A second order Newton method for sound soft inverse obstacle scattering

Abstract

Summary: A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by Kress and Serranho [\textit{R. Kress}, Inverse Probl. 19, No.~6, S91--S104 (2003; Zbl 1052.65055); \textit{R. Kress} and \textit{P. Serranho} Inverse Probl. 21, No.~2, 773--784 (2005; Zbl 1070.35126); \textit{R. Kress} and \textit{P. Serranho} J. Comput. Appl. Math. 204, No.~2, 418--427 (2007; Zbl 1350.35139); \textit{P. Serranho}, Inverse Probl. 22, No.~2, 663--680 (2006; Zbl 1094.35145); \textit{P. Serranho}, Inverse Probl. Imaging 1, No.~4, 691--712 (2007; Zbl 1149.35082)] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by \textit{F. Hettlich} and \textit{W. Rundell} [SIAM J. Numer. Anal. 37, No.~2, 587--620 (2000; Zbl 0946.35115)]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data.