The domain derivative in inverse obstacle scattering with nonlinear impedance boundary condition

In this paper an inverse obstacle scattering problem for the Helmholtz equation
with nonlinear impedance boundary condition is considered. For a certain class
of nonlinearities, well-posedness of the direct scattering problem is proven.
Furthermore, differentiability of solutions with respect to the boundary is
shown by the variational method. A characterization of the derivative allows
for iterative regularization schemes in solving the inverse problem, which
consists in reconstructing the scattering obstacle from the far field pattern of a
scattered wave. An all-at-once Newton-type regularization method is developed
to illustrate the use of the domain derivative by some numerical examples.