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

Abstract

An important but missing component in the application of the kernel independent fast multipole method (KIFMM) is the capability for flexibly and efficiently imposing singly, doubly, and triply periodic boundary conditions. In most popular packages such periodicities are imposed with the hierarchical repetition of periodic boxes, which may give an incorrect answer due to the conditional convergence of some kernel sums. Here we present an efficient method to properly impose periodic boundary conditions using a near-far splitting scheme. The nearfield contribution is directly calculated with the KIFMM method, while the far-field contribution is calculated with a multipole-to-local (M2L) operator which is independent of the source and target point distribution. The M2L operator is constructed with the far-field portion of the kernel function to generate the far-field contribution with the downward equivalent source points in KIFMM. This method guarantees the sum of the near-field & far-field converge pointwise to results satisfying periodicity and compatibility conditions. The computational cost of the far-field calculation observes the same (N ) complexity as FMM and is designed to be small by reusing the data computed by KIFMM for the near-field. The far-field calculations require no additional control parameters, and observes the same theoretical error bound as KIFMM. We present accuracy and timing test results for the Laplace kernel in singly periodic domains and the Stokes velocity kernel in doubly and triply periodic domains.

Complex dynamics of long, flexible fibers in shear

“…This approach is straightforward to implement, but is limited to a common value of ε for all points. Further, if periodic boundary condition are necessary, the periodizing operator T M 2L must be recalculated for each different ε [24]. Another approach is to regard each source point in S ε as a four dimensional vector (w 1 , w 2 , w 3 , ε).…”

“…After w equiv k is found, the singular kernel S 0 is used throughout the tree traversal. This approach allows the regularization parameter ε to vary for different source points, and allows the direct reuse of T M 2L computed for S 0 [24] to implement various types of periodic boundary conditions. The computation code is implemented based on the parallel KIFMM library PVFMM [14].…”

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

“…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].…”

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

“…In this section we present numerical results using the periodic KIFMM method developed in our previous work [17]. It works by splitting the infinite periodic domain into a near field and a far field.…”

“…The kernel sum could be simply written as g = Kq, where the indices s, t are suppressed. For Stokes and Laplace kernel sums, recently developed optimal fast periodic kernel summation methods with flexible periodic boundary conditions can be used, including the Spectral Ewald methods [14; 15; 16], which scale as O(N log N), and the periodic Kernel Independent Fast Multipole Method (KIFMM) method by Yan and Shelley [17], which scales as O(N). However the image systems developed by Gimbutas et al [11] does not work in this framework, because the partially periodic (i.e., simply or doubly periodic) summations for the Stokeslet and the Laplace monopole kernel do not allow a net force or a net monopole in a periodic box, as otherwise the infinite periodic summations diverge.…”