BAUDOUIN, Lucie; DE BUHAN, Maya; ERVEDOZA, Sylvain; OSSES, Axel

doi:10.1137/20M1315798

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BAUDOUIN, Lucie
Équipe Méthodes et Algorithmes en Commande [LAAS-MAC]

DE BUHAN, Maya
Mathématiques Appliquées Paris 5 [MAP5 - UMR 8145]
Centre National de la Recherche Scientifique [CNRS]

ERVEDOZA, Sylvain
Institut de Mathématiques de Bordeaux [IMB]
Centre National de la Recherche Scientifique [CNRS]

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Article de revue

Ce document a été publié dans

SIAM Journal on Numerical Analysis. 2021-04, vol. 59, n° 2, p. 998–1039

Society for Industrial and Applied Mathematics

Résumé en anglais

We present a globally convergent numerical algorithm based on global Carleman estimates to reconstruct the speed of propagation of waves in a bounded domain with Dirichlet boundary conditions from a single measurement of the boundary flux of the solutions in a finite time interval. The global convergence of the proposed algorithm naturally arises from the proof of the Lipschitz stability of the corresponding inverse problem for both sufficiently large observation time and boundary using global Carleman inequalities. The speed of propagation is supposed to be independent of time but varying in space with a trace and normal derivative known at the boundary and belonging to a certain admissible set that limits the speed fluctuations with respect to a given exterior point x0. In order to recover the speed, we also require a single experiment with null initial velocity and initial deformation having some monotonicity properties in the direction of x − x0. We perform numerical simulations in the discrete setting in order to illustrate and to validate the feasibility of the algorithm in both one and two dimensions in space. As proved theoretically, we verify that the numerical reconstruction is achieved for any admissible initial guess, even in the presence of small random disturbances on the measurements.< Réduire

Mots clés en anglais

hyperbolic equation

inverse problem

reconstruction algorithm

Carleman estimates AMS subject classifications

URI

https://oskar-bordeaux.fr/handle/20.500.12278/192572

DOI

http://dx.doi.org/10.1137/20M1315798

Project ANR

Synthèse d'observateur pour des systèmes de dimension infinie - ANR-19-CE48-0004

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251