Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform (Communications in Computational Physics, 2021)
We develop efficient and accurate sum-of-exponential (SOE) approximations for the Gaussian using rational approximation of the exponential function on thenegative 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 inapproximation 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 particularlystraightforward and easy to implement, requiring only twenty-four lines of MATLABcode. The two-dimensional version requires some care with data structures, but is significantly more efficient than existing FGTs. Following a detailed presentation of thetheoretical foundations, we demonstrate the performance of the fast transforms withseveral numerical experiments.