Fourier Based Fast Multipole Method for The
نویسنده
چکیده
The fast multipole method (FMM) has had great success in reducing the computa4 tional complexity of solving the boundary integral form of the Helmholtz equation. We present a 5 formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. 6 By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of 7 the algorithm are accelerated by causing the interpolation operators to become straightforward ap8 plications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing 9 a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori 10 determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and 11 verified numerically. Various optimizations are considered to reduce the number of quadrature points 12 and reduce the cost of computing the transfer function. 13
منابع مشابه
A Comparative Study of Multipole and Empirical Relations Methods for Effective Index and Dispersion Calculations of Silica-Based Photonic Crystal Fibers
In this paper, we present a solid-core Silica-based photonic crystal fiber (PCF) composed of hexagonal lattice of air-holes and calculate the effective index and chromatic dispersion of PCF for different physical parameters using the empirical relations method (ERM). These results are compared with the data obtained from the conventional multipole method (MPM). Our simulation results reveal tha...
متن کاملFast Multipole Method based filtering of non-uniformly sampled data
Non-uniform fast Fourier Transform (NUFFT) and inverse NUFFT (INUFFT) algorithms, based on the Fast Multipole Method (FMM) are developed and tested. Our algorithms are based on a novel factorization of the FFT kernel, and are implemented with attention to data structures and error analysis.
متن کاملParticle Simulation Based on Nonequispaced Fast Fourier Transforms
The fast calculation of long-range interactions is a demanding problem in particle simulation. The main focus of our approach is the decomposition of the problem in building blocks and present efficient numerical realizations for these blocks. For that reason we recapitulate the fast Fourier transform at nonequispaced nodes and the fast summation method. We describe the application of these alg...
متن کاملA New Algorithm for the Nonequispaced Fast Fourier Transform on the Rotation Group
We develop an approximate algorithm to efficiently calculate the discrete Fourier transform on the rotation group SO(3). Our method needs O ( L3 logL log(1/ε) + log(1/ε)3Q ) arithmetic operations for a degree-L transform at Q nodes free of choice and with desired accuracy ε. Our main contribution is a method that allows to replace finite expansions in Wigner-d functions of arbitrary orders with...
متن کاملFourier-Based Fast Multipole Method for the Helmholtz Equation
The multilevel fast multipole method (MLFMM) is an algorithm that has had great success in reducing the computational time required to find the solution to the Galerkin boundary integral form of the Helmholtz equation. We present a new formulation of the MLFMM using Fourier basis functions rather than spherical harmonics in order to accelerate and simplify the time-critical stages of the algori...
متن کاملA Fast Algorithm for Filtering and Wavelet Decomposition on the Sphere
Abstract. This paper introduces a new fast algorithm for uniform-resolution filtering of functions defined on the sphere. We use a fast summation algorithm based on Nonequispaced Fast Fourier Transforms, building on previous work that used Fast Multipole Methods. The resulting algorithm performs a triangular truncation of the spectral coefficients while avoiding the need for fast spherical Four...
متن کامل