Simultaneous reconstruction of the solution and the source of hyperbolic equations from boundary measurements: a robust numerical approach. (19th October 2016) - Cîndea, Nicolae; Münch, Arnaud
Abstract: We introduce a direct method that makes it possible to solve numerically inverse type problems for linear hyperbolic equations posed in Ω × ( 0, T ) − Ω, a bounded subset of R N . We consider the simultaneous reconstruction of both the state and the source term from a partial boundary observation. We employ a least-squares technique and minimize the L 2 -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 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 element discretization. We prove the strong convergence of the approximation and then discuss several examples in the one- and two-dimensional cases.