A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

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 va...

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Veröffentlicht in:Journal of scientific computing Jg. 103; H. 3; S. 85
Hauptverfasser: Aleotti, Stefano, Donatelli, Marco, Krause, Rolf, Scarlato, Giuseppe
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.06.2025
Springer Nature B.V
Schlagworte:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Convergence
Convex analysis
Iterative algorithms
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Optimization
Preconditioning
Theoretical
65K10
Ill-posed problems
Image deblurring
Convex optimization
65F22
Preconditioning
ISSN:0885-7474, 1573-7691
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Zusammenfassung: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 ℓ 2 –norm on the fidelity term.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-02863-8