On the convergence of a new Newton-type method ininverse scattering

Abstract

We investigate the reconstruction of the domain D fromthe measured far field pattern for scattering of some incidentfield ui, e.g. for the inverse two- or three-dimensionalacoustic sound-soft or two-dimensional electromagneticperfect-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 orspectral cut-off. We prove superlinear local convergence of regularized Newton updates towards a regularized solution and the convergence of the regularizedsolution towards the true solution when the data error tends tozero, both under the condition that the true solution has ananalytic boundary and the normal derivative of the total fieldhas no zeros on the boundary. In addition, the strong relationof Newton's method to a version of the point-sourcemethod is shown when Newton updates are used to find theunknown domain from the reconstructed scattered field us.