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

Lekbir Afraites

doi:10.3934/dcdss.2021069

Article Contents

2022, Volume 15, Issue 1: 23-40. Doi: 10.3934/dcdss.2021069

This issue Previous Article Shape optimization method for an inverse geometric source problem and stability at critical shape Next Article Extended Krylov subspace methods for solving Sylvester and Stein tensor equations

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

Lekbir Afraites^,

LMA, Faculty of Sciences and Technology, University of Sultan Moulay Slimane, Béni Mellal, Morroco

^* Corresponding author: Lekbir Afraites
^* Corresponding author: Lekbir Afraites

Received: October 2020

Revised: April 2021

Early access: June 2021

Published: January 2022

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Abstract

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.

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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

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

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Figure 3. Reconstruction of different shapes with medium configuration

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Figure 4. Reconstruction of more complex configurations

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Figure 5. Reconstruction of more complex shapes

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

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

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

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

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

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

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