Controllability of partial differential equations and applications / Contrôle des EDP et applications

Collection Controllability of partial differential equations and applications / Contrôle des EDP et applications

Organizer(s) Dermenjian, Yves ; Cristofol, Michel ; Gaitan, Patricia ; Le Rousseau, Jérôme ; Yamamoto, Masahiro

Date(s) 09/11/2015 - 13/11/2015

linked URL http://conferences.cirm-math.fr/1368.html

00:00:00 / 00:00:00

1 5

Inverse problems for linear PDEs using mixed formulations

By Arnaud Münch

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.

Information about the video

Date of recording 10/11/2015
Date of publication 15/12/2015
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.18892703
Cite this video Münch, Arnaud (10/11/2015). Inverse problems for linear PDEs using mixed formulations. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18892703
URL https://dx.doi.org/10.24350/CIRM.V.18892703

Domain(s)

Optimization and Control

Bibliography

Münch, A., & Souza, D. (2015). Inverse problems for linear parabolic equations using mixed formulations - Part 1 : Theoretical analysis. <arXiv:1508.07854> - http://arxiv.org/abs/1508.07854
Nicolae, C., & Münch, A. (2015). Inverse problems for linear hyperbolic equations using mixed formulations. <arXiv:1502.00114> - http://arxiv.org/abs/1502.00114
Nicolae, C., & Münch, A. (2015). Reconstruction of the solution and the source of hyperbolic equations from boundary measurements: mixed formulations. <arXiv:1505.02566> - http://arxiv.org/abs/1505.02566

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