Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
184 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring (2403.16352v1)

Published 25 Mar 2024 in math.NA and cs.NA

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 $\tau$ 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.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (47)
  1. High-order schemes and their error analysis for generalized variable coefficients fractional reaction–diffusion equations. Mathematical Methods in the Applied Sciences, 46(16):16521–16541, 2023.
  2. On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties. Numerical Linear Algebra with Applications, 30(1):e2462, 2023.
  3. Finite difference approximations for fractional advection–dispersion flow equations. Journal of computational and applied mathematics, 172(1):65–77, 2004.
  4. Preconditioning technique based on sine transformation for nonlocal Helmholtz equations with fractional Laplacian. Journal of Scientific Computing, 97(1):17, 2023.
  5. Generalized locally Toeplitz sequences: theory and applications, volume 1. Springer, 2017.
  6. Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: Exposition and review. Archives of Computational Methods in Engineering, 26(5):1639–1690, 2019.
  7. Spectral analysis and structure preserving preconditioners for fractional diffusion equations. Journal of Computational Physics, 307:262–279, 2016.
  8. Multigrid preconditioners for anisotropic space-fractional diffusion equations. Advances in Computational Mathematics, 46(3):1–31, 2020.
  9. A circulant preconditioner for fractional diffusion equations. Journal of Computational Physics, 242:715–725, 2013.
  10. Spectral analysis for preconditioning of multi-dimensional Riesz fractional diffusion equations. Numerical Mathematics: Theory, Methods and Applications, 15(3):565–591, 2022.
  11. Preconditioners for fractional diffusion equations based on the spectral symbol. Numerical Linear Algebra with Applications, page e2441, 2022.
  12. Any circulant-like preconditioner for multilevel matrices is not superlinear. SIAM Journal on Matrix Analysis and Applications, 21(2):431–439, 2000.
  13. A rational preconditioner for multi-dimensional Riesz fractional diffusion equations. Computers & Mathematics with Applications, 143:372–382, 2023.
  14. Multigrid method for fractional diffusion equations. Journal of Computational Physics, 231(2):693–703, 2012.
  15. Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. Journal of Computational Physics, 350:992–1011, 2017.
  16. Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations. SIAM Journal on Scientific Computing, 40(6):A4007–A4039, 2018.
  17. Finite difference approximations for two-sided space-fractional partial differential equations. Applied numerical mathematics, 56(1):80–90, 2006.
  18. A direct O⁢(N⁢l⁢o⁢g2⁢N)𝑂𝑁𝑙𝑜superscript𝑔2𝑁O(Nlog^{2}N)italic_O ( italic_N italic_l italic_o italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_N ) finite difference method for fractional diffusion equations. Journal of Computational Physics, 229(21):8095–8104, 2010.
  19. Fast numerical contour integral method for fractional diffusion equations. Journal of Scientific Computing, 66(1):41–66, 2016.
  20. Stefano Serra-Capizzano. On the extreme spectral properties of Toeplitz matrices generated by L1 functions with several minima/maxima. BIT, 36(1):135–142, 1996.
  21. Stefano Serra-Capizzano. On the extreme eigenvalues of Hermitian (block) Toeplitz matrices. Linear algebra and its applications, 270(1-3):109–129, 1998.
  22. On the condition numbers of large semidefinite Toeplitz matrices. Linear algebra and its applications, 279(1-3):285–301, 1998.
  23. On the rate of convergence of the preconditioned conjugate gradient method. Numerische Mathematik, 48(5):499–523, 1986.
  24. Stefano Serra-Capizzano. Toeplitz preconditioners constructed from linear approximation processes. SIAM Journal on Matrix Analysis and Applications, 20(2):446–465, 1998.
  25. Stefano Serra-Capizzano. A note on antireflective boundary conditions and fast deblurring models. SIAM Journal on Scientific Computing, 25(4):1307–1325, 2004.
  26. Anti-reflective boundary conditions and re-blurring. Inverse Problems, 21(1):169, 2004.
  27. Robust and optimal multi-iterative techniques for IgA Galerkin linear systems. Computer Methods in Applied Mechanics and Engineering, 284:230–264, 2015.
  28. Algebraic multigrid. In Multigrid methods, pages 73–130. SIAM, 1987.
  29. Multigrid. Elsevier, 2000.
  30. Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM Journal on Scientific Computing, 17(5):1068–1081, 1996.
  31. Multigrid method for ill-conditioned symmetric Toeplitz systems. SIAM Journal on Scientific Computing, 19(2):516–529, 1998.
  32. A V-cycle multigrid for multilevel matrix algebras: proof of optimality. Numerische Mathematik, 105(4):511–547, 2007.
  33. Marco Donatelli. An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices. Numerical Linear Algebra with Applications, 17(2-3):179–197, 2010.
  34. Giuseppe Fiorentino and S Serra-Capizzano. Multigrid methods for Toeplitz matrices. Calcolo, 28(3):283–305, 1991.
  35. V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM Journal on Matrix Analysis and Applications, 26(1):186–214, 2004.
  36. Stefano Serra-Capizzano. Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numerische mathematik, 92(3):433–465, 2002.
  37. A smoothing analysis for multigrid methods applied to tempered fractional problems. Linear and Multilinear Algebra, pages 1–24, 2023.
  38. Stefano Serra-Capizzano. An ergodic theorem for classes of preconditioned matrices. Linear Algebra and its Applications, 282(1-3):161–183, 1998.
  39. Tony F Chan. An optimal circulant preconditioner for Toeplitz systems. SIAM journal on scientific and statistical computing, 9(4):766–771, 1988.
  40. Spectral and computational properties of band symmetric Toeplitz matrices. Linear Algebra and its Applications, 52:99–126, 1983.
  41. An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients. Applied Mathematics and Computation, 402:126091, 2021.
  42. Fractional calculus in image processing: a review. Fractional Calculus and Applied Analysis, 19(5):1222–1249, 2016.
  43. Spectral approximation of fractional PDEs in image processing and phase field modeling. Computational Methods in Applied Mathematics, 17(4):661–678, 2017.
  44. Fractional graph Laplacian for image reconstruction. Applied Numerical Mathematics, 2023.
  45. Marco Donatelli. A multigrid for image deblurring with Tikhonov regularization. Numerical linear algebra with applications, 12(8):715–729, 2005.
  46. Per Christian Hansen. Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, 1998.
  47. Regularization preconditioners for frame-based image deblurring with reduced boundary artifacts. SIAM Journal on Scientific Computing, 38(1):B164–B189, 2016.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now