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 and Ken Trotti
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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

Mathematics, 2024, vol. 12, issue 12, 1-21

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.

Keywords: fractional equations; multigrid methods; image deblurring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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