The Regularization Theory For Ill-posed Problems And Application

Inverse problems and ill-posed problems are very hot in mathematics nowadays. The well-posed problems is that the solution of problems is existent, unique and stable, if there is one or more to be dissatisfied, it is ill-posed. However, the main difficulty about the solution of ill-posed problems lies in the instability of solution, which will produce a great error between the approximate solution and the correct solution where the observation data in sourcebook are small error (which in practice is inevitable), this is the nature of difficulties of ill-posed problems. A general way that we solute ill-posed problems is regularization method. 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 makes a conclusion that the Moore-Penrose generalized solution of linear compact operator equations is unstable. The article introduces general regularization theories about ill-posed problems and a variety of regularization method besides Tikhonov Regularization and Landweber Iterative Method, it also discusses the calculation of regularization resolution's error and the choice of regularization parameter.Landweber iterative methods for solving large-scale problems is very beneficial, and relatively stable. Currently, Landweber iterative method has been further development in the ill-posed nonlinear problems. However, Landweber iterative sequence convergence speed (especially when the real solution "smoothness" very poor) is very slow. This paper presents a new iteration format, this format can greatly accelerate the iterative convergence rate.This paper is returned back applying Landweber iterative method to numerical differential, numerical differential will be converted into a special category I Fredholm integral equation and given specific numerical experiments.Boundary function identification in linear differential equation is common and very important problem in mathematical physics, the difficulty of the problem is determining endpoint. Using Tikhonov regularization method, the identification of boundary function is written as the minimum problem of functional, the regularization parameters is chosen by Morzove principle, and the corresponding minimum is obtained based on evolutionary algorithm and numerical experiment is given.