Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Figure(9) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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 $

  • [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.
  • 加载中



Article Metrics

HTML views(1242) PDF downloads(271) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint