Iterative Regularization Methods For Linear Discrete Ill-posed Problems

The study of ill posed problem or inverse problem has become a hot issue internationally at the end of the 20 th century,and it has become a research area that is widely concerned by modern mathematicians.With the development of production and science and technology and the urgent need for applications,the ill-posed problems have widely applications arising in image processing,automatic control,physical property detection and so on.Based on related research results at home and abroad,this paper studies the improved iterative regularization methods for linear discrete ill-posed problems.This paper mainly includes the following contents: firstly,based on the least squares problem and the linear complementary problem,the Fractional Tikhonov regularization method is combined with non-negative constraints to propose a modulus-based iterative methods for constrained Fractional Tikhonov regularization for solving discrete ill-posed problems.Secondly,the Fractional Tikhonov regularization of non-stationary iterations is studied.In combination with the Morozov discrepancy principle,a projected non-stationary iterated Fractional Tikhonov regularization algorithm for large-scale discrete ill-posed problems is proposed.All the proposed algorithms are programmed and the numerical examples are carried out for classic examples and image processing applications.Numerical examples illustrate the effectiveness of the proposed methods.