On the numerical solution of the inverse obstacle scattering problem

The subject of this thesis is the numerical solution of the inverse obstacle scattering problem governed by the Helmholtz equation in two dimensions. The problem is to determine the shape of an obstacle from scattering measurements corresponding to time-harmonic plane incident waves. The inverse obstacle problem, as many other inverse problems, is nonlinear and ill-posed.; As it is well known, a nonlinear problem FG=u, can be effectively solved with an iterative method, such as Newton's method, provided that we start with a good initial guess. The highly nonlinear and oscillatory nature of the inverse scattering problem makes the initial guess extremely difficult to obtain.; In this work, we present a continuation method to reliably solve the inverse obstacle problem numerically without the need of a good initial guess.; The continuation method starts at the lowest wavenumber for which the scattering data is available, and at which the inverse problem is nearly linear. We first solve this nearly linear problem to obtain an approximate solution. Then, the continuation method recursively refines the approximate solution G obtained at wavenumber k by solving the linearized equation F'G dG=u-FG at wavenumber k+dk . This upward march in wavenumber ends at the highest wavenumber at which the scattering measurement is available.; Our numerical experiments show that the scheme is stable and convergent for obstacles difficult to reconstruct: shapes with complex features and concavities. Various numerical and technical issues related to the implementation of the continuation method, such as regularization of ill-possedness and efficient frequency stepping have been systematically investigated and resolved in this work.