Ph.D. student under the direction of A. Massa

Thesis title: Bayesian inversion and sparsity in the framework of wavefield inverse problems
Thesis abstract: The PhD subject is developed according to the research axes of the Inverse Problems Group of the Signal Center (Pôle Signaux) of the Laboratoire des Signaux et Systèmes (L2S) UMR8506 under the supervision of M. Lambert, and it is led in the framework of the DIGITEO Chair SIRENA held at L2S by A. Massa, ELEDIA, University of Trento. The work focuses on the characterization of objects from electromagnetic waves that they scatter. Such a problem is known to be non-linear and ill-posed, leading to either no solution or multiple solutions. The idea proposed here is to add information on the sparsity of the solution in order to better constrain it from well-chosen prior information. This work extends the investigations already carried out within ELEDIA under the direction of A. Massa on the implementation methods of probabilistic Bayesian inversion, incorporating constraints of sparsity on the solutions sought in case of anisotropic objects in free space and then stratified environments. Particular attention will be paid to the upstream notions of representations by suitable bases of descriptors of the structures under test by electromagnetic means. After completing a thorough review of the state of the art of inversion and imaging involving compressed acquisition methods (compressive sensing, CS), the thesis will be concerned by what is a compressible problem as part of the characterization of objects by electromagnetic waves. The implementation or customization (if available) of inversion algorithms or imaging "based CS" will then be considered. At first, only the two-dimensional case will be studied with both conventional polarizations (TE and TM), and once theoretical and numerical aspects properly identified, understood and validated via suitable numerical experimentation, the three-dimensional case will be considered. More generally, it is expected that significant algorithmic advances will take place within the framework outlined above, and that, beyond, proofs of concept will be given, allowing, PhD completed, to imagine extending this approach to other applications/configurations of interest.