Regularization Method Based On Singular Systems And The Variational Principle And Its Application

There are many inverse problems in the fields of science and engineering technology.Most inverse problems are ill-posed. Inverse and ill-posed problems must be solvedby regularization method. We chieffy consider the unstability of ill-posed problems.Effective regularization methods are constructed and choice of regularization parameterand realization of the algorithm are the main problems for inverse problems study.The main work of this thesis includes two parts. The first part analyzes the ill-posedness of the first kind operator equation. Basing on spectral analysis and accordingto the theory of singular system of compact operator, a new regularization filter is putforward. And then a new regularization method for solving the first kind of operatorequation in the presence of noisy date is constructed. By a priori and a posteriorichoice of the regularization parameter for new regularization method respectively, theoptimum asymptotic convergence order of the regularized solution is obtained. Andfinally, a numerical example is presented by solving the first kind of Fredholm integralequation, and we compare the numerical results between Li's paper and this thesis.The second part is about image restoration. Many scholars proposed the imagerestoration method which were based on variational method . The main points of thosemethods are to protect the edge. This thesis is based on variational theory by aids ofadaptive regularized parameter method , which uses adaptive regularization parameterfollowing the change of gradient. We solve the variational problem by a 5-point differencescheme of the partial differential equation, and we get the discrete solution of variationmodel.