Simultaneous reconstruction of an obstacle and its wavenumber for the Helmholtz equation: a robust optimization approach

In this paper, we propose a method to recover simultaneously the shape and wavenumber of an object from noisy radar or sonar measurements. The physical problem is modeled in a bounded domain with the Helmholtz equation and appropriate transmission conditions. The approach relies on a Kohn–Vogelius formulation of the inverse problem and uses Nesterov’s accelerated gradient descent combined with shape optimization techniques. To address the robustness of the method with respect to measurement noise, we are looking to minimize a convex combination of the expected value and variance, which we achieve by using a Karhunen–Loéve expansion. The effectiveness of the approach is demonstrated through two-dimensional numerical experiments.