Stability of an inverse problem for the discrete wave equation and convergence results

Lucie Baudouin (1) , Sylvain Ervedoza (2) , Axel Osses (3)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 DIMEC - Departamento de Ingeniería Mecánica

Using uniform global Carleman estimates for discrete elliptic and semi-discrete hyperbolic equations, we study Lipschitz and logarithmic stability for the inverse problem of recovering a potential in a semi-discrete wave equation, discretized by finite differences in a 2-d uniform mesh, from boundary or internal measurements. The discrete stability results, when compared with their continuous counterparts, include new terms depending on the discretization parameter h. From these stability results, we design a numerical method to compute convergent approximations of the continuous potential.

Mots clés: wave equation; inverse problem; discrete Carleman estimates; stability estimates

Domaines: Equations aux dérivées partielles [math.AP]

Lucie Baudouin, Sylvain Ervedoza, Axel Osses. Stability of an inverse problem for the discrete wave equation and convergence results. Journal de Mathématiques Pures et Appliquées, 2015, 103 (6), pp.1475. ⟨10.1016/j.matpur.2014.11.006⟩. ⟨hal-00874565v2⟩