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 - <p style="text-align: justify;">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.</p>