A Fast Multipole Method based on Band-limited Approximations for Radial Basis Functions

TL;DR

This paper introduces a novel kernel-independent fast multipole method for radial basis function interpolation, utilizing band-limited approximations and quadrature rules to significantly improve computational efficiency for large datasets.

Contribution

The paper presents a new FMM approach based on band-limited approximations, extending the applicability and efficiency of fast summation in RBF interpolation.

Findings

Reduces computational complexity from O(N^2) to near O(NlogN)

Enables efficient large-scale RBF interpolation

Extends FMM applicability through band-limited approximation

Abstract

The meshless/meshfree radial basis function (RBF) method is a powerful technique for interpolating scattered data. But, solving large RBF interpolation problems without fast summation methods is computationally expensive. For RBF interpolation with $N$ points, using a direct method requires $\mathcal{O}(N^2)$ operations. As a fast summation method, the fast multipole method (FMM) has been implemented in speeding up the matrix-vector multiply, which reduces the complexity from $\mathcal{O}(N^2)$ to $\mathcal{O}(N^{1.5})$ and even to $\mathcal{O}(NlogN)$ for the multilevel fast multipole method (MLFMM). In this paper, we present a novel kernel-independent fast multipole method for RBF interpolation, which is used in combination with the evaluation of point-to-point interactions by RBF and the fast matrix-vector multiplication. This approach is based on band-limited approximation and quadrature rules, which extends the range of applicability of FMM.