Year: 2021
Author: Yong Zhang, Chijie Zhuang, Shidong Jiang
Communications in Computational Physics, Vol. 29 (2021), Iss. 5 : pp. 1570–1582
Abstract
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 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 $\mathcal{O}(M+N)$ 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 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2020-0116
Communications in Computational Physics, Vol. 29 (2021), Iss. 5 : pp. 1570–1582
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: One dimensional convolution sum of exponentials singular kernel discrete density.