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A new coupled complex boundary method (CCBM) for an inverse obstacle problem

  • * Corresponding author: Lekbir Afraites

    * Corresponding author: Lekbir Afraites
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  • In the present work we introduce and study a new method for solving the inverse obstacle problem. It consists in identifying a perfectly conducting inclusion $ \omega $ contained in a larger bounded domain $ \Omega $ via boundary measurements on $ \partial \Omega $. In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [16]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion $ \omega $. Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain $ \omega $. We characterize the gradient of the cost functional in order to make a numerical resolution. We then investigate the stability of the optimization problem and explain why this inverse problem is severely ill-posed by proving compactness of the Hessian of cost functional at the critical shape. Finally, some numerical results are presented and compared with classical methods.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Reconstruction of circular shape and evolution of cost function and shape gradient with respect to iterations

    Figure 2.  Reconstruction of ellipse shape and evolution of cost function and shape gradient with respect to iterations

    Figure 3.  Reconstruction of different shapes with medium configuration

    Figure 4.  Reconstruction of more complex configurations

    Figure 5.  Reconstruction of more complex shapes

    Figure 6.  Reconstruction of simple shapes with noise 3$ \% $

    Figure 7.  Reconstruction of different configurations with noise 3$ \% $

    Figure 8.  Reconstruction of more complex shapes with noise 3$ \% $

    Figure 9.  Reconstruction for more complex shapes with noise 5$ \% $

    Figure 10.  Reconstruction of more complex shapes with noise 10$ \% $

    Figure 11.  The comparison between the evolution of the cost function and the gradient with respect the iteration number

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