SIAM Journal on Scientific and Statistical Computing
The fast Gauss transform with variable scales
SIAM Journal on Scientific and Statistical Computing
Fast Fourier transforms for nonequispaced data
SIAM Journal on Scientific Computing
Accelerating Fast Multipole Methods for the Helmholtz Equation at Low Frequencies
IEEE Computational Science & Engineering
Higher-Order Fourier Approximation in Scattering by Two-Dimensional, Inhomogeneous Media
SIAM Journal on Numerical Analysis
Short Note: The type 3 nonuniform FFT and its applications
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
A wideband fast multipole method for the Helmholtz equation in three dimensions
Journal of Computational Physics
Multiresolution quantum chemistry in multiwavelet bases
ICCS'03 Proceedings of the 2003 international conference on Computational science
Hi-index | 31.45 |
We construct an approximation of the free space Green's function for the Helmholtz equation that splits the application of this operator between the spatial and the Fourier domains, as in Ewald's method for evaluating lattice sums. In the spatial domain we convolve with a sum of decaying Gaussians with positive coefficients and, in the Fourier domain, we multiply by a band-limited kernel. As a part of our approach, we develop new quadratures appropriate for the singularity of Green's function in the Fourier domain. The approximation and quadratures yield a fast algorithm for computing volumetric convolutions with Green's function in dimensions two and three. The algorithmic complexity scales as , where is selected accuracy, κ is the number of wavelengths in the problem, d is the dimension, and C is a constant. The algorithm maintains its efficiency when applied to functions with singularities. In contrast to the Fast Multipole Method, as , our approximation makes a transition to that of the free space Green's function for the Poisson equation. We illustrate our approach with examples.