The Study Of Frozen Landweber Iteration For Nonlinear Ill-Posed Problems

Hadmard introduced the concept of a well-posed problems, originating from the philosophy that the mathematical model of a physical problems has to have the properties of uniqueness, existence and stability of solution. If one of the properties fails to hold it, he calls the problems ill-posed. Many interesting and important inverse problems in science lead to ill-posed problems. Therefor, it is important to research on the ill-posed problems.Devoting to solve the ill-posed problems, regularization method is very useful. It include the well known Tikhonov regularization and Landweber iteration method. and comparing the two method, we note that Landweber'method is more stable than Tikhonov regularization, but it is considerably higher, especiallyδis small. So there are several ways of generalizing the accelerated Landweber iteration, whereas most of there mainly applied to linear ill-posed problems. It is just the beginning of applying to the nonlinear ill-posed problems.The Landweber iteration for ill-posed problems is considered in this paper. Motivated by the modified Newton iteration, this paper combines Samaskii technique to the Landweber iteration proposes a Frozen Landweber iteration, and a convergence analysis of it is also presented. Without destroying the stability, the Landweber iteration is accelerated through saving computation quantity. Absorbed by multilevel idea, the paper adopts multilevel method in order to choose the adaptive disperse parameters. At the last part of the paper, a numerical example about Hammerstein integral equation in nonlinear ill-posed problems is given to test the theoretical assertions. The result shows the advantage of the Frozen Landweber iteration and the multilevel Frozen Landweber iteration.