The Study Of The Regularization Methods For Two Classes Of Ill-posed Problems

This thesis investigates the regularization methods for two classes of ill-posed problems. The first one is the problem of numerical analytic continuation in com-plex analysis, which is a linear and severely ill-posed problem. Over the years, advances in this research are not remarkable and the earlier results mostly focus on the conditional stability. However, it seems that there are few applications of modern theory of regularization methods which have been developed intensively in the last few decades. In this thesis, we consider the problem of numerical analytic continuation on a strip domain. Firstly, we transform it into a linear operator equation by using Fourier transform and analyze the ill-posedness. Moreover, the optimal error bound of regularization method is also given. Based on this, for stably computing it, we provide a modified Tikhonov regularization method and a mollification regularization method, and give error estimate under a-priori and a-posteriori parameter choice rules, respectively. Several numerical examples are provided, which show the two methods work effectively. The other kind is that some related problems in the general theory on a nonlinear ill-posed operator equa-tion are considered. Imposing some nonlinear conditions on the nonlinear operator, we provide a-priori and two a-posteriori parameter choice rules which are specific to Tikhonov regularization method under logarithmic-type source condition and give the convergence rates, respectively. In addition, an unknown boundary iden-tification problem, which has important application background, is also discussed. Numerical experiments support our theoretical results and show the effectiveness of the methods.