Nonlinear Implicit Iterative Method And Regularization GMRES Method For Solving Ill-posed Problems

In this thesis, we have discussed how to solve the nonlinear ill-posed problems mainly. Presently, the theory of linear ill-posed problem has been relatively perfect and has favorable effect in the actual application. However, the theory and the practice of nonlinear ill-posed problem need to be perfected. Indeed, most models of mathematics and physics problems are nonlinear, for example, parameter identification problem, inverse scatting problem, inverse Sturm-Liouville problem and the first nonlinear Fredholm equation, etc.. Therefore, it is a very important and useful work to study the nonlinear ill-posed problems in theory and numerical algorithms.Many methods and techniques in linear ill-posed problems have been used in nonlinear ill-posed problems successfully. We extend the implicit iterative method in linear ill-posed operator equation to solve nonlinear ill-posed problems and present the nonlinear implicit iterative method. From the three aspects of theory analysis, numerical implementation and examples, we study the nonlinear implicit iterative method and get favorable effect too. The main contents of the dissertation are as following:Firstly, the nonlinear implicit iterative method is proposed. Since the Tikhonov functional is coercive, it has a minimizer and the minimizer is bounded. We prove that the error consequences of solutions of the nonlinear implicit iterative method are monotone decreasing and with this monotone get the convergence of the nonlinear implicit iterative method for exact and inexact equations.Secondly, the main part in implementation of the nonlinear implicit iterative method is how to minimize the Tikhonov functional in each iteration. When regularization parameter is fixed, how to minimize the Tikhonov functional is a well-posed optimization problem. In principle, all of nonlinear optimization methods can be implied to implement the nonlinear implicit iterative method. But, the convergence may not be hold since the local convergence of some methods and the non strictly convex of the Tikhonov functional. We apply the steepest descent method and the modified Gauss-Newton method to implement the nonlinear implicit iterative method and present two algorithms: IIGRA and IIMGN. We prove that every two minimizations can be connected spontaneously only requiring to add some simple restrictions on iterative initial value, and have the convergence without changing iterative initial value in interspace. Combined with nonlinear conjugate gradient method, we propose IINCG algorithm but do not analyse it in theory. Three numerical examples show the effectiveness of the three algorithms.Thirdly, we present a modification method of the nonlinear implicit iterative method—replacement functional method. One hand, two parameters (C_n and n) in this method are changed and it is some likely non-stationary implicit iterative method. On the other hand, contraction mapping is used to minimize Tikhonov replacement functional, so it has simple iterative format and converges fast.Finally, We turn our sight back to linear ill-posed problems. Regarded as a modification of GMRES method, a kind of double regularization GMRES(m) method is proposed to buildup the regularization properties of GMRES method. We have good effect for this method in numerical examples. It is like walking on two feet to destination more stably and fast with double regularization. We also present a hybrid regularized GMRES (m) with modified L-curve for image restoration. With examples in two-value image and gray image, numerical results illustrate good restoration effect.