Simultaneous reconstruction of the solution and the source of hyperbolic equations from boundary measurements: a robust numerical approach. (19th October 2016)

• Record Type: Journal Article
• Authors: Cîndea, Nicolae; Münch, Arnaud
• Is Part Of: Inverse problems. Volume 32:Number 11(2016:Nov.)
• Publication Date: 2016-10-19
• Subjects: wave equation -- inverse problems -- mixed formulation; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/11/115020
• Languages: English
• ISSNs: 0266-5611

Abstract

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.