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

Laurent Bourgeois; Dmitry Ponomarev; Jérémi Dardé

doi:10.3934/ipi.2019019

Article Contents

2019, Volume 13, Issue 2: 377-400. Doi: 10.3934/ipi.2019019

This issue Previous Article Electrical networks with prescribed current and applications to random walks on graphs Next Article Regularization of a backwards parabolic equation by fractional operators

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: April 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 35R25, 35R30, 35R35; Secondary: 65M60.

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Figure 1. Notations

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Figure 2. Illustration of the non-monotonicity of the mapping $ O \mapsto D(\Omega,\Gamma) $

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Figure 3. 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 $

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Figure 4. Two discs. Left: function $ u_ \varepsilon $. Right: function $ |u_ \varepsilon -u| $

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Figure 5. Validation of the level set method ($ T = 25 $)

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Figure 6. Two discs and exact data. Top left: $ T = 10 $. Top right: $ T = 15 $. Bottom: $ T = 25 $

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Figure 7. Two discs and noisy data. Top left: $ \delta = 0 $ (exact data). Top right: $ \delta = 0.02 $. Bottom: $ \delta = 0.05 $

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Figure 8. 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} $

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Figure 9. Boomerang obstacle. Left: $ \delta = 0 $ (exact data). Right: $ \delta = 0.02 $

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