Abdelkrim Chakib

The multiframe super-resolution (SR) techniques are considered as one of the active research fields. More precisely, the construction of the desired high resolution (HR) image with less artifacts in the SR models, which are always ill-posed problems, requires a great care. In this paper, we propose a new fourth-order equation based on a diffusive tensor that takes the benefit from the diffusion model of Perona-Malik in the flat regions and the Weickert model near boundaries with a high diffusion order. As a result, the proposed SR approach can efficiently preserve image features such as corner and texture much better with less blur near edges. The existence and uniqueness of the proposed partial differential equation (PDE) are also demonstrated in an appropriate functional space. Finally, the given experimental results show the effectiveness of the proposed PDE compared to some competitive methods in both visually and quantitatively.

In a previous article of the authors [Journal of Convex Analysis 21 (2014) 67–87], we extended to the case of W^{1,1}_{\mathrm{loc}} W l o c 1 , 1 functions a formula, known for C^1 C 1 functions, for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains. The interest in that formula was that, due to the convexity context, it uses the support functions of the domains to express the shape derivative. Unfortunately, we noticed after the publication that we made a mistake in the proof of the main result. In this Erratum we give an alternative argument so that all the results in the above-mentioned article remain valid.

We extend to the case of W ^1,1 _loc functions a formula, known for C ¹ functions, for the computation of the shape derivative of an integral cost functional with respect to a class of admissible convex domains. The convexity context allows one to use the support functions of the domains to express the shape derivative. We shall also illustrate the formula by applying it to the computation of the shape derivative for a shape optimization problem and by giving an algorithm

We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.

A non-overlapping domain decomposition method for elasticity equations based on an optimal control formulation is presented. The existence of a solution is proved and the convergence of a subsequence of the approximate solutions to a solution of the continuous problem is shown. The implementation based on lagrangian method is discussed. Finally, numerical results showing the efficiency of our approach and confirming the convergence result are given.

This work was carried out when the first and the third authors was visiting Professors at Laboratoire de Mathématiques Jean Leray, UMR662, CNRS/Université de Nantes/Ecole Centrale de Nantes.

Shape optimization amounts to find the optimal shape of a domain which minimizes a given criterion, often called a cost functional. Here, we are interested in the case where the criterion is computed through the solution of a partial differential equation, the so-called state equation, which makes the optimization problem non-trivial. We use a general parameterization of the unknown boundary in order to preserve the physical general information and we prove the continuity of the cost functional.