Section: New Results

Inverse problems

Quasi-Reversibility method and exterior approach for evolution problems

Participants : Eliane Bécache, Laurent Bourgeois.

This work is a collaboration with Jérémi Dardé from Toulouse University and has been the object of the internship of Lucas Franceschini, student at ENSTA. We address some linear ill-posed problems involving the heat or the wave equation, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using classical Lagrange finite elements. We have also designed a new approach called the “exterior approach” to solve inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation. It is based on a combination of an elementary level set method and the quasi-reversibility methods we have just mentioned. Some numerical experiments have proved the feasibility of our strategy in all those situations.

Uniqueness and non-uniqueness results for the inverse Robin problem

Participant : Laurent Bourgeois.

This work is a collaboration with Laurent Baratchart and Juliette Leblond (Inria, APICS). We consider the classical Robin inverse problem, which consists in finding the ratio between the normal derivative and the trace of the solution (the Robin coefficient) on a subset of the boundary, given the Cauchy data (both the normal derivative and the trace of the solution) on the complementary subset. More specifically, we consider a Robin coefficient which is merely in $L^{\infty }$ and a Neumann data in $L^2$. In the $2D$ case we prove uniqueness of the Robin coefficient for a problem governed in a Lipschitz domain by a conductivity equation with a conductivity chosen in $W^{1,r}$, where $r>2$. We also prove a non-uniqueness result in the $3D$ case. In two dimensions, the proof relies on complex analysis, while in higher dimension, the proof relies on a famous counterexample to unique continuation by Bourgain and Wolff.

Higher-order expansion of misfit functional for defect identification in elastic solids

Participants : Marc Bonnet, Rémi Cornaggia.

This work, done in the context of the PhD of Rémi Cornaggia, concerns the defect identification by time-harmonic elastodynamic measurements. We propose a generalization to higher orders of the concept of topological derivative, by expanding the least-squares functional in powers of the small radius of a trial inclusion. This expansion is facilitated by resorting to an adjoint state. With this approach, a region of interest may be exhaustively probed at reasonable computational cost.

Inverse scattering and invisibility with a finite set of emitted-received waves

Participant : Anne-Sophie Bonnet-Ben Dhia.

In collaboration with Lucas Chesnel from CMAP at Ecole Polytechnique and Sergei Nazarov from Saint-Petersburg University, we investigate a time harmonic acoustic scattering problem by a compactly supported penetrable inclusion in the free space. We consider cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we have shown that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, for a given real wavenumber, we built a constructive technique (which provides a numerical algorithm) to prove that there exist inclusions for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements.

Energy-based cost functional for three-dimensional transient elastodynamic imaging

Participant : Marc Bonnet.

This work is a collaboration with Wilkins Aquino (Duke University, USA). It is concerned with large-scale three-dimensional inversion under transient elastodynamic conditions by means of the modified error in constitutive relation (MECR), an energy-based, cost functional. Each evaluation of a time-domain MECR cost functional involves the solution of two elastodynamic problems (one forward, one backward), which moreover are coupled (unlike the case of $L^2$ misfit functionals). This coupling creates a major computational bottleneck, making MECR-based inversion difficult for spatially 2D or 3D configurations. To overcome this obstacle, we propose an approach whose main ingredients are (a) setting the entire computational procedure in a consistent time-discrete framework that incorporates the chosen time-stepping algorithm, and (b) using an iterative successive over-relaxation-like method for the resulting stationarity equations. The resulting MECR-based inversion algorithm is formulated under quite general conditions, allowing for 3D transient elastodynamics, straightforward use of available parallel solvers, a wide array of time-stepping algorithms commonly used for transient structural dynamics, and flexible boundary conditions and measurement settings. The proposed MECR algorithm is then demonstrated on computational experiments involving 2D and 3D transient elastodynamics and up to over 500 000 unknown elastic moduli.