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

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

Laurent Bourgeois 1, , Dmitry Ponomarev 1, and  Jérémi Dardé 2,

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.

Keywords: Inverse obstacle problem, quasi-reversibility, level set method, unique continuation, wave equation, lateral Cauchy data.
Mathematics Subject Classification: Primary: 35R25, 35R30, 35R35; Secondary: 65M60.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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