Section: New Results

Iterative Methods for Non-linear Inverse Problems

Inverse medium problem for axisymmetric eddy current models

Participants : Houssem Haddar, Zixian Jiang, Mohamed 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 finalized a 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 finalized 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 addressed two issues:

  • We developed asymptotic models to identify thin highly conducting deposits. We derived three possible asymptotic models that can be exploited in the inverse problem. We are about to finalize a preprint on this topic.

  • We extended the inverse scheeme to 3D configurations with axisymmetry at infinity: this includes exact charactherization of the shape derivative for a mixed formulation of eddy current problems and a parametric inversion scheme based on a pre-defined discrete grid for deposit location [14] .

The conformal mapping method and free boundary problems

Participant : Houssem Haddar.

Together with R. Kress we 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 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. We are currently investigating the extention of this method to solve free boundary value problems.

A steepest descent method for inverse electromagnetic scattering problems

Participant : Houssem Haddar.

Together with N. Chaulet, we proposed the application of a 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 [4] .

A posteriori error estimates: Application to Electrical Impedance Tomography

Participants : Olivier Pantz, Matteo Giacomini.

One of the main problem in shape optimization problems is due to the fact that the gradient is never computed exactly. When the current solution is far from a local optimum, this is not a problem: even a rough approximation of the gradient enable us to exhibit a descent direction. On the contrary, when close to a local optimal, a very precise computation of the gradient is needed. Together with Karim Trabelsi, we propose to use a-posteriori error estimates to evaluate the error made on the computation of the gradient. This enables us to ensure that at each step, a genuine descent direction is used in the gradient method. Our method has been applied to the minimization of the Kohn-Vogelius functional in the context of electrical impedance tomography. An article is currently in preparation.

A robust stopping rule for EM algorithm with applications to SAXS measurements

Participants : Federico Benvenuto, Houssem Haddar.

The aim of this work was to develop a fully automatic method for the reconstruction of the volume distribution of diluted polydisperse non-interacting nanoparticles with identical shape from Small Angle X-ray Scattering measurements. The described method solves a maximum likelihood problem with a positivity constraint on the solution by means of an Expectation Maximization iterative scheme coupled with an innovative type of regularization. Such a regularization, together with the positivity constraint results in high fidelity quantitative reconstructions of particle volume distributions making the method particularly effective in real applications. The performance of the method on synthetic data in the case of uni- and bi-modal particle volume distributions are shown. Moreover, the reliability of the method is tested when applied to real data provided by a Xenocs device prototype. Finally, the method can be extended to the analysis of the particle distribution for different types of nano-structures.