TY - INPR
A1 - Pereverzyev, S.S.
A1 - Pinnau, R.
A1 - Siedow, N.
T1 - Regularized Fixed-Point Iterations for Nonlinear Inverse Problems
N2 - 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.
T3 - Berichte der Arbeitsgruppe Technomathematik (AGTM Report) - 262
KW - nonlinear inverse problem
KW - regularization
KW - derivative-free iterative method
KW - nonlinear heat equation
KW - initial temperature
KW - heat radiation
Y1 - 2005
UR - https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1653
UR - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:hbz:386-kluedo-13860
ER -