Author: Otmar Scherzer
Abstract: Landweber iteration x_{k+1} = x_k - F'(x_k)^*(F(x_k)-y) for the solution of a nonlinear operator equation F(x_0)=y_0 can be viewed as a fixed point iteration with fixed point operator x-F'(x)^*(F(x)-y). Especially for nonlinear ill-posed problems, it seems impossible to verify that this fixed point operator is of contractive type, which is a typical assumption for proving (weak) convergence of fixed point iteration schemes. However, for specific examples of nonlinear ill-posed problems it is possible to verify conditions of quasi-contractive type. Weak convergence of Landweber iteration can be proven by application of general results for fixed iterations, based on quasi-contractive type conditions. In Hanke et al. a condition on the operator F has been investigated, which guarantees convergence of the Landweber's method. A geometrical interpretation of this condition is given and is compared with well-known conditions in the theory of fixed point iterations.