Fast non-iterative methods for defect identification

This communication summarizes recent investigations on the identification of defects (cavities, inclusions) of unknown geometry and topology by means of the concept of topological sensitivity. This approach leads to the fast computation (equivalent to performing a few direct solutions), by means of ordinary numerical solution methods such as the BEM (used here), the FEM or the FDM, of defect indicator functions. Substantial further acceleration is obtained by using fast multipole accelerated BEMs. Possibilities afforded by this approach are demonstrated on numerical examples. The paper concludes with a discussion of further research on theoretical and numerical issues

Multifrequency Topological Derivative Approach to Inverse Scattering Problems in Attenuating Media

Detecting objects hidden in a medium is an inverse problem. Given data recorded at detectors when sources emit waves that interact with the medium, we aim to find objects that would generate similar data in the presence of the same waves. In opposition, the associated forward problem describes the evolution of the waves in the presence of known objects. This gives a symmetry relation: if the true objects (the desired solution of the inverse problem) were considered for solving the forward problem, then the recorded data should be returned. In this paper, we develop a topological derivative-based multifrequency iterative algorithm to reconstruct objects buried in attenuating media with limited aperture data. We demonstrate the method on half-space configurations, which can be related to problems set in the whole space by symmetry. One-step implementations of the algorithm provide initial approximations, which are improved in a few iterations. We can locate object components of sizes smaller than the main components, or buried deeper inside. However, attenuation effects hinder object detection depending on the size and depth for fixed ranges of frequencies

Analyse et applications

Le document comporte quatre thèmes de recherche: 1. Méthode de la dérivée topologique en optimisation de formes 2. Convexité en dimension infinie 3. Inégalité de Wente pour l'opérateur de Helmholtz modifié 4. Théorie du point fixe et application

Estimates and asymptotic expansions for condenser p-capacities. The anisotropic case of segments

International audienceWe provide estimates and asymptotic expansions of condenser p-capacities and focus on the anisotropic case of segments. After preliminary results, we study p-capacities of points with respect to asymptotic approximations, positivity cases and convergence speed of descending continuity. We introduce equidistant condensers to point out that the anisotropy caused by a segment in the p-Laplace equation is such that the Pólya-Szegö rearrangement inequality for Dirichlet type integrals yields a trivial lower bound. Moreover, when p > N, one cannot build an admissible solution for the segment, however small its length may be, by extending the case of a punctual obstacle. Our main contribution is to provide a lower bound to the N-dimensional condenser p-capacity of a segment, by means of the N-dimensional and of the (N −1)-dimensional condenser p-capacities of a point. The positivity cases follow for p-capacities of segments. Our method could be extended to obstacles with codimension ≥ 2 in higher dimensions, such as surfaces in R^4. Introducing elliptical condensers, we obtain an estimate and the asymptotic expansion for the condenser 2-capacity of a segment in the plane. The topological gradient of the 2-capacity is not an appropriate tool to separate curves and obstacles with non-empty interior in 2D. In the case p ≠ 2, elliptical condensers should prove useful to obtain further estimates of p-capacities of segments

Contour Detection and Completion for Inpainting and Segmentation Based on Topological Gradient and Fast Marching Algorithms

We combine in this paper the topological gradient, which is a powerful method for edge detection in image processing, and a variant of the minimal path method in order to find connected contours. The topological gradient provides a more global analysis of the image than the standard gradient and identifies the main edges of an image. Several image processing problems (e.g., inpainting and segmentation) require continuous contours. For this purpose, we consider the fast marching algorithm in order to find minimal paths in the topological gradient image. This coupled algorithm quickly provides accurate and connected contours. We present then two numerical applications, to image inpainting and segmentation, of this hybrid algorithm

Mathematical challenges in shape optimization

International audienceThis paper covers some theoretical investigations performed in France, in the framework of the CNRS programme GDR {\it Applications Nouvelles de l'Optimisation de Forme}. The programme included also some activities in Poland. We do not restrict the presentation to the French community in the research field, the list of references includes all recent monographs on the shape optimization. The outline of the paper is the following. First we present some main fields of the activity in shape optimization. To present some precise results, from mathematical point of view, we include two sections. The first is devoted to the eigenvalues, the second to the drag minimization. Many theoretical questions related to these problems are still open