Section: New Results

Inverse problems for heterogeneous systems

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

• In [32], we deal with a problem of observability for waves propagating in two environments with different speeds of propagation. We give an explicit construction of the regions of observability in the two-dimensional case. This allows us to determine in which locations we have to make some measurements in order to obtain the solution within the domain.

• In [33], we deal with the observability of the 1-D wave equation. The semi discretization of the waves problem leads to some uniform observability problems. This is due to the bad approximation of the high frequencies of discrete solutions. Some remedies are known, which involve finite element methods. In this paper, we give three methods allowing to retrieve the uniform observability when the approximations are made with a Galerkin method.

• In [15], Ramdani et al. proposed an algorithm for estimating from partial measurements the population for a linear age-structured population diffusion model. In this work, the physical parameters of the model were assumed to be known. The authors investigate the inverse problem of simultaneously estimating the population and the spatial diffusion coefficient for an age-structured population model. The measurement used is the time evolution of the population on a subdomain in space and age. The proposed method is based on the generalization to the infinite dimensional setting of an adaptive observer originally proposed for finite dimensional systems.

• In [13], Munnier and Ramdani proposed an explicit reconstruction formula for a two-dimensional cavity inverse problem. The proposed method was limited to the case of a single cavity due to the use of conformal mappings. In [13], Munnier and Ramdani consider the case of a finite number of cavities and aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non iterative and uses two main ingredients. First, the authors show how to compute so-called generalized Pólia-Szegö tensors (GPST) of the cavities from the DtN of the cavities. Secondly, the authors shows that the obtained shape from GPST inverse problem can be transformed into a shape from moments problem, for some particular configurations. However, numerical results suggest that the reconstruction method is efficient for arbitrary geometries.

• In [2], we show that, generically, a (finite dimensional) sampled system is observable provided that the number of outputs is at least equal to the number of inputs plus 2. This work complements some previous works on the subject.

• In [18], we design a state observer for a coupled two dimensional partial differential equations (PDEs) system used to describe the heat transfer in a membrane distillation system for water desalination.

  In [23], we deal with uniqueness and stability issues for the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from some asymptotic knowledge of the boundary spectral data of the corresponding Schrödinger operator under Dirichlet boundary conditions.