A hybrid method for two-dimensional crack reconstruction

Abstract

We present a new method for solving the time-harmonic inverse scattering problem for sound-soft or perfectly conducting cracks in two dimensions. Our approach extends a method that was recently suggested by one of us for inverse obstacle scattering. It can be viewed as a hybrid between a regularized Newton iterationmethod applied to a nonlinear operator equation involving the operator that, for a fixed incident wave, maps the crack onto the far-field pattern of the scattered wave and a decomposition method due to Kirsch and Kress. As an important feature, in contrast to the traditional Newton iterations for solving inverse scattering problems, our method does not require a forward solver for each iteration step. The theoretical background of the method is based on the minimization of a cost function containing an additional penalty term to deal with reconstructing the full crack. Numerical examples illustrate the feasibility of the method and its stability with respect to noisy data. We expect that the method can also be extended to sound-hard cracks.