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Error analysis of some nonlocal diffusion discretization schemes

Gonzalo Galiano

TL;DR

This work analyzes three discretization strategies for nonlocal diffusion problems inspired by bilateral image filtering: a pointwise scheme, a functional rearrangement (RR) approach with quantized ranges, and a Fourier-transform (FFT) method exploiting convolution structure. An explicit Euler semi-discrete framework is established, with rigorous error bounds showing convergence to the continuous solution and quantifiable dependencies on time step $\tau$, spatial mesh $h$, and range-quantization parameters. The RR and FFT methods incorporate range and spatial quantization or spectral techniques, yielding $O(\tau + h + q + r)$ error (RR) or $O(\tau + h + q)$ (FFT) under suitable assumptions, with periodic data enabling stronger results. Numerical experiments confirm the theoretical bounds and reveal practical trade-offs: Ptw is highly accurate for small radii and linear range, while FFT-based methods excel for larger radii, especially when augmented by oversampling, highlighting the balance between accuracy and runtime in nonlocal diffusion discretizations.

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.

Error analysis of some nonlocal diffusion discretization schemes

TL;DR

This work analyzes three discretization strategies for nonlocal diffusion problems inspired by bilateral image filtering: a pointwise scheme, a functional rearrangement (RR) approach with quantized ranges, and a Fourier-transform (FFT) method exploiting convolution structure. An explicit Euler semi-discrete framework is established, with rigorous error bounds showing convergence to the continuous solution and quantifiable dependencies on time step , spatial mesh , and range-quantization parameters. The RR and FFT methods incorporate range and spatial quantization or spectral techniques, yielding error (RR) or (FFT) under suitable assumptions, with periodic data enabling stronger results. Numerical experiments confirm the theoretical bounds and reveal practical trade-offs: Ptw is highly accurate for small radii and linear range, while FFT-based methods excel for larger radii, especially when augmented by oversampling, highlighting the balance between accuracy and runtime in nonlocal diffusion discretizations.

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.

Paper Structure

This paper contains 15 sections, 2 theorems, 104 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A general approximation scheme
  3. Error bound for the common terms
  4. Discretization methods
  5. The Pointwise method
  6. Using functional rearrangements
  7. Using the Fourier transform
  8. Numerical experiments
  9. Execution time
  10. Experiment 1: Box spatial kernel and linear range kernel
  11. Experiment 2: Box spatial kernel and power-like range kernel
  12. Experiment 3: Gaussian spatial kernel and power-like range kernel
  13. Error
  14. Experiment 4: Mixed orders of the size meshes
  15. Experiment 5: Same orders of size meshes

Key Result

Lemma 1

Let $v\in BV(\Omega)$ and consider the function $v_h^{\text{\tiny P}}$ defined by (def:vG) and (def:trozos). Then there exists a constant $C>0$, independent of $v$ and $h$, such that where $| Dv |(\Omega)$ is the variation of $v$ in $\Omega$. Similarly, we have where $J_h^{\text{\tiny P}}(\mathbf{x},\mathbf{y}) = J_h^{\text{\tiny P}}[\mathbf{j},\mathbf{k}]$ if $\mathbf{x}\in\Omega_\mathbf{j}$ an

Figures (6)

  • Figure 1: Initial datum and its profiles at $x_1=0.45$ and $x_2=0.5$.
  • Figure 2: Experiment 1: Execution times (seconds, log$_{10}$ scale) versus radius. Legends are the same for all the plots, see the top-left plot.
  • Figure 3: Experiment 1: Relative errors (log$_{10}$ scale) versus radius. We show the case $J_\Omega = 100$ nodes. The cases $J_\Omega=200,300$ practically reproduce the same figures.
  • Figure 4: Experiment 2: Execution times (seconds, log$_{10}$ scale) versus radius. Legends are the same for all the plots, see the top-left plot.
  • Figure 5: Experiment 3: Execution times (seconds, log$_{10}$ scale) versus radius. Legends are the same for all the plots, see the top-left plot.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2