The fast Gauss transform with all the proofs
نویسندگان
چکیده
In this report we discuss some technical aspects of the fast Gauss transform (FGT). The FGT is a special case of the single level FMM for the Gaussian potential. We give detailed derivation of the correct error bounds which is rather terse in the original paper. We also briefly discuss about the data structures to implement the space subdivision in higher dimensional spaces. [ FGT ]: April 8, 2006
منابع مشابه
The fast Gauss transform with complex parameters q
We construct a fast method, OðN logNÞ, for the computation of discrete Gauss transforms with complex parameters, capable of dealing with unequally spaced grid points. The method is based on Fourier techniques, and in particular it makes use of a modified unequally spaced fast Fourier transform algorithm, in combination with previously suggested divide and conquer strategies for ordinary fast Ga...
متن کاملQuadratic Reciprocity and Sign of Gauss Sum via the Finite Weil Representation
We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p, q and pq. 0. Introduction Two basic results due to Gauss are the quadratic reciprocity law and the sign of th...
متن کاملFast Gauss transforms with complex parameters using NFFTs
at the target knots yj ∈ [−14 , 1 4 ], j = 1, . . . ,M , where σ = a + ib, a > 0, b ∈ R denotes a complex parameter. Fast Gauss transforms for real parameters σ were developed, e.g., in [15, 8, 9]. In [12], we have specified a more general fast summation algorithm for the Gaussian kernel. Recently, a fast Gauss transform for complex parameters σ with arithmetic complexity O(N log N + M) was int...
متن کاملThe Fast Generalized Gauss Transform
The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N + M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with O(q) distinct Gaus...
متن کاملA sharp error estimate for the fast Gauss transform
We report an error estimate of the multi-dimensional fast Gauss transform (FGT), which is much sharper than that previously reported in the literature. An application to the Karhunen–Loeve decomposition in the three-dimensional physical space is also presented that shows savings of three orders of magnitude in time and memory compared to a direct solver. 2006 Elsevier Inc. All rights reserved.
متن کامل