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A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring

Danyal Ahmad, Marco Donatelli, Mariarosa Mazza, Stefano Serra-Capizzano, Ken Trotti

TL;DR

This paper analyzes the convergence of multigrid methods for discretizations of the one- and two-dimensional Riesz fractional diffusion equations, whose linear systems form dense, symmetric Toeplitz (or banded) matrices. It extends convergence theory from two-grid to V-cycle and W-cycle multigrid, introduces a band-approximation Galerkin variant, and compares against $\tau$ and circulant preconditioners using both 1D and 2D RFDE tests with variable coefficients. The study shows level-independent convergence of the multigrid schemes under careful smoothing and projection choices, and demonstrates that the $\tau$ preconditioner consistently offers the best performance for these problems, including an application to image deblurring with Tikhonov regularization. Overall, the results provide practical, scalable solvers for large, ill-conditioned fractional diffusion systems and highlight effective preconditioning strategies for real-world imaging tasks.

Abstract

In the past decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz-space FDE whose theoretical convergence analysis of such multigrids is currently limited to the two-grid method. Here we provide a detailed theoretical convergence study in the case of V-cycle and W-cycle. Moreover, we discuss its use combined with a band approximation and we compare the result with both $τ$ and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz-FDE with variable coefficients. Finally, we apply the best-performing method to an image deblurring problem with Tikhonov regularization.

A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring

TL;DR

This paper analyzes the convergence of multigrid methods for discretizations of the one- and two-dimensional Riesz fractional diffusion equations, whose linear systems form dense, symmetric Toeplitz (or banded) matrices. It extends convergence theory from two-grid to V-cycle and W-cycle multigrid, introduces a band-approximation Galerkin variant, and compares against and circulant preconditioners using both 1D and 2D RFDE tests with variable coefficients. The study shows level-independent convergence of the multigrid schemes under careful smoothing and projection choices, and demonstrates that the preconditioner consistently offers the best performance for these problems, including an application to image deblurring with Tikhonov regularization. Overall, the results provide practical, scalable solvers for large, ill-conditioned fractional diffusion systems and highlight effective preconditioning strategies for real-world imaging tasks.

Abstract

In the past decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz-space FDE whose theoretical convergence analysis of such multigrids is currently limited to the two-grid method. Here we provide a detailed theoretical convergence study in the case of V-cycle and W-cycle. Moreover, we discuss its use combined with a band approximation and we compare the result with both and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz-FDE with variable coefficients. Finally, we apply the best-performing method to an image deblurring problem with Tikhonov regularization.
Paper Structure (14 sections, 6 theorems, 71 equations, 7 figures, 4 tables)

This paper contains 14 sections, 6 theorems, 71 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multigrid methods and level independency
  4. Behaviour of symbols $f_{k,\alpha}$
  5. Smoothing Property
  6. Approximation property and level independency
  7. Band Approximation
  8. Numerical Examples
  9. Example 1.
  10. 2D Problems
  11. Example 2.
  12. Example 3.
  13. Application to image deblurring
  14. Conclusions

Key Result

Lemma 2.1

For $1<\alpha<2$, the coefficients $g_{k}^{(\alpha)\,}$ defined in eq:g_Coef satisfy

Figures (7)

  • Figure 1: Plot of $C_{k,\alpha}$ vs the level $k$, for different values of $\alpha$.
  • Figure 2: Plots of $f_{k+1,\alpha}/c(\alpha)$ for $\alpha=1.5$ and $k=0,\dots,8$, with $C_{k,\alpha}$ computed according to \ref{['eq:C_values']}.
  • Figure 3: Plot of $\mathbf{g}_{n,\alpha}(x)$ for different values of $\alpha$.
  • Figure 4: Plots of $f$ and ${g}_{s}$.
  • Figure 5: Plots of ${\delta}_{s}$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Lemma 2.1
  • Definition 1
  • Lemma 2.2
  • Theorem 3.1: Ruge-Stüben ruge1987algebraic
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1