CONVERGENCE CRITERIA OF ITERATIVE METHODS BASED ON LANDWEBER ITERATION FOR SOLVING NONLINEAR PROBLEMS

Landweber iteration X(k+1) = X(k) - F'(X(k))*(F(X(k)) - y) for the solution of a nonlinear operator equation F(X(o)) = y(o) 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 er 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. (C) 1995 Academic Press, Inc.