(untitled) (Communications in Computational Physics, 2021)
Based on the recently-developed sum-of-exponential (SOE) approximation, in this article, we propose a fast algorithm to evaluate the one-dimensional convolution potential φ(x)=K∗ρ=∫^1_0K(x−y)ρ(y)dy at (non)uniformly distributed target grid points x_i^M_i=1, where the kernel K(x) might be singular at the origin and the source density function ρ(x) is given on a source grid y_i ^N_j=1 which can be different from the target grid. It achieves an optimal accuracy, inherited from the interpolation of the density ρ(x), within O(M+N) operations. Using the kernel's SOE approximation K_ES, the potential is split into two integrals: the exponential convolution φ_ES=K_ES∗ρ and the local correction integral φ_cor=(K−K_ES)∗ρ. The exponential convolution is evaluated via the recurrence formula that is typical of the exponential function. The local correction integral is restricted to a small neighborhood of the target point where the kernel singularity is considered. Rigorous estimates of the optimal accuracy are provided. The algorithm is ideal for parallelization and favors easy extensions to complicated kernels. Extensive numerical results for different kernels are presented.