Lekbir AfraitesUniversité Beni Mellal, Maroc.

Titre : New and non-conventional shape optimization methods for solving an inverse obstacle problem


Résumé :

In this work, we propose a new and non-conventional shape optimization approaches for the resolution of shape inverse problems inspired by non-destructive testing and evaluation. It consists in identifying a perfectly conducting inclusion ω contained in a larger bounded domain Ω via boundary measurements on ∂Ω. Our main objective is to improve the detection of the concave parts or regions of the unknown inclusion/obstacle/boundary through two different strategies and under shape optimization settings.

Firstly, we will introduce the new coupled complex boundary method (CCBM), originally proposed in [3]. 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 ω. Mathematically, thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain ω. 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 (see [1]).

Secondly, we will introduce the so-called alternating direction method of multipliers or ADMM method in shape optimization framework to solve our inverse problem using a single boundary measurement. We will illustrate the effectiveness of the proposed schemes by testing them to some shape detection problems with pronounced concavities and under noisy data. Finally, some numerical results are presented and compared with classical methods in two and three dimensions (see [2]).


[1] Lekbir Afraites. A new coupled complex boundary method (ccbm) for an inverse obstacle problem. Discrete & Continuous Dynamical Systems-Series S, 15(1), 2022.

[2]  Lekbir Afraites, Aissam Hadri, and Julius Fergy T Rabago. An admm numerical approach in shape optimization setting for geometric inverse problems. arXiv preprint arXiv:2301.10355, 2023.

[3]  Xiaoliang Cheng, Rongfang Gong, Weimin Han, and Xuan Zheng. A novel coupled complex boundary method for solving inverse source problems. Inverse Problems, 30(5):055002, 2014.