Convergence Criteria of Iterative Methods Based on Landweber Iteration for Solving Nonlinear Problems

Landweber iteration xk+1 = xk − F′(xk)*(F(xk) − y) for the solution of a nonlinear operator equation F(x0) = y0 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 point iterations, based on quasi-contractive type conditions. In a recent paper by 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.