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Approximate solution of nonlinear inverse problems by fixed-point iteration

: Pereverzyev, S.; Pinnau, R.; Siedow, N.


Bonnet, M.:
ICIPE 2008, 6th International Conference on Inverse Problems in Engineering : 15-19 June 2008, Dourdan, Paris, France
Bristol: IOP Publishing, 2008 (Journal of physics. Conference series 135)
ISSN: 1742-6588
Art. 01208, 8 S.
International Conference on Inverse Problems in Engineering (ICIPE) <6, 2008, Dourdan>
Konferenzbeitrag, Zeitschriftenaufsatz
Fraunhofer ITWM ()

In this paper we propose a derivative-free iterative method for the approximate solution of a nonlinear inverse problem Fx = y. In this method the iterations are defined as Gx(sub k+1) = Gx(sub k) + (Sy - SFx(sub k)), where G is an easily invertible operator and S is an operator from a data space to a solution space. We give general suggestions for the choice of operators G and S and show a practically relevant example of an inverse problem where such a method is succesfully applied. We carry out analysis of the proposed method for linear inverse problems. Using the recently introduced balancing principle we construct a stopping rule. Under reasonable assumptions, we show that this stopping rule leads to the regularization algorithm. Numerical results for a test example show its satisfactory behavior.