# American Institute of Mathematical Sciences

April  2019, 13(2): 377-400. doi: 10.3934/ipi.2019019

## An inverse obstacle problem for the wave equation in a finite time domain

 1 Laboratoire POEMS, ENSTA ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France 2 Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, GF-31062 Toulouse Cedex 9, France

Received  June 2018 Revised  September 2018 Published  January 2019

We consider an inverse obstacle problem for the acoustic transient wave equation. More precisely, we wish to reconstruct an obstacle characterized by a Dirichlet boundary condition from lateral Cauchy data given on a subpart of the boundary of the domain and over a finite interval of time. We first give a proof of uniqueness for that problem and then propose an "exterior approach" based on a mixed formulation of quasi-reversibility and a level set method in order to actually solve the problem. Some 2D numerical experiments are provided to show that our approach is effective.

Citation: Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019
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Notations
Illustration of the non-monotonicity of the mapping $O \mapsto D(\Omega,\Gamma)$
Radial case. Discrepancy $|u_ \varepsilon -u|$ as a function of $|x|$, for $t = 2.5$, $t = 3$, $t = 3.5$, $t = 4$ and $t = 4.5$
Two discs. Left: function $u_ \varepsilon$. Right: function $|u_ \varepsilon -u|$
Validation of the level set method ($T = 25$)
Two discs and exact data. Top left: $T = 10$. Top right: $T = 15$. Bottom: $T = 25$
Two discs and noisy data. Top left: $\delta = 0$ (exact data). Top right: $\delta = 0.02$. Bottom: $\delta = 0.05$
Partial (exact) data and one disc. Left: obstacle located far away from $\partial G \setminus \overline{\Gamma}$. Right: obstacle located close to $\partial G \setminus \overline{\Gamma}$
Boomerang obstacle. Left: $\delta = 0$ (exact data). Right: $\delta = 0.02$
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