Inverse problems for linear hyperbolic equations using mixed formulations. (19th May 2015)

• Record Type: Journal Article
• Authors: Cîndea, Nicolae; Münch, Arnaud
• Is Part Of: Inverse problems. Volume 31:Number 7(2015:Jul.)
• Publication Date: 2015-05-19
• Subjects: inverse problems -- reconstruction of solution -- hyperbolic systems -- mixed formulations; 35L10 -- 65M12 -- 93B40; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/7/075001
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We introduce a direct method which allows the solving of numerically inverse problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the equation posed in — Ω being a bounded subset of —from a partial distributed 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 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 element discretization. We prove the strong convergence of the approximation and then discuss several examples for N = 1 and N = 2. The problem of the reconstruction of both the state and the source terms is also addressed.