Regularized Fixed-Point Iterations for Nonlinear Inverse Problems
نویسندگان
چکیده
In this paper we introduce a derivative-free, iterative method for solving nonlinear illposed problems Fx = y, where instead of y noisy data yδ with ‖y − yδ‖ ≤ δ are given and F : D(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 Auk+1 = yδ − Guk. 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 behavior 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 δ, 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 behavior. The theoretical assumptions are illustrated by a computational experiment. AMS MSC: 65J15, 65J20, 80A23.
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