Search results for " 65K10"
showing 5 items of 5 documents
A product space reformulation with reduced dimension for splitting algorithms
2021
AbstractIn this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra’s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assump…
CLEAR: Covariant LEAst-Square Refitting with Applications to Image Restoration
2017
International audience; In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data. Total variation regularizations and non-local means are special cases of interest. We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we provide an approach that has a ``twicing'' flavor a…
Regularity of sets under a reformulation in a product space of reduced dimension
2023
Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze how some regularity properties are preserved under a reformulation in a product space of reduced dimension. This allows us to establish local linear convergence of parallel projection methods which are constructed through this reformulation.
Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems
2019
In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to lar…
Weak and strong convergence of an inertial proximal method for solving bilevel monotone equilibrium problems
2022
In this paper, we introduce an inertial proximal method for solving a bilevel problem involving two monotone equilibrium bifunctions in Hilbert spaces. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the iterative method are established. Two particular cases illustrating the proposed method are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under saddle point constraint. Furthermore, a numerical example is given to demonstrate the implementability of our algorithm. The algorithm and its convergence results improve and develop previous results i…