## Section: New Results

### Iterative Methods for Non-linear Inverse Problems

#### Inverse medium problem for axisymmetric eddy current models

Participants : Houssem Haddar, Zixian Jiang, Armin Lechleiter.

We are interested in 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 investigated the validity of the eddy current model for these kinds of problems and developed numerical methods for the solution of the direct problem in weighted Sobolev spaces. Then we computed the shape derivative of an inclusion which allows to use regularized iterative methods to solve the inverse problem.

#### Hybrid methods for inverse scattering problems

Participants : GrĂ©goire Allaire, Houssem Haddar, Dimitri Nicolas.

It is well admitted that optimization methods offer in general a good accuracy but are penalized by the cost of solving the direct problem and by requiring a large number of iterations due to the ill-posedness of the inverse problem. However, profiting from good initial guess provided by sampling methods these method would become viable. Among optimization methods, the Level Set method seems to be well suited for such coupling since it is based on capturing the support of the inclusion through an indicator function computed on a cartesian grid of probed media. Beyond the choice of an optimization method, our goal would be to develop coupling strategies that uses sampling methods not only as an initialization step but also as a method to optimize the choice of the incident (focusing) wave that serves in computing the increment step.

We investigated a coupling approach between the level set method and LSM where the initialization is done using a crude estimate provided by the linear sampling method. The obtained results validate the efficiency of this coupling in the case of simply and multiply connected obstacles that are well separated. Incorporating this coupling in a multi-frequency approach is under investigations.

#### Inverse Scattering by means of shape optimization

Participant : Olivier Pantz.

We have investigated a new strategy based on the shape optimization method in order to recover the shape of a perfect conductor based on measures of the fields it scattered when enlightened by planar waves. The method consists in minimizing a cost function with respect to the shape of a "guessed" conductor. In a previous work, we have implemented this method using the standard quadratic differences between the scattered fields of the "target" and "guessed" conductors as cost-function. We have extended this method to another cost-function that could be seen as a relaxation of the previous one. This work has been conducted by V. Vostrikov under the supervision of O. Pantz in the context of an internship. A code has been developed to implement this new approach and has been concluded by a final report.

#### The conformal mapping method for the inverse conductivity problem

Participant : Houssem Haddar.

In a series of recent papers Akduman, Haddar and Kress have developed
a new simple and fast numerical scheme for solving two-dimensional inverse boundary value problems
for the Laplace equation that model non-destructive testing and evaluation
via electrostatic imaging.
In the fashion of a decomposition method, the
reconstruction of the boundary shape _{0}
of a perfectly conducting or a nonconducting inclusion
within a doubly connected conducting medium from over-determined
Cauchy data on the accessible exterior boundary _{1}
is separated into a nonlinear well-posed problem and a linear ill-posed problem.
The approach is based on a conformal map :BD that takes an annulus
B bounded by two concentric circles onto D .
In the first step, in terms of the
given Cauchy data on _{1} , by successive approximations one has to solve a nonlocal and nonlinear
ordinary differential
equation for the boundary values |_{C1} of this mapping on the exterior
boundary circle of B .
Then in the second step
a Cauchy problem for the holomorphic function in B has to be solved
via a regularized Laurent expansion to obtain the unknown boundary
_{0} = (C_{0}) as the image of the interior boundary circle C_{0} .

In a joint work with R. Kress we proposed an extension of this approach to two-dimensional inverse electrical impedance tomography with piecewise constant conductivities. A main ingredient of our method is the incorporation of the transmission condition on the unknown interior boundary via a nonlocal boundary condition in terms of an integral equation. We present the foundations of the method, a local convergence result and exhibit the feasibility of the method via numerical examples [20] . We currently investigate the extention to the case of electrode-type measurements.

#### Inverse source problem related to contaminants transport with adsorption in porous media

Participants : Aziz Darouichi, Houssem Haddar.

Soil and groundwater contamination by organic compounds have increasingly become a major environmental concern. This contamination can typically arise from an accidental or deliberate spills of some substances, like an industrial solvent, gasoline or other organic products, during the production, the transport or the storage of these constituents. These substances constitute a real threat to groundwater contamination since they constitute a long term source of contamination by dissolving during infiltrating rainwater, flooding ... , and furthermore presenting a health hazard and peril to inhabitants living next to these areas. The modelling of this phenomena necessitates the knowledge of certain physical parameters which are very often unknown or poorly known (dispersion, initial concentration of the pollutant, adsorption isotherms, ... ).

As first investigations, we considered the inverse source problem related to contaminant transport in porous media with adsorption in equilibrium mode, where the data are final measurements of the contaminant concentration. The underlying forward model is an advective?dispersive transport equation coupled with a mass balance equation for the stationary water phase. We solved the inverse problem using a weighted cost functional combined with a regularization of the gradient. The convergence of the method is proved for simplified configurations. The unique identification is however still an open problem for this model. The extension to nonlinear transport problems is under study.