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Volume 31, Issue 1
Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform

Shidong Jiang & Leslie Greengard

Commun. Comput. Phys., 31 (2022), pp. 1-26.

Published online: 2021-12

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  • 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.

  • Keywords

Fast Gauss transform, sum-of-exponential approximation, best rational approximation, model reduction.

  • AMS Subject Headings

31A10, 65F30, 65E05, 65Y20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1, author = { and Shidong Jiang and and 21548 and and Shidong Jiang and Leslie and Greengard and and 21549 and and Leslie Greengard}, title = {Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform}, journal = {Communications in Computational Physics}, year = {2021}, volume = {31}, number = {1}, pages = {1--26}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0031}, url = {http://global-sci.org/intro/article_detail/cicp/20015.html} }
TY - JOUR T1 - Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform AU - Shidong Jiang , AU - Greengard , Leslie JO - Communications in Computational Physics VL - 1 SP - 1 EP - 26 PY - 2021 DA - 2021/12 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0031 UR - https://global-sci.org/intro/article_detail/cicp/20015.html KW - Fast Gauss transform, sum-of-exponential approximation, best rational approximation, model reduction. AB -

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.

Shidong Jiang & Leslie Greengard. (2021). Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform. Communications in Computational Physics. 31 (1). 1-26. doi:10.4208/cicp.OA-2021-0031
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