SIAM Journal on Scientific and Statistical Computing
The fast Gauss transform with variable scales
SIAM Journal on Scientific and Statistical Computing
Numerical simulations of unsteady crystal growth
SIAM Journal on Applied Mathematics
Improved Fast Gauss Transform and Efficient Kernel Density Estimation
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
IEEE Transactions on Pattern Analysis and Machine Intelligence
Application of the Fast Gauss Transform to Option Pricing
Management Science
A fast method for solving the heat equation by layer potentials
Journal of Computational Physics
High Order Accurate Methods for the Evaluation of Layer Heat Potentials
SIAM Journal on Scientific Computing
IEEE Transactions on Image Processing
Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
Fast Approximation of the Discrete Gauss Transform in Higher Dimensions
Journal of Scientific Computing
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The fast Gauss transform allows for the calculation of the sum of $N$ Gaussians at $M$ points in $\mathcal{O}(N+M)$ 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 $\mathcal{O}(q)$ distinct Gaussian-type kernels in order to obtain $q$th-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the “diagonal translation” version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of $N$ and $M$.