Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices

TL;DR

This paper introduces fast, parallelizable algorithms for computing the Fourier spectra of spherically symmetric nonlocal diffusion operators with weak algebraic singularities, improving efficiency and accuracy in multiple dimensions.

Contribution

It develops novel algorithms with linear complexity for eigenvalue computation of nonlocal operators, including asymptotic series and stability enhancements, applicable across all spatial dimensions.

Findings

Algorithms are trivially parallelizable and scalable.

Eigenvalue computation accuracy is individually controllable.

Linear recurrence relations improve efficiency and stability.

Abstract

We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source and target tends to $0$. Rapid algorithms are derived for their Fourier spectra with the computation of each eigenvalue independent of all others. The algorithms are trivially parallelizable, capable of leveraging more powerful compute environments, and the accuracy of the eigenvalues is individually controllable. The algorithms include a Maclaurin series and a full divergent asymptotic series valid for any $d$ spatial dimensions. Using Drummond's sequence transformation, we prove linear complexity recurrence relations for degree-graded sequences of numerators and denominators in the rational approximations to the divergent asymptotic series. These relations are important to ensure that the algorithms are efficient, and also increase the numerical stability compared with the conventional algorithm with quadratic complexity.