Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Goodwill, Tristan, Melia, Owen
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funder: Flatiron Health
OpenAlex ID: W7221564401