Research On Some Iterative Algorithms For Nonlinear Ill-posed Problems

In recent years,driven by the development of many practical fields such as geophysics,life science and material science,the research on inverse problems theory and algorithm has attracted the attention of researchers at home and abroad.Inverse problems are ill-posed,so they need to be solved by using regularization methods.At present,iterative regularization methods have become the focus because of their easy implementation.In solving practical problems,in addition to the ill-posedness,the scale of the problem,the diversity of solutions and the complexity of noisy data should also be considered.Motivated by this,the dissertation is devoted to studying several classes of more effective iterative regularization methods.To solve large-scale nonlinear ill-posed problems in Hilbert spaces,the effective Kaczmarz-type iterative regularization methods are constructed.Focusing on the nonlinear ill-posed problems in Banach spaces,a fast algorithm is designed for dealing with different types of noisy data.The convergence and convergence rate of the inexact Newton-Landweber iteration are analyzed under the H ¨older stability.Focusing on large-scale nonlinear ill-posed problems in Hilbert spaces,based on the averaged Kaczmarz method,this dissertation introduces a non-smooth function and an appropriate step size to propose a fast averaged Kaczmarz method with convex penalty term.The method has a simple format and is easy to be realized numerically.Moreover,it is highly stable.Under the general assumptions and stopping rule of the Kaczmarz type method,the convergence and regularity of the proposed method are analyzed.Photoacoustic tomography and elliptic parameter identification are used to verify the capability of reconstructing non-smooth solutions.At the same time,the efficiency of the method is demonstrated by comparing with other Kaczmarz type methods.To solve nonlinear ill-posed problems in Banach spaces,the acceleration of homotopy perturbation iteration is studied.This dissertation introduces the extended form of Nesterov acceleration strategy into the homotopy perturbation iteration to propose an accelerated homotopy perturbation iteration with uniformly convex penalty term.Under the assumptions of nonlinear operators and the discrepancy principle,the convergence and regularity of the method are proved.In order to further optimize the acceleration effect,the discrete backtracking search algorithm is selected to calculate the combined parameter in the method.The Robin coefficient reconstruction and elliptic parameter identification problems are selected for numerical simulation to verify the acceleration effect of the proposed method and the ability to deal with kinds of noisy data.Considering to solve large-scale nonlinear ill-posed inverse problems in Hilbert spaces,based on the homotopy perturbation Kaczmarz iteration,this dissertation utilize the sequential subspace optimization strategy to construct the accelerated homotopy perturbation Kaczmarz iteration with multiple search directions,and the step size is calculated by projecting.Under the assumptions of Kaczmarz type iterative regularization assumptions,the convergence of the method in the case of exact data is analyzed,and the regularity of the method in the noisy data case is proven.The effectiveness and acceleration of the proposed method are verified by one and two dimensional parameter identification problems.Considering the inexact Newton-Landweber iteration that can solve nonlinear illposed problems.Existing literatures analyze the convergence and regularity analysis of the method by using nonlinear conditions.But the convergence rate of the method has not been discussed.Motivated by this,the dissertation use the H ¨older stability to discuss the convergence analysis of the inexact Newton-Landweber iteration.The convergence and convergence rates are studied in Hilbert spaces and Banach spaces,respectively.