Study On Numerical Methods Of Discrete Ill-Posed Problem In Inverse Problem

Inverse problem is an interdisciplinary and frontier science problem. It has great significance not only in theory but also in applied field. Inverse problem's theory and computation are more difficult than direct problem's, due to its high nonlinear and ill-posed. How to get the numeric solution of these discrete ill-posed problems has been become a special course and mathematicans and scholars are studying in this field.According to some practical examples, this thesis gives the basic definition and uniform form of inverse problems and ill-posed problems, and then operator equation is got by discursion. In order to find a stable approximate solution of inverse problem, the general theory about regularization of ill-posed problems is introduced. Considering that the numerical computation of inverse problem must be discreted and be defined in the finite setting. The discrete ill-posed regularization method is discussed in this paper. By use of the singular values decomposition, the expression of solution is got. And the property of ill-posedness of operator equation roots that the singular values trend to zero is explained. Thereout, it is available that the regularization method is used to solve discrete ill-posed problems and choose priori regularization parameters. In the end, the first kide Fredholm integral equation and initial-value inverse problem of heat conduction equation are provided. The methods proposed in the paper are used to carry out numerical simulation. Both the singular values decomposition and Tikhonov regularization are analysed during the numerical computation .The numerical computed examples that show these methods are feasible and effective.