Fast one-dimensional convolution with general kernels using sum-of-exponential approximation
Document Type
Article
Publication Date
3-1-2021
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=R01K(x−y)ρ(y)dyat (non)uniformly distributed target grid points {xi}iM=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 {yj}Nj=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 KES, the potential is split into two integrals: the exponential convolution φES =KES∗ρ and the local correction integral φcor = (K−KES)∗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.
Identifier
85104872978 (Scopus)
Publication Title
Communications in Computational Physics
External Full Text Location
https://doi.org/10.4208/CICP.OA-2020-0116
e-ISSN
19917120
ISSN
18152406
First Page
1570
Last Page
1582
Issue
5
Volume
29
Recommended Citation
Zhang, Yong; Zhuang, Chijie; and Jiang, Shidong, "Fast one-dimensional convolution with general kernels using sum-of-exponential approximation" (2021). Faculty Publications. 4282.
https://digitalcommons.njit.edu/fac_pubs/4282