Section: New Results

Inverse problems for heterogeneous systems

Participants : David Dos Santos Ferreira, Alexandre Munnier, Karim Ramdani, Julie Valein, Jean-Claude Vivalda.

Many inverse problems (IP) appearing in fluid-structure interaction and wave propagation problems have been investigated in the team.

In [14], Munnier and Ramdani consider the 2D inverse problem of recovering the positions and the velocities of slowly moving small rigid disks in a bounded cavity filled with a perfect fluid. Using an integral formulation, they first derive an asymptotic expansion of the DtN map of the problem as the diameters of the disks tend to zero. Then, combining a suitable choice of exponential type data and the DORT method (French acronym for Diagonalization of the Time Reversal Operator), a reconstruction method for the unknown positions and velocities is proposed. Let us emphasize here that this reconstruction method uses in the context of fluid-structure interaction problems a method which is usually used for waves inverse scattering (the DORT method).

In [13], Munnier and Ramdani propose a new method to tackle a geometric inverse problem related to Calderón's inverse problem. More precisely, they propose an explicit reconstruction formula for the cavity inverse problem using conformal mapping. This formula is derived by combining two ingredients: a new factorization result of the DtN map and the so-called generalized Pólia-Szegö tensors of the cavity.

In [9], P. Caro (Department of Mathematics and Statistics, Helsinki, Finland), D. Dos Santos Ferreira and Alberto Ruiz (Instituto de Ciencias Matematicas, Madrid, Spain) obtained stability estimates for potentials in a Schrödinger equation in dimension higher than 3 from the associated Dirichlet-to-Neumann map with partial data. The estimates are of log-log type and represent a quantitative version of the uniqueness result of Kenig, Sjöstrand and Uhlmann. The proof is based on a reduction to a stability estimate on the attenuated geodesic ray transform on the hypersphere.

In [15], Ramdani, Tucsnak (Institut de Mathématiques de Bordeaux, France) and Valein tackle a state estimation problem for a system of infinite dimension arising in population dynamics (a linear model for age-structured populations with spatial diffusion). Assume the initial state to be unknown, the considered inverse problem is to estimate asymptotically on time the state of the system from a locally distributed observation in both age and space. This is done by designing a Luenberger observer for the system, taking advantage of the particular spectral structure of the problem (the system has a finite number of unstable eigenvalues).

In [2], Ammar (Faculté des Sciences de Sfax, Tunisia), Massaoud (Faculté des Sciences de Sfax, Tunisia) and Vivalda characterize the globally Lipschitz continuous systems defined on ${ℝ}^n$ whose observability is preserved under time sampling.