A Convergence Analysis Of The Landweber Iteration For Nonlinear Ill-posed Problems

When apply Landweber regularization to solve linear ill-posed problems: Landweber,Friedman,Bialy suggested rewriting the equation in the form and iterating this equation,i e., computingThis iteration scheme can be interpreted as the steepest descent algorithm applied to the quadratic function x→||Fx-y||2.Besides these,we have got the conclusion of the convergence and conver-gence rates of iterative solution xm,δto the exact solution x as below:(1) Assume and||z||≤E, choose then we have|(2)Assume and||z||≤E, choose then we have(3) When use Landweber regularization for linear ill-posed problems,if we have stronger restrict to x,the iterative sequence{xm,δ}converges to the exact solution with the speed of O(δ2r/2r+1)at most. When we apply the above methods to nonlinear ill-posed problems, be-cause of the ill-posedness of nonlinear problems, the solution of equation usu-ally doesn't depend on the data condition continuously or isn't unique or doesn't exist.At present the study of Landweber regularization for nonlinear ill-posed problems are all based on several assumptions on initial conditions or special Hilbert spaces to discuss the convergence and convergence rates. This paper summarizes the former study, and analyze, study and compare the unification of linear and nonlinear problems. On the base of it,we use a method with parameter to solve the problem.In order to overcome the first problems, we make several assumptions as below:(1)The Frechet derivative F'(·) of F is locally uniformly bounded;Then the Landweber iterative equation for nonlinear ill-posed problems is defined as below: for a is a parameter.This is different to [13].This condition is enough to ensure at least local convergence to a solution of F(x)=y inβp/2(x0).We have the convergence conclusion of Landweber regularization as below: TheoremAssume x, is a solution of the nonlinear ill-posed problems F(x)=y,the disturbed data yδsatisfies||yδ-y||≤δ.The iteration is stopped after k*= k*(δ) steps according to a generalized discrepancy principle,i.e. then according to the assumptions,we use to solve the problem, we have the iterated solution xδk*δconverges to x*,andSo we proved that the improved method is available. For perturbed data with noise levelδwe propose a stopping rule that also yields the convergence rate O(δ1/2), but with weaker conditions(η<1).This paper is organized as below:we introduce the ill-posed problems and Landweber regularization in chapter l;we result the convergence and conver-gence rates of Landweber regularization under the assumptions for nonlinear ill-posed problems in chapter 2;in chapter 3,we will prove the conclusions in chapter 2; in chapter 4 we discuss the application in numerical computation of Landweber regularization for nonlinear ill-posed problems in order to prove the availability of the method.