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A linear regularization method for a nonlinear parameter identification problem

  • M. Thamban Nair EMAIL logo and Samprita Das Roy

Abstract

In order to obtain regularized approximations for the solution q of the parameter identification problem -.(qu)=f in Ω along with the Neumann boundary condition quν=g on Ω, which is an ill-posed problem, we consider its weak formulation as a linear operator equation with operator as a function of the data uW1,(Ω), and then apply the Tikhonov regularization and a finite-dimensional approximation procedure when the data is noisy. Here, Ω is a bounded domain in d with Lipschitz boundary, fL2(Ω) and gH-1/2(Ω). This approach is akin to the equation error method of Al-Jamal and Gockenback (2012) wherein error estimates are obtained in terms of a quotient norm, whereas our procedure facilitates to obtain error estimates in terms of the regularization parameters and data errors with respect to the norms of the spaces under consideration. In order to obtain error estimates when the noisy data belongs to L2(Ω) instead of W1,(Ω), we shall make use of a smoothing procedure using the Clement operator under additional assumptions of Ω and u.

Funding statement: The first version of this paper was presented to the International Work Shop on Recent Developments in Inverse Problems held at the Weierstrass Institute of Applied Analysis and Stochastics, Berlin, during September 17–18, 2015. Also, the first author acknowledges the DST, Government of India for the support for the work through the project grant No. SERB/F/1808/2014-15 dated 17.06.2014 (MAT/14-15/042/DSTX/MTHA).

Acknowledgements

The authors thank Neela Nataraj and S. Kesavan for fruitful discussions, and the referees for many useful suggestions on its earlier versions which greatly improved the content and presentation of this paper.

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Received: 2015-10-3
Revised: 2016-9-22
Accepted: 2016-9-28
Published Online: 2016-11-23
Published in Print: 2017-12-1

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