Nonlinear Generalized Impedance Boundary Conditions in Inverse Scattering

Nonlinear impedance boundary conditions in acoustic scattering are used as a model for perfectly conductive objects covered with a thin layer of nonlinear medium. In this work, we consider a scattering problem modeled by the Helmholtz equation with a nonlinear generalized impedance boundary condition. For a certain class of nonlinearities, we prove the well-posedness of the nonlinear direct problem and of the associated linearized problem.
Furthermore, we show the differentiability of the solution to the scattering problem with respect to the boundary of the scatterer and prove the existence of a domain derivative that satisfies the Helmholtz equation together with a linear impedance boundary condition.
A characterization of the domain derivative enables iterative regularization schemes to solve the inverse obstacle scattering problem, which consists of reconstructing the obstacle from the far-field pattern of a scattered wave. Since each iteration step requires solving a nonlinear boundary value problem, we suggest an all-at-once method based on linearizing the scattering problem and applying a regularized Newton step for the reconstruction. ... mehr
We introduce an integral equation method for solving linearized scattering problems and use the Nyström method to compute approximate solutions of the boundary integral equations.
Finally, we supplement our theoretical results and the description of the reconstruction algorithm with several examples to illustrate the performance of the proposed scheme.