IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i12p1916-d1419085.html
   My bibliography  Save this article

A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring

Author

Listed:
  • Danyal Ahmad

    (Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy)

  • Marco Donatelli

    (Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy)

  • Mariarosa Mazza

    (Dipartimento di Matematica, Università Degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy)

  • Stefano Serra-Capizzano

    (Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy)

  • Ken Trotti

    (Facoltà di Informatica, Università della Svizzera Italiana, Via Giuseppe Buffi 13, CH-6900 Lugano, Switzerland)

Abstract

In recent 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 multigrid methods is currently limited in the relevant literature to the two-grid method. Here we provide a detailed theoretical convergence study in the multilevel setting. 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 investigate the use of a Riesz–Space FDE in a variational model for image deblurring, comparing the performance of specific preconditioning strategies.

Suggested Citation

  • Danyal Ahmad & Marco Donatelli & Mariarosa Mazza & Stefano Serra-Capizzano & Ken Trotti, 2024. "A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring," Mathematics, MDPI, vol. 12(12), pages 1-21, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1916-:d:1419085
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/12/1916/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/12/1916/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Pan, Kejia & Sun, Hai-Wei & Xu, Yuan & Xu, Yufeng, 2021. "An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 402(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1916-:d:1419085. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.