Inverse problems for linear hyperbolic equations via a mixed formulation

Nicolae Cindea (1) , Arnaud Münch (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

We introduce in this document a direct method allowing to solve nu-merically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in Ω × (0, T) -Ω a bounded subset of R N -from a partial distributed observation. We employ a least-squares technic and minimize the L 2 -norm of the distance from the observa-tion 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 partic-ular 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 discussed several example for N = 1 and N = 2. The problem of the reconstruction of both the state and the source term is also addressed.

Domaines: Optimisation et contrôle [math.OC]

Nicolae Cindea, Arnaud Münch. Inverse problems for linear hyperbolic equations via a mixed formulation. 2015. ⟨hal-01104217v3⟩