The Fast Generalized Gauss Transform

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The 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 Gaussian-type kernels in order to obtain qth 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 avoided. ...
Marina Spivak, Shravan K. Veerapaneni, Leslie Gree
Added 21 May 2011
Updated 21 May 2011
Type Journal
Year 2010
Authors Marina Spivak, Shravan K. Veerapaneni, Leslie Greengard
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