Inverse problems for the wave equation using mixed formulations and space-time finite elements
Thursday, March 24, 2016 - 11:00am - 12:00pm
Nicolae Cindea (Blaise Pascal University)
The aim of this talk is to introduce a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the equation from a partial distributed observation. We employ a least-squares technique and minimize the norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discuss several examples. The problem of the reconstruction of both the state and the source term is also addressed.