ANTOINE LAURAIN
PHD IN MATHEMATICS
INVERSE PROBLEMS

Keywords: inverse problems

The inverse source problem

The inverse source problem consists in reconstructing a mass distribution in a geometrical domain from boundary measurements of the associated potential and its normal derivative. In this paper the inverse source problem is reformulated as a topology optimization problem, where the support of the mass distribution is the unknown variable. The Kohn-Vogelius functional is minimized. It measures the misfit between the solutions of two auxiliary problems containing information about the boundary measurements. The Newtonian potential is used to complement the unavailable information on the hidden boundary. The resulting topology optimization algorithm is based on an analytic formula for the variation of the Kohn-Vogelius functional with respect to a class of mass distributions consisting of a finite number of ball-shaped trial anomalies. The resulting reconstruction algorithm is non-iterative and very robust with respect to noisy data. Finally, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments in two and three spatial dimensions are presented.

> A new reconstruction method for the inverse source problem from partial boundary measurements
> A new reconstruction method for the inverse potential problem
> A Non-Iterative Method for the Inverse Potential Problem Based on the Topological Derivative

Numerical algorithms for inverse problems

Two approaches are proposed for solving inverse problems in shape optimization. We are looking for the unknown position of a small hole in a domain ­. First, the asymptotic analysis of the underlying p.d.e. defined in a perturbed domain is performed and the so-called topological derivative is defined. Then, in the first approach, the self-adjoint extensions of elliptic operators are used to model the solution of a partial differential equation defined in the singularly perturbed domain. A least-square functional is then minimized to identify the hole. In the second approach, neural networks are used to determine the inverse of the mapping which associates a set of shape functionals to the position of the unknown hole. In both approaches the topological derivatives are used to approximate the shape functionals.

> Numerical algorithms for an inverse problem in shape optimization

Self-adjoint extensions

Self-adjoint extensions of elliptic operators are used to model the solution of a partial differential equation defined in a singularly perturbed domain. The asymptotic expansion of the solution of a Laplacian with respect to a small parameter " is first performed in a domain perturbed by the creation of a small hole. The resulting singular perturbation is approximated by choosing an appropriate self-adjoint extension of the Laplacian, according to the previous asymptotic analysis. The sensitivity with respect to the position of the center of the small hole is then studied for a class of functionals depending on the domain. A numerical application for solving an inverse problem is presented. Error estimates are provided and a link to the notion of topological derivative is established.

> Using self-adjoint extensions in shape optimization
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