Inverse scattering for planar cracks via nonlinear integral equations

We present a Newton-type method for reconstructing planar sound-soft or perfectly conducting cracks from far-field measurements for one time-harmonic scattering with plane wave incidence. Our approach arises from a method suggested by Kress and Rundell (Inv. Probl. 2005; 21(4):1207-1223) for an inverse boundary value problem for the Laplace equation. It was extended to inverse scattering problems for sound-soft obstacles (Mathematical Methods in Scattering Theory and Biomedical Engineering. World Scientific: Singapore, 2006; 39-50) and for sound-hard cracks (Inv. Probl. 2006; 22(6)). In both cases it was shown that the method gives accurate reconstructions with reasonable stability against noisy data. The approach is based on a pair of nonlinear and ill-posed integral equations for the unknown boundary. The integral equations are solved by linearization, i.e. by regularized Newton iterations. Numerical reconstructions illustrate the feasibility of the method.