On the use of medium frequencies in the solution of the inverse scattering problem
Résumé
In this article we study the presence of multiple critical points in the usual topology optimization formulation of the classical inverse scattering problem. We consider a very simple example of the two-dimensional problem: the scatterer is a disk composed of a homogeneous material that is within a homogeneous medium of different material properties. The example considers the most favorable conditions for solving the inverse problem: measurements are taken on a whole circle surrounding the scatterer, for incident plane waves coming from all directions. In addition, these measurements have no error. However, in the case of medium frequencies (wavelengths from one-third to one scatterer diameter), we show that multiple critical points exist when we consider the optimality criteria provided by the shape gradient and also when we consider the optimality criteria provided by the topological derivative of the usual objective functional of the problem. These critical points have the same radial symmetry as the circular scatterer, which allows the application of a very accurate semi-analytical method based on series expansions to obtain the solutions of the forward and adjoint problems, as well as to compute the shape and topological derivatives. The critical points are obtained by solving the nonlinear system of equations that provide the optimality conditions, and are reported for different values of the material properties that define the inverse problem.
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|---|
| Licence |