Two-step Methods For Nonlinear Ill-posed Problems

Since the 1960s,the study of ill-posed problems has attracted much attention in many academic fields.Many problems in physics,geography,information science and other fields can result in ill-posed problems.However,direct methods can not be used to solve the ill-posed problems,otherwise,the error of the solution will be caused in the case of observation data error.One of the main ideas to solve ill-posed problems is regularization.This idea is to use a series of solutions of well-posed problems to approximate the solutions of the original ill-posed problems.Furthermore,the idea of numerical iteration and regularization has a good performance in solving ill-posed prob-lems,and it is also one of the basic means to solve ill-posed problems.In this paper,the main method we discussed is regularized two-step iterative method for solving ill-posed problems.For the nonlinear ill-posed equation F(x)=y~?,there are many classical iterative algorithms,such as Landweber method,Levenberg Mar-quardt and so on.In this paper,we first propose a two-step Levenberg-Marquardt method.Compared with the one-step Levenberg Marquardt method,the main advan-tage of this method is that it effectively reduces the computation of the derivation op-erator,and increases the selection of different regularization parameters and adjustable step factors,which makes the two-step Levenberg Marquardt method more flexible and practical.Secondly,this paper proposes a two-step regularized Gauss Newton method,which can reduce the amount of computation and improve the efficiency of computation.This paper is divided into five chapter.In Chapter 1 we introduce the development history and research status of ill-posed problems.Then in Chapter 2,some prelimi-nary knowledge is given including the basic knowledge of some analytical and linear algebra,which paves the way for the following theoretical chapters.In Chapter 3,we propose a two-step Levenberg-Marquardt method and analyze its convergence.The local convergence theorem,the error monotonicity theorem and the estimation of iter-ation steps are also given.In Chapter 4,another two-step regularized Gauss-Newton method is proposed and its error estimates are analyzed.In Chapter 5,two numeri-cal examples are given:parameter identification and convolution equation.By these numerical examples,the two-step Levenberg-Marquardt method and the two-step reg-ularized Gauss-Newton method are compared with some classical regularized iterative algorithms,so that we can obtain the conclusion that the two-step methods can reduce CPU time and the iterative errors.