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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> contained in a larger bounded domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> via boundary measurements on <inline-formula><tex-math id="M3">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula>. In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [<xref ref-type="bibr" rid="b16">16</xref>]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion <inline-formula><tex-math id="M4">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>. Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain <inline-formula><tex-math id="M5">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>. 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.</p>