## Regularized Fixed-Point Iterations for Nonlinear Inverse Problems

• In this paper we introduce a derivative-free, iterative method for solving nonlinear ill-posed problems $$Fx=y$$, where instead of $$y$$ noisy data $$y_\delta$$ with $$|| y-y_\delta ||\leq \delta$$ are given and $$F:D(F)\subseteq X \rightarrow Y$$ 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 $$A u_{k+1}=y_\delta-Gu_k$$. 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 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 $$\delta$$, 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 behavior. The theoretical assumptions are illustrated by a computational experiment.

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Verfasserangaben: S.S. Pereverzyev, R. Pinnau, N. Siedow urn:nbn:de:hbz:386-kluedo-13860 Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (262) Preprint Englisch 2005 2005 Technische Universität Kaiserslautern 10.08.2005 derivative-free iterative method ; heat radiation; initial temperature ; nonlinear heat equation ; nonlinear inverse problem ; regularization Preprint Version von gleichnamigen Artikel, der in Inverse Problems, 22 (2006), pp. 1–22 erschienen ist Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 65-XX NUMERICAL ANALYSIS / 65Jxx Numerical analysis in abstract spaces / 65J15 Equations with nonlinear operators (do not use 65Hxx) 65-XX NUMERICAL ANALYSIS / 65Jxx Numerical analysis in abstract spaces / 65J20 Improperly posed problems; regularization 80-XX CLASSICAL THERMODYNAMICS, HEAT TRANSFER (For thermodynamics of solids, see 74A15) / 80Axx Thermodynamics and heat transfer / 80A23 Inverse problems Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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