| In this thesis, we have discussed how to solve the nonlinear ill-posed problems. The problems in many applied domains often can be formulated as a nonlinear inverse problem, for example, parameter identification problem, inverse scatting problem, inverse Sturm-Liouville problem and the first nonlinear Fredholm equation, etc.. 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. There are finite aspects to use for reference because only few theories of nonlinear inverse problems have been developed. The difficulties in the research of the nonlinear inverse problems are the nonlinearity, ill-posedness and infinity. Now the key for solving the problems is how to construct the regularization operator and how to choose the parameter to make the method into a regularization method.Several Newton-type regularization methods with double parameters are presented in the thesis. Firstly we presented Newton-implicit iterative method and the inner regularization parameter is inner iterative number which is determined by Hanke rule. At the same time, a Newton-type method with double regularization parameters is given. The double regularization parameters are determined by the modified Hanke's rule. And Newton-implicit iterative method can be regarded as a special case of this Newton-type method with double regularization parameters. The convergence and the stability of the two methods are proved. And the numerical examples show the effectiveness of the method with double parameters. However, the convergence rates of the regularization method with Hanke rule can not be proved now.Secondly, after we use the Tikhonov regularization to solve the linearized equation and use the idea of Samanskii to combine the one-step Newton iterate with the more-step simplified Newton iterate, we can derive the asymptotic simplified Newton method with double parameters. The ratio of the two parameters is determined by Bakushinskii rule. In the outer iteration, we firstly adoptαpriori rule determining the outer iteration number and analyze the convergence of the approximation solution and the optimal convergence rates of the approximation under the proper source condition.Thirdly, under the condition with noαpriori smoothness, in order to obtain the optimal convergence rates, it is not practical to use theαpriori rule. And it is necessary to use the a posteriori stopping rule which not only depends on the error boundδbut also on the perturbed data y~δ. In the following, we introduce the Kaltenbacher-typeαposteriori stopping rule and the Lepskij-type a posteriori stopping rule. In the Kaltenbacher-typeαposteriori stopping rule, we can only give the optimal convergence rate in v∈(0,1/2]. Under this stopping rule, in order to obtain the convergence rate in v > 1/2, the assumptions on the nonlinearity of the nonlinear operator has to be strengthened. However, the strengthened condition can not be verified in the actual problems. To conquer this difficulty, we introduce the Lepskij-typeαposteriori stopping rule and prove the optimal convergence rates in v∈(0, 1/2]∪N.In the end, it is worthwhile to mention that we present a new inner stopping rule in the last chapter because one of the difficulties to resolve the nonlinear ill-posed problem is how to choose the inner regularization parameter. And the new rule can simulate the error level of the righthand in the linearized equation nicely. Combine the new stopping rule and using the implicit iterative method in the inner iteration, we get a new method. We compare the new method with Tikhonov method and Bakushinskii method and the numerical examples show the superiority of the new method. Finally, we use implicit iterative method or Tikhonov method for solving the linearized equation and compare the new rule with Hanke rule and Bakushinskii rule, the superiority of the new rule is taken on.
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