A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring
<|reference_start|>A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring: 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.<|reference_end|>
@article{ahmad2024a,
    title={A note on the convergence of multigrid methods for the Riesz-space
  equation and an application to image deblurring},
    author={Danyal Ahmad, Marco Donatelli, Mariarosa Mazza, Stefano
  Serra-Capizzano, Ken Trotti},
    journal={arXiv preprint arXiv:2403.16352},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.16352},
    primaryClass={math.NA cs.NA}
}