Reconstruction of the Interior Interface of a Two‐Layered Cavity by a Two‐Stage Method

ABSTRACT A two‐stage method is proposed to reconstruct the interior interface of a two‐layered cavity. In the first stage, the incident, scattered and transmitted fields are decomposed from the measurements of the total fields on two closed manifolds inside the inner layer and the transmitted field on one closed manifold inside the outer layer. This decomposition is based on the single‐layer potential representations of the three fields and the resulting linear integral system. Conditional stability of the decomposition is provided. In the second stage, motivated by the transmission boundary conditions on the interior interface, we formulate an optimization problem by using the Cauchy data of the incident, scattered and transmitted fields on admissible manifolds and employ the Levenberg–Marquardt (L–M) method to solve it. The convergence property of this method is also established. We finally present several numerical examples to demonstrate the viability and effectiveness of the proposed two‐stage method.