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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Author:S.S. Pereverzyev, R. Pinnau, N. Siedow
Serie (Series number):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (262)
Document Type:Preprint
Language of publication:English
Year of Completion:2005
Year of Publication:2005
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2005/08/10
Tag:derivative-free iterative method; heat radiation; initial temperature; nonlinear heat equation; nonlinear inverse problem; regularization
Source:Preprint Version von gleichnamigen Artikel, der in Inverse Problems, 22 (2006), pp. 1–22 erschienen ist
Faculties / Organisational entities:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):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
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011