ISP-EDP Associate Team

Identification of singular parameters in PDEs

Inria Associate Team program (2022-2024). 2023 poster.


From Inria (IDEFIX project team):

From LAMHA (Laboratory of Applied Mathematics and Harmonic Analysis):

From LAMSIN (Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur):


The goal of this associate team is to contribute to the analysis of inverse problems where the sought parameters lack regularity. A typical example is the geometric inverse problem, where the geometry to be recovered from given data represents the discontinuity set for some physical coefficients in a PDE model. This problem arises in a variety of applications such as geophysics (e.g. the parameter being the sound velocity), non-destructive testing (e.g. the parameter being the crack impedance, dielectric properties of deposits), medical imaging (e.g. the parameter being the conductivity). For this type of problem, a classical formulation of the inverse problem as an optimization problem would generally be confronted with the lack of differentiability of the state variable with respect to the discontinuity location. We explore two main strategies to address this issue. The first one is based on the design of a suitable misfit functional that would be differentiable even if the state variable is not. This is the case, for example, of the Kohn-Vogelius cost function for self-adjoint operators. The second strategy would be to develop optimization-free inversion procedures that avoid the derivative of the state variable. This is the case, for instance, of sampling methods that have been developed for cracks.



Scientific progress

T. A. Vu, M. Bonazzoli and H. Haddar have analyzed two variants of the so-called multi-step one-shot methods to solve general linear inverse problems for parameter identification. In these methods, we iterate simultaneously on the inverse problem parameter (by the gradient descent method), and on the state/adjoint variables (by a fixed-point method). We have established sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. We have performed several numerical experiments to illustrate the convergence of these methods in comparison with the gradient descent method. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm. Poster.

The latter point has connections with the work of R. Jerbi, S. Chaabane and H. Haddar on geometric inverse problems coupling the use of Kohn-Vogelius functional with a domain decomposition method (optimized Schwartz algorithm). We have analyzed the convergence of this method in the simplified case of cylindrically invariant geometries and proved the effectiveness of this coupling through numerical application to inverse conductivity problems. Poster.

In connection with the PhD work of N. Jenhani, we have revisited the differential sampling method introduced by Haddar-Nguyen (2017) for the identification of a periodic domain and some local perturbation. We have provided a theoretical justification of the method that avoids the assumption that the local perturbation is also periodic. The theoretical framework uses functional spaces with continuous dependence with respect to the Floquet-Bloch variable. The cornerstone of the analysis is the justification of the Generalized Linear Sampling Method in this setting for a single Floquet-Bloch mode. Poster.

In connection with the PhD work of A. Labidi, we have derived conditional stability estimates for inverse scattering problems related to the time-harmonic magnetic Schrödinger equation. We have proved logarithmic-type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near-field or far-field maps. The approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions. Poster.