Fast One-Dimensional Convolution with General Kernels Using Sum-of-Exponential Approximation (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 $φ(x)=K∗ρ=∫^1_{0}K(x−y)ρ(y)dy$ at (non)uniformly distributed target gridpoints {$x_i$}$^M_{i=1}$, where the kernel $K(x)$ might be singular at the origin and the sourcedensity function $ρ(x)$ is given on a source grid ${{{y_i}}}^N_{j=1}$ which can be different fromthe target grid. It achieves an optimal accuracy, inherited from the interpolation ofthe density $ρ(x)$, within $\mathcal{O}(M+N)$ 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 isevaluated via the recurrence formula that is typical of the exponential function. Thelocal correction integral is restricted to a small neighborhood of the target point wherethe kernel singularity is considered. Rigorous estimates of the optimal accuracy areprovided. The algorithm is ideal for parallelization and favors easy extensions to complicated kernels. Extensive numerical results for different kernels are presented.