A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring: A preconditioned version of a nested primal-dual algorithm for image deblurring
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| Titel: | A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring: A preconditioned version of a nested primal-dual algorithm for image deblurring |
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| Autoren: | Stefano Aleotti, Marco Donatelli, Rolf Krause, Giuseppe Scarlato |
| Quelle: | Journal of Scientific Computing. 103 |
| Publication Status: | Preprint |
| Verlagsinformationen: | Springer Science and Business Media LLC, 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | ill-posed problems, Numerical optimization and variational techniques, convex optimization, Ill-posedness and regularization problems in numerical linear algebra, preconditioning, Numerical aspects of computer graphics, image analysis, and computational geometry, FOS: Mathematics, 65F22 (Primary) 65K10 (Secondary), Mathematics - Numerical Analysis, Numerical Analysis (math.NA), image deblurring |
| Beschreibung: | Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy proposed in Aleotti et al. (Comput. Optim. Appl., 2024) can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical $$\ell ^2$$ ℓ 2 –norm on the fidelity term. |
| Publikationsart: | Article |
| Dateibeschreibung: | application/xml |
| Sprache: | English |
| ISSN: | 1573-7691 0885-7474 |
| DOI: | 10.1007/s10915-025-02863-8 |
| DOI: | 10.48550/arxiv.2409.13454 |
| Zugangs-URL: | http://arxiv.org/abs/2409.13454 |
| Rights: | CC BY |
| Dokumentencode: | edsair.doi.dedup.....24a8434d1027ac6acaf17b0c88bac3c2 |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy proposed in Aleotti et al. (Comput. Optim. Appl., 2024) can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical $$\ell ^2$$ ℓ 2 –norm on the fidelity term. |
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| ISSN: | 15737691 08857474 |
| DOI: | 10.1007/s10915-025-02863-8 |