Landweber Iterative Regularization Methods And Algorithms For Several Kinds Of Ill-posed Problems

In this thesis,we consider several kinds of ill-posed problems,i.e.,the unknown source identification problem for time-fractional diffusion equation,the inverse initial value problem for time-fractional diffusion equation,the unknown source identification problem for the modified Helmholtz equation in strip domain.Although these problems have been discussed before,but most of the numerical results are much affected by the unknown a priori information.In this paper,the Landweber iterative regularization method is used to solve these ill-posed problems,the corresponding a posteriori regularization methods are studied,and the reatively complete theoretical knowledge is established.In the second chapter of this paper,we consider identifying an unknown source problem for time-fractional diffusion equation with variable coefficients in a general bounded domain.This is an ill-posed problem.We use the Landweber iterative regularization method to recovery the ill-posedness of the problem.Moreover,the convergence estimates are given under an a priori and an a posteriori regularization choice rules,respectively.Finally,numerical examples show that the effectiveness of the regularization method.In Chapter 3,we discuss the initial value problem of the time-fractional diffusion equation in a general bounded domain.This is an ill-posed problem.The Landweber iterative regularization method is proposed to solve this problem and obtain the regularization solution.Under the a priori and the a posteriori regularization parameters choice rules,the H¨older type error estimates between the exact solution and the regularization solution are obtained.In Chapter 4,we study the unknown source identification problem for the modified Helmholtz equation in strip domain.We propose the Landweber iterative regularization method to solve this problem and obtain the regularization solutions.Under the a priori and a posteriori regularization parameters choice rules,we all obtain the H¨older type error estimates between the exact solution and the regularization solution.Several numerical examples are also provided to show that the Landweber iterative regularization method works well for solving this problem.Finally,the theoretical results and the numerical results in this paper can fully and effectively show that Landweber iterative regularization method can solve the ill-posed problems well.