Regularization Bases On Spectral Theory And Moore-Penrose Generalized Inverse

In mathematical physics inverse problems' researching is very hot nowadays. However, the main difficulty about the solution of inverse problems lies in ill-posedness, which is the instability of approximate solution, that is, the solution of a equation (if existing) incontinuously rely on the data in right hand side. There will produce a great error between the approximate solution and the correct solution where the data in right hand side are errant. A general way that we solute ill-posed problems is regularization method. Therefore, how to build up effective regularization method and algorithm are very important parts of ill-pose problems researching in inverse problems field.Beginning with some cases, the article gives basic definitions of inverse problems and ill-posed problems. Then it discusses the Moore-Penrose generalized solution and the Moore-Penrose generalized inverse, and makes a conclusion that linear compact operator equations are ill-posed, that is, the Moore-Penrose generalized solution is unstable. In order to find a stable approximate solution of linear compact operator equation, the article introduces general theories about ill-posed problems, it bases on spectral theory of self-adjiont compact operators and the singular value decomposition for compact operators, avails singular system to give expression of the solution, and explains ill-posedness of compact operator equation roots in the property that the singular values trends to zero. Thereout, it is provided with theoretic support of building up regularization method by inducting regularization filter to weaken or filtrate the influence that the nature of the singular value being very close to zero has on the solution's stability. On the basis of that, the article gives a regularization filter and builds up a quite new method of regularization. It also discusses the calculation of regularization resolution's enor and the choice of regularization parameter, and proves that this method is the best to make the resolution have order optimality. Besides, the article presents two important regularization filter and also discusses corresponding Tikhonov Regularization and Landweber Iterative Method. These two methods avoid calculating singular system and are well applied in the fields of all kinds of inverse problems' researching. Note that regularized problems are usually defined in an infinite setting and have to be discTetized for an implementation and there have been many comparatively adult methods of calculating singular system. Therefore, there is no need to avoid the calculation of singular system in inverse problems' calculating of numeric value. Here TSVD Regularization Method is very simple and quite effective one. The article discusses the calculation of TSVD regularization resolution's error and the choice of TSVD regularization parameter in details. By prior and posterior choices of regularization parameter, it is proved that the error of regularization solution has order optimality, and through cases it is also explained that TSVD Regularization Method is a effective one to resolve ill-pose problems with characteristics of little amount of calculating and of easier regularization parameter confirming.