A method to compute periodic sums

Abstract

In a number of problems in computational physics, a finite sum of kernel functions centered at N particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even though the finite sum can be efficiently computed via fast summation algorithms, such as the fast multipole method (FMM), the periodized extension is usually treated via a different algorithm, Ewald summation, accelerated via the fast Fourier transform (FFT). A different approach to compute this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper. The method splits the periodized sum in to two parts. The first, comprising the contribution of all points outside a large sphere enclosing the box, and some of its neighbors, is approximated inside the box by a collection of kernel functions ("sources") placed on the surface of the sphere or using an expansion in terms of spectrally convergent local basis functions. The second part, comprising the part inside the sphere, and including the box and its immediate neighborhood, is treated via available summation algorithms. The coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential, imposed on a circumspherical surface for the box. While the method is presented in general, details are worked out for the case of evaluating electrostatic potentials and forces. Results show that when used with the FMM, the periodized sum can be computed to any specified accuracy, at an additional cost of the order of the free-space FMM. Several technical details and efficient algorithms for auxiliary computations are provided, as are numerical comparisons.

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

“…While we restricted our attention to static geometry BVPs, the scheme can handle moving geometry problems, such as the flow of bubbles, vesicles, or bacteria, in a straightforward manner (e.g., see [58]). Note that in (2.25) one need not sum all 3 3 copies of source quadrature nodes when applying A, but just the ones falling inside the proxy circle (or sphere) [30]. Note that in (2.25) one need not sum all 3 3 copies of source quadrature nodes when applying A, but just the ones falling inside the proxy circle (or sphere) [30].…”

“…The use of point sources as a particular solution basis that is efficient for smooth solutions is known as the "method of fundamental solutions" (MFS) [8,21], "method of auxiliary sources" [46], "charge simulation method" [44], or, in fast solvers, "equivalent source" [11] or "proxy" [59] representations. This is also used in the recent 3D periodization schemes of Gumerov-Duraiswami [30] and Yan-Shelley [76]. Finally, the low-rank perturbations that enlarge the range of Q are inspired by the low-rank perturbation methods for singular square systems of Sifuentes et al [73].…”

Flexibly imposing periodicity in kernel independent FMM: A multipole-to-local operator approach

“…For the randomly distributed points in Figure 5 B, N B N and adding periodic boundary condition has little cost in τ as shown by Table 3. By reusing the octree data of B 0 , the triply periodic cases only cost about 10% ∼ 40% more time than the free-space cases even in high precision cases, much more efficient than the previously proposed method [12] where the near-field B 0 is calculated without reusing the octree for B 0 .…”

Harmonic surface mapping algorithm for fast electrostatic sums

“…Let α be the ratio between R s and the radius of the circumsphere of the central box. An error bound of the spherical harmonic expansion for the electric potential truncated at n = P is given by [25],…”

“…The HSMA removes the difficulty of solving boundary integral equations by introducing an auxiliary surface away from the central box. The idea of the auxiliary surface has been used in scattering problems [23], multiphase flows [24], and electrostatics [25,26], which is successful because it allows that the nearest-neighbor interactions within the surface is summed directly and the distant interactions can be approximated by a small number of basis functions using least-squares fittings. The harmonic surface mapping developed in this work maps the contribution of the distant interactions into a surface integral such that it can be approximated by discrete images on surfaces with a high order of convergence due to the use of the Fibonacci integration.…”