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Regularization of a backwards parabolic equation by fractional operators
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 |
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. |
| [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 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 5.
Validation of the level set method ($ 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} $
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