Section: New Results

Iterative Methods for Non-linear Inverse Problems

Inverse medium problem for axisymmetric eddy current models

Participants : Houssem Haddar, Zixian Jiang, Kamel Riahi.

We continued our developments of shape optimization methods for inclusion detection in an axisymmetric eddy current model. This problem is motivated by non-destructive testing methodologies for steam generators. We are finalizing our joint work with A. Lechleiter on numerical methods for the solution of the direct problem in weighted Sobolev spaces using approriate Dirichlet-to-Neumann mappings to bound the computational domain. We are also finalizing jointly with M. El Guedri the work on inverse solver using a regularized steepest descent method for the problem of identifying a magnetite deposits using axial eddy current probe.

We are currently investigating two research directions:

• The development of asymptotic models to identify thin highly conducting deposits. We derived three possible asymptotic models that can be exploited in the inverse problem. The numerical validation is under study.

• The extension of this work to 3D configurations with axisymmetric configuration at infinity, which has been started with the PostDoc of K. Riahi.

A min-max formulation for inverse scattering problems

Participants : Grégoire Allaire, Houssem Haddar, Dimitri Nicolas.

After having developed an inverse solver combining the use of Level-Set method and topological garadient method for multistatic inverse scattering problem and numerically showed how convergence can be achieved with intial guess provided by the Linear Sampling Method, we explored the use of an objective function that would lead to quicker and more stable reconstructions. This has been achieved through maximizing the least-square difference with respect to the Herglotz kernel of used incident wave while minimizing with respect to the geometrical parameters. Premliminary numerical experimentations showed that this procedure is viable and lead to quicker inversion algorithms [5] .

The conformal mapping method and inverse scattering at low frequencies

Participant : Houssem Haddar.

Together with R. Kress we have employed a conformal mapping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for electrostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. In a recent work [41] we proposed an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to obtain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We presented the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples.

A steepest descent method for inverse electromagnetic scattering problems

Participants : Houssem Haddar, Nicolas Chaulet.

In a continuation of our earlier work jointly with L. Bourgeois [13] , we studied the application of non linear optimization techniques to solve the inverse scattering problems for the $3D$ Maxwell's equations with generalized impedance boundary conditions. We characterized the shape derivative in the case where the GIBC is defined by a second order surface operator. We then applied a boundary variation method based on a regularized steepest descent to solve the 3-D inverse problem with partial farfield data. The obtained numerical results demonstrated the possibility of identifying the shape of coated objects as well as the parameters of the coating in the $3D$ Maxwell case.