On the convergence of a new Newton-type method in inverse scattering

We investigate the reconstruction of the domain D from the measured far field pattern for scattering of some incident field u(i), e.g. for the inverse two- or three-dimensional acoustic sound-soft or two-dimensional electromagnetic perfect-conductor obstacle scattering problems. Both a perfect Newton scheme, a partially regularized and a fully regularized Newton scheme with stopping rule are investigated. As regularization methods we combine quasi-solutions with either the point-source method, minimum norm solutions, Tikhonov regularization or spectral cut-off. We prove superlinear local convergence of regularized Newton updates towards a regularized solution and the convergence of the regularized solution towards the true solution when the data error tends to zero, both under the condition that the true solution has an analytic boundary and the normal derivative of the total field has no zeros on the boundary. In addition, the strong relation of Newton's method to a version of the point-source method is shown when Newton updates are used to find the unknown domain from the reconstructed scattered field u(s).