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.