Error analysis of some nonlocal diffusion discretization schemes

TL;DR

This paper analyzes the convergence and error estimates of two nonlocal diffusion discretization schemes inspired by image filtering algorithms, incorporating functional rearrangements and Fourier transform, with numerical experiments on performance and execution time.

Contribution

It introduces and analyzes two novel nonlocal diffusion discretization algorithms, providing convergence proofs, error estimates, and performance evaluation.

Findings

Discrete approximations converge to continuous solutions

Error estimates are established for the schemes

Numerical experiments demonstrate efficiency and execution time

Abstract

We study two numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and on the Fourier transform. Apart from the usual time-space discretization, these algorithms also use the discretization of the range of the solution (quantization). We show that the discrete approximations converge to the continuous solution in suitable functional spaces, and provide error estimates. Finally, we present some numerical experiments illustrating the performance of the algorithms, specially focusing in the execution time.