present error estimates and a parameter selection scheme for all parameters of the method, including a new estimate for the shape parameter of the Kaiser-Bessel window. Finally, we consider methods for force computation and compare the runtime of the SE method with that of the fast multipole method.

Midtown splines: An optimal charge assignment for electrostatics calculations

Transferring particle charges to and from a grid plays a central role in the particle-mesh algorithms widely used to evaluate the electrostatic energy in molecular dynamics (MD) simulations. The computational cost of this transfer process represents a substantial part of the overall time required for simulation and is primarily determined by the size of the support (the set of grid nodes at which the transfer function is evaluated). The accuracy of the resulting approximation depends on the form of the transfer function, of which several have been proposed, as well as the size and shape of its support. Here, we show how to derive the transfer function that yields maximal asymptotic accuracy for a given support in the limit of fine grid resolution, finding that all such functions are splines, and we determine these functions (which we refer to as midtown splines) for a variety of choices of support to find optimally efficient transfer functions at accuracy levels relevant to MD simulations. We describe midtown splines that achieve fourth- and sixth-order accuracy in the grid spacing while requiring a support size of 32 and 88 grid nodes, respectively, compared to the 64 and 216 nodes required by the most widely used transfer functions (B-splines). At accuracy levels typically used in MD simulations, the use of midtown splines thus cuts the time required for charge spreading by roughly a factor of two.