A quasi-boundary method for solving an inverse diffraction problem

In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary $ z = 0 $, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.