Section: New Results

Iterative Methods for Non-linear Inverse Problems

Convergence Analysis of Newton type methods

Participant : Armin Lechleiter.

Despite Newton-like methods are among the classical techniques for solving non-linear inverse problems, their convergence analysis is still incomplete. In a joint project with Andreas Rieder, we develop a general convergence analysis for an entire class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. The methods under consideration consists of two components: the outer Newton iteration (stopped by a discrepancy principle) and an inner regularization scheme which provides the update of the iteration. In this paper we give a novel and unified convergence analysis which is not restricted to a specific inner regularization scheme. Indeed, our analysis applies to a variety of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients [19] .

Hybrid methods for inverse scattering problems

Participants : Grégoire Allaire, Houssem Haddar, Olivier Pantz, 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.

Dimitri Nicolas started his PhD on September 2009 on this topic under the supervision of G. Allaire. Preliminary 2-d numerical experiments have been conducted by initializing a geometric optimization algorithm with the shape provided by the linear sampling method. The obtained results validate the efficiency of this coupling in the case of simply connected obstacles. More complex configurations are under investigations.

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 = (C₀) as the image of the interior boundary circle C₀ .

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 [35] .