The paper extends investigations of identification problems by shape optimization methods for perfectly conducting inclusions to the case of perfectly insulating material. The Kohn and Vogelius criteria as well as a tracking type objective are considered for a variational formulation. In case of problems in dimension two, the necessary condition implies immediately a perfectly matching situation for both formulations. Similar to the perfectly conducting case, the compactness of the shape Hessian is shown and the illposedness of the identification problem follows. That is, the second order quadratic form is no longer coercive. We illustrate the general results by some explicit examples and we present some numerical results.
In denoising problems, it is always hard to deal with a mixture of two noise densities. In this paper, we introduce a nonlinear constrained-PDE based on the fractional order tensor diffusion to approximate the clean image and also the impulse component in a mixture of Gaussian and impulse noise. Our model offers an ideal compromise between the edge preservation, staircasing creation and the loss of image contrasts. The proposed constrained-PDE is formulated using a variational model that features a L1 data discrepancy encoding the impulse noise and an L2 term of the clean image, which is a solution of a high-order PDE. Then, a rigorous analysis of the existence of the solution to the proposed model is checked in a suitable functional framework. In addition, to solve the constrained-PDE, we consider an extension of the primal-dual algorithm to nonlinear operators with an accelerate Bregman iteration. Numerical experiments show that the proposed model produces pleasant results in terms of restoration quality and solution efficiency compared to some competitive regularizations.
This paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion ω inside a body Ω from boundary measurement. This inverse problem -known as electrical impedance tomography -has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state u with respect to perturbations of the shape of the interface ∂ω, then we choose a cost function in order to recover the geometry of ∂ω and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.
In this paper, we are interested in the mathematical and numerical study of a variational model derived as Reaction-Diffusion System for image denoising. We use a nonlinear regularization of total variation (TV) operator's, combined with a decomposition approach of H −1 norm suggested by Guo and al. ([19],[20]). Based on Galerkin's method, we prove the existence and uniqueness of the solution on Orlicz space for the proposed model. At last, compared experimental results distinctly demonstrate the superiority of our model, in term of removing noise while preserving the edges and reducing staircase effect.
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