Muriel Boulakia

In this work, we are interested by the identification in a wave equation of a space dependent source term multiplied by a known time and space dependent function, from internal velocity or field measurements. The first part of the work consists in proving stability inequalities associated with this inverse problem from adapted Carleman estimates. Then, we present a sequential reconstruction strategy which is proved to be equivalent to the minimization of a cost functional with Tikhonov regularization. Based on the obtained stability estimates, the reconstruction error is evaluated with respect to the noise intensity. Finally, the proposed method is illustrated with numerical simulations, both in the case of regular source terms and of piecewise constant source terms.

We revisit the direct method for the reconstruction of coefficients in second order elliptic partial differential equations. Two model problems are considered: first the reconstruction of the diffusion coefficient in a scalar elliptic problem and second the reconstruction of the shear modulus in the elastography problem. To highlight the versatility of the framework, different notions of stability are exploited in the two situations. In the scalar case, the system is interpreted as a hyperbolic transport equation and an inf-sup condition on the discrete level is leveraged for the analysis of the numerical method. We obtain error estimates on the reconstruction coefficient that are suboptimal by half an order, which is known to be sharp on general meshes. In the vector case, the minimization of the residual in dual norm and a stability result on the continuous problem lead to error estimates that are optimal compared to the approximation. For both problems, the theoretical results are illustrated by some numerical examples.

This paper studies the unique continuation problem for the Stokes equations given a database of population measurements. The problem is set up as a minimization problem under a PDE constraint and discretized using the finite element method. It is then regularized by the population data, by imposing that the solution lives near a finite-dimensional subspace generated by the database. This study examines how the inclusion of population data in the resolution improves both theoretical and numerical results. Using the proposed method, global error estimates for the velocity and pressure are obtained at an improved rate of convergence. The inclusion of the population data also has a positive impact on the numerical test cases, in both 2D and 3D, and especially when the measurements are scarce.