Approximating the Gaussian as a Sum of Exponentials and its Applications to the Fast Gauss Transform

Shidong Jiang, Leslie Greengard

Research output: Contribution to journalArticlepeer-review

Abstract

We develop efficient and accurate sum-of-exponential (SOE) approximations for the Gaussian using rational approximation of the exponential function on the negative real axis. Six digit accuracy can be obtained with eight terms and ten digit accuracy can be obtained with twelve terms. This representation is of potential interest in approximation theory but we focus here on its use in accelerating the fast Gauss transform (FGT) in one and two dimensions. The one-dimensional scheme is particularly straightforward and easy to implement, requiring only twenty-four lines of MATLAB code. The two-dimensional version requires some care with data structures, but is significantly more efficient than existing FGTs. Following a detailed presentation of the theoretical foundations, we demonstrate the performance of the fast transforms with several numerical experiments.

Original languageEnglish (US)
Pages (from-to)1-26
Number of pages26
JournalCommunications in Computational Physics
Volume31
Issue number1
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy (miscellaneous)

Keywords

  • Best rational approximation
  • Fast gauss transform
  • Model reduction
  • Sum-of-exponential approximation

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