A numerical method for the shape reconstruction problem in acoustic scattering*

The inverse problem under consideration is to determine the shape of a two-dimensional obstacle from the knowledge of the far-field pattern of the acoustic scattered wave. A nonlinear optimization scheme for solving the ill-posed and nonlinear problem is developed which is shown to be applicable for all positive wave number of the incident plane wave. Specifically, we show that the optimization scheme can reproduce efficiently the shape of an unknown obstacle from the knowledge of only one Fourier coefficient of the far-field pattern. An adaptive iteration algorithm with two steps at each loop for numerically implementing the optimization scheme is presented and verified to be efficient and practical by some numerical examples.