Regularization Methods For Some Ill-posed Problems

In recent years, more and more people are concerning about inverse problems, the main reason is the fast development of applied mathematics driven by practical problems in industrial production and engineering. In the present paper, we consider some inverse problems related to fractional diffusion equation and elliptic equation, besides, we also study the general theory for ill-posed problems under the framework of numerical pseudo-differential operator. In chapter 2, we will give some optimal analysis on an inverse problem for time fractional diffusion equation, and then we will use a particular iterative regularization method to obtain some optimal error es-timates, where the regularization parameters are, respectively, chosen by a-prior and a-posterior rule. Meanwhile, we also get the similar results for space fractional diffu-sion equation. In the third chapter, we will use an iterative regularization method to study numerical analytic continuation and Cauchy problem for Laplace equation. For these two problems, we will give some comparison between the proposed method and other regularization methods, the numerical results show that the iterative regulariza-tion method can be an alternative to the existing regularization methods. In the last chapter, we conclude that many well known regularization methods can be presented in an unified framework of numerical pseudo-differential operator, particularly, we will use Meyer wavelet regularization method to deal with some ill-posed problems in the general framework.