A New Version of the Adaptive Fast Gauss Transform for Discrete and Continuous Sources

Leslie F. Greengard, Shidong Jiang, Manas Rachh, Jun Wang

Research output: Contribution to journalArticlepeer-review


We present a new version of the fast Gauss transform (FGT) for discrete and continuous sources. Classical Hermite expansions are avoided entirely, making use only of the planewave representation of the Gaussian kernel and a new hierarchical merging scheme. For continuous source distributions sampled on adaptive tensor product grids, we exploit the separable structure of the Gaussian kernel to accelerate the computation. For discrete sources, the scheme relies on the nonuniform fast Fourier transform (NUFFT) to construct near field plane-wave representations. The scheme has been implemented for either freespace or periodic boundary conditions. In many regimes, the speed is comparable to or better than that of the conventional FFT in work per grid point, despite being fully adaptive.

Original languageEnglish (US)
Pages (from-to)287-315
Number of pages29
JournalSIAM Review
Issue number2
StatePublished - 2024


  • fast Gauss transform
  • Fourier spectral approximation
  • level-restricted adaptive tree
  • nonuniform fast Fourier transform

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics


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