The fast generalized Gauss transform

Marina Spivak, Shravan K. Veerapaneni, Leslie Greengard

Research output: Contribution to journalArticle

Abstract

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 mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of N and M.

Original languageEnglish (US)
Pages (from-to)3092-3107
Number of pages16
JournalSIAM Journal on Scientific Computing
Volume32
Issue number5
DOIs
StatePublished - 2010

Keywords

  • Fast algorithms
  • Gauss transform
  • Heat potentials
  • High-order accuracy
  • Tensorproduct grids

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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