Cauchy Fast Multipole Method for General Analytic Kernels

Abstract

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between N points in O(N ) or O(N log N ) steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Standard FMMs are derived from analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs have been extended by the introduction of black-box and kernel independent techniques. In these approaches, the user provides only a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal changes to the computer program. This paper presents a novel kernel independent FMM, which leads to diagonal multipoleto-local operators. The result is a significant reduction in the computational cost, particularly when high accuracy is needed. The approach is based on Cauchy's integral formula and the Laplace transform. We will present a numerical analysis of the convergence and numerical results in the case of a multilevel one-dimensional FMM.

Analytical low-rank compression via proxy point selection

“…The work can serve as a starting point to study the proxy point method for more general kernels and higher dimensions. Some possible strategies in future work will be based on other kernel expansions or Cauchy FMM ideas [24]. Various results here are already applicable to more general kernels and other approximation methods.…”

“…Unlike fully algebraic compression, there are also various analytical compression methods that take advantage of degenerate approximations like in (1.1) to compute low-rank approximations. The degenerate approximations may be obtained by Taylor expansions, multipole expansions [12], spherical harmonic basis functions [36], Fourier transforms with Poisson's formula [1,26], Laplace transforms with the Cauchy integral formula [24], Chebyshev interpolations [9], etc. Various other polynomial basis functions may also be used [33].…”

Gaussian Quadrature Rule using ε-Quasiorthogonality

“…They can form the starting point for various fast algorithms such as the fast multipole method, hierarchical matrices (H-matrices), etc. Such techniques are particularly desirable when it comes to the solution of integral equations with translation-invariant kernels (see e.g., [9,1]). Very powerful techniques based on dynamical systems and recursion ideas were recently introduced by Beylkin & Monzón [1,2] in order to approach this problem.…”

A Kernel-Independent Treecode Based on Barycentric Lagrange Interpolation

“…An alternative method employed in the kernel-independent FMM (KIFMM) uses equivalent densities [60,61], while other kernel-independent FMMs use Legendre expansions [22], matrix compression based on skeletonization [42], and truncated Fourier series [62]. Recently an FMM based on the Cauchy integral formula and Laplace transform was proposed for general analytic functions [35], and a kernelindependent treecode was developed using approximate skeletonization for particle systems in high dimensions [41].…”