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
References:
[1]

M. Bonnet, Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain, Comput. Methods Appl. Mech. Engrg., 195 (2006), 5239-5254. doi: 10.1016/j.cma.2005.10.026.

[2]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Problems and Imaging, 4 (2010), 351-377. doi: 10.3934/ipi.2010.4.351.

[3]

L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system, Inverse Problems and Imaging, 8 (2014), 23-51. doi: 10.3934/ipi.2014.8.23.

[4]

L. Bourgeois and J. Dardé, The "exterior approach" applied to the inverse obstacle problem for the heat equation, SIAM Journal on Numerical Analysis, 55 (2017), 1820-1842. doi: 10.1137/16M1093872.

[5]

L. Bourgeois and A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems, M2AN, 52 (2018), 123-145. doi: 10.1051/m2an/2018008.

[6]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104. doi: 10.1088/0266-5611/21/3/018.

[7]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.

[8]

Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001, 17pp. doi: 10.1088/0266-5611/26/8/085001.

[9]

J. Dardé, Quasi-reversibility and Level Set Methods Applied to Elliptic Inverse Problems, PhD Université Paris-Diderot - Paris Ⅶ, 2010.

[10]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Problems and Imaging, 10 (2016), 379-407. doi: 10.3934/ipi.2016005.

[11]

A. Doubova and E. Fernández-Cara, Some geometric inverse problems for the linear wave equation, Inverse Problems and Imaging, 9 (2015), 371-393. doi: 10.3934/ipi.2015.9.371.

[12]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[14]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, Springer, Paris, 2005. doi: 10.1007/3-540-37689-5.

[15]

V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002, 18pp. doi: 10.1088/0266-5611/25/12/123002.

[16]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[17]

R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.

[18]

C. D. Lines and S. N. Chandler-Wilde, A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180. doi: 10.1007/s00607-004-0109-8.

[19]

J.-L. Lions, E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 2, Dunod, Paris, 1968.

[20]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation De Systèmes Distribués, Tome 1, Masson, Paris, 1988.

[21]

E. Lunéville and N. Salles, eXtended Library of Finite Elements in C++, http://uma.ensta-paristech.fr/soft/XLiFE++.

[22]

L. Oksanen, Solving an inverse obstacle problem for the wave equation by using the boundary control method, Inverse Problems, 29 (2013), 035004, 12pp. doi: 10.1088/0266-5611/29/3/035004.

[23]

S. Y. Oudot, L. J. Guibas, J. Gao and Y. Wang, Geodesic delaunay triangulations in bounded planar domains, ACM Trans. Algorithms, 6 (2010), Art. 67, 47 pp. doi: 10.1145/1824777.1824787.

[24]

L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, Communications in Partial Differential Equations, 16 (1991), 789-800. doi: 10.1080/03605309108820778.

show all references

References:
[1]

M. Bonnet, Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain, Comput. Methods Appl. Mech. Engrg., 195 (2006), 5239-5254. doi: 10.1016/j.cma.2005.10.026.

[2]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Problems and Imaging, 4 (2010), 351-377. doi: 10.3934/ipi.2010.4.351.

[3]

L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system, Inverse Problems and Imaging, 8 (2014), 23-51. doi: 10.3934/ipi.2014.8.23.

[4]

L. Bourgeois and J. Dardé, The "exterior approach" applied to the inverse obstacle problem for the heat equation, SIAM Journal on Numerical Analysis, 55 (2017), 1820-1842. doi: 10.1137/16M1093872.

[5]

L. Bourgeois and A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems, M2AN, 52 (2018), 123-145. doi: 10.1051/m2an/2018008.

[6]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104. doi: 10.1088/0266-5611/21/3/018.

[7]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.

[8]

Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001, 17pp. doi: 10.1088/0266-5611/26/8/085001.

[9]

J. Dardé, Quasi-reversibility and Level Set Methods Applied to Elliptic Inverse Problems, PhD Université Paris-Diderot - Paris Ⅶ, 2010.

[10]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Problems and Imaging, 10 (2016), 379-407. doi: 10.3934/ipi.2016005.

[11]

A. Doubova and E. Fernández-Cara, Some geometric inverse problems for the linear wave equation, Inverse Problems and Imaging, 9 (2015), 371-393. doi: 10.3934/ipi.2015.9.371.

[12]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[14]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, Springer, Paris, 2005. doi: 10.1007/3-540-37689-5.

[15]

V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002, 18pp. doi: 10.1088/0266-5611/25/12/123002.

[16]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[17]

R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.

[18]

C. D. Lines and S. N. Chandler-Wilde, A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180. doi: 10.1007/s00607-004-0109-8.

[19]

J.-L. Lions, E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 2, Dunod, Paris, 1968.

[20]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation De Systèmes Distribués, Tome 1, Masson, Paris, 1988.

[21]

E. Lunéville and N. Salles, eXtended Library of Finite Elements in C++, http://uma.ensta-paristech.fr/soft/XLiFE++.

[22]

L. Oksanen, Solving an inverse obstacle problem for the wave equation by using the boundary control method, Inverse Problems, 29 (2013), 035004, 12pp. doi: 10.1088/0266-5611/29/3/035004.

[23]

S. Y. Oudot, L. J. Guibas, J. Gao and Y. Wang, Geodesic delaunay triangulations in bounded planar domains, ACM Trans. Algorithms, 6 (2010), Art. 67, 47 pp. doi: 10.1145/1824777.1824787.

[24]

L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, Communications in Partial Differential Equations, 16 (1991), 789-800. doi: 10.1080/03605309108820778.

Figure 1.  Notations
Figure 2.  Illustration of the non-monotonicity of the mapping $ O \mapsto D(\Omega,\Gamma) $
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 $
Figure 4.  Two discs. Left: function $ u_ \varepsilon $. Right: function $ |u_ \varepsilon -u| $
Figure 5.  Validation of the level set method ($ T = 25 $)
Figure 6.  Two discs and exact data. Top left: $ T = 10 $. Top right: $ T = 15 $. Bottom: $ T = 25 $
Figure 7.  Two discs and noisy data. Top left: $ \delta = 0 $ (exact data). Top right: $ \delta = 0.02 $. Bottom: $ \delta = 0.05 $
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} $
Figure 9.  Boomerang obstacle. Left: $ \delta = 0 $ (exact data). Right: $ \delta = 0.02 $
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