Regularization Methods For A Problem Of Analytic Continuation

The problem of analytic continuation is severely ill-posed. It is impossible tosolve using numerical methods, which are used to deal with direct problem. There-fore some efective regularization methods are necessary to solve them. In thisthesis, firstly we describe and analyze essential reasons of the problem in detail,discuss conditional stability, give and prove the convergence error estimate betweenthe exact solution and the generalized Tikhonov regularization solution. Then wereview some diferent regularization schemes from the angle of the generalized reg-ularization theory. Finally we also give some numerical experiments to show thatthe feasibility and efciency of the proposed regularization method.This paper is divided into four sections:in Section1, we briefly introduce important theories related to this issue, includ-ing ill-posed problems, inverse problems, the research of the analytic continuationproblem;in Section2, stability estimate is proved, a generalized Tikhonov regularizationis provided and the corresponding error estimate is obtained, this part has beenpublished in the Journal of Mathematics and Computers in Simulation;Section3is built on the basis of the previous part, moreover, we propose a newa posteriori parameter choice and obtain the corresponding error estimate;in Section4, we consider some other regularization methods to stabilize thisproblem.