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

2016/06/24 by Wei Zhao, Martin Stoll, Zhao, Wei +1
Engineering · Physics and Astronomy · #Numerical methods in engineering #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods

paper · pdf · doi:10.48550/arxiv.1606.07652

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 O(N²) 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 O(N²) to O(N^1.5) and even to 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.

Related

• Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation
• A Fast Summation Method for translation invariant kernels
• Optimized M2L Kernels for the Chebyshev Interpolation based Fast Multipole Method
• The Inverse Fast Multipole Method
• A Simplified Fast Multipole Method Based on Strong Recursive Skeletonization