Numerical solution of an inverse obstacle scattering problem with near-field data

Consider the scattering of an arbitrary time-harmonic incident wave by a sound soft obstacle. In this paper, a novel method is presented for solving the inverse obstacle scattering problem of the two-dimensional Helmholtz equation, which is to reconstruct the obstacle surface by using the near-field data. The obstacle is assumed to be a small and smooth perturbation of a disc. The method uses the transformed field expansion to reduce the boundary value problem into a successive sequence of one-dimensional problems which are solved in closed forms. By dropping the higher order terms in the power series expansion and truncating the infinite linear system for the first order term, the inverse problem is linearized and an approximate but explicit formula is obtained between the Fourier coefficients of the solution and data. A nonlinear correction algorithm is introduced to improve the accuracy of the reconstructions for large deformations. Numerical examples show that the method is simple, efficient, and stable to reconstruct the obstacle with subwavelength resolution.