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H 1 Inverse problems for linear PDEs using mixed formulations

Auteurs : Münch, Arnaud (Auteur de la Conférence)
CIRM (Editeur )

Résumé : We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic 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 elements discretization. We show the strong convergence of the approximation and then discussed several examples for $N = 1$ and $N = 2$. The reconstruction of both the state and the source term is also discussed, as well as the boundary case. The parabolic case - more delicate as it requires the use of appropriate weights - will be also addressed. Joint works with Nicolae Cîndea and Diego Araujo de Souza.

Codes MSC :
35L10 - General theory of second-order, hyperbolic equations
65M12 - Stability and convergence of numerical methods (IVP of PDE)
93B40 - Computational methods

 Informations sur la Vidéo Langue : Anglais Date de publication : 15/12/15 Date de captation : 10/11/15 Collection : Research talks ; Control Theory and Optimization ; Partial Differential Equations Format : MP4 Durée : 00:48:26 Domaine : PDE ; Control Theory & Optimization Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-11-10_Munch.mp4 Informations sur la rencontre Nom de la rencontre : Controllability of partial differential equations and applications / Contrôle des EDP et applicationsOrganisateurs de la rencontre : Dermenjian, Yves ; Cristofol, Michel ; Gaitan, Patricia ; Le Rousseau, Jérôme ; Yamamoto, MasahiroDates : 09/11/15 - 13/11/15 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1368.htmlCitation Data DOI : 10.24350/CIRM.V.18892703 Cite this video as: Münch, Arnaud (2015). Inverse problems for linear PDEs using mixed formulations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18892703 URI : http://dx.doi.org/10.24350/CIRM.V.18892703

Bibliographie

1. Münch, A., & Souza, D. (2015). Inverse problems for linear parabolic equations using mixed formulations - Part 1 : Theoretical analysis. - http://arxiv.org/abs/1508.07854

2. Nicolae, C., & Münch, A. (2015). Inverse problems for linear hyperbolic equations using mixed formulations. - http://arxiv.org/abs/1502.00114

3. Nicolae, C., & Münch, A. (2015). Reconstruction of the solution and the source of hyperbolic equations from boundary measurements: mixed formulations. - http://arxiv.org/abs/1505.02566

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