Regularized fixed-point iterations for nonlinear inverse problems

In this paper, we introduce a derivative-free, iterative method for solving nonlinear ill-posed problems Fu = y, where instead of y, noisy data y(delta) with parallel to y - y(delta)parallel to <= delta are given and F : X -> Y is a nonlinear operator between Hilbert spaces X and Y. This method is defined by splitting the operator F into a linear part A and a nonlinear part G, such that F = A + G. Then iterations are organized as AU(k+1) = y(delta) - GU(k). In the context of ill-posed problems, we consider the situation when A does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators A and G, we study the behaviour of the iteration error. We obtain its stability with respect to the iteration number k as well as the optimal convergence rate with respect to the noise level delta, provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behaviour. The theoretical assumptions are illustrated by a computational experiment.