A kernel-independent precorrected-FFT method
Abstract
The precorrected-FFT has been used to rapidly evaluate N-body sums arising in electromagnetics and acoustics settings. It has, however, been historically limited to kernels that satisfy a Green's identity. In this work, we demonstrate that, using an improved version of the method of proxy points, the precorrected-FFT can be used to efficiently compute N-body sums with a wide variety of kernels. This algorithm requires no analytic information beyond pointwise access to the kernel. Our examples include kernels from non-local PDEs, Matern kernels, anisotropic kernels, and the Green's functions of high order PDEs. We also provide a detailed algorithm for choosing the proxy points and spreading grid, as implemented in the new pcfft package. We show these algorithmic choices result in favorable scaling when compared to other kernel-independent fast summation methods. Finally, we use the algorithm to accelerate a wide variety of integral equations solvers involving non-standard kernels, with applications in surface scattering, geophysics, and microbial community formation.
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BibTeX
@misc{goodwill2026kernelindependent,
title = {{A kernel-independent precorrected-FFT method}},
author = {Tristan Goodwill and Owen Melia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12821},
eprint = {2610.12821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12821},
}
RIS
TY - UNPB
TI - A kernel-independent precorrected-FFT method
AU - Tristan Goodwill
AU - Owen Melia
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12821
UR - https://arxiv.org/abs/2610.12821
AB - The precorrected-FFT has been used to rapidly evaluate N-body sums arising in electromagnetics and acoustics settings. It has, however, been historically limited to kernels that satisfy a Green's identity. In this work, we demonstrate that, using an improved version of the method of proxy points, the precorrected-FFT can be used to efficiently compute N-body sums with a wide variety of kernels. This algorithm requires no analytic information beyond pointwise access to the kernel. Our examples include kernels from non-local PDEs, Matern kernels, anisotropic kernels, and the Green's functions of high order PDEs. We also provide a detailed algorithm for choosing the proxy points and spreading grid, as implemented in the new pcfft package. We show these algorithmic choices result in favorable scaling when compared to other kernel-independent fast summation methods. Finally, we use the algorithm to accelerate a wide variety of integral equations solvers involving non-standard kernels, with applications in surface scattering, geophysics, and microbial community formation.
ER -