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Inverse Problems and Imaging (IPI)
 

Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise

Pages: 271 - 290, Volume 2, Issue 2, May 2008      doi:10.3934/ipi.2008.2.271

 
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Thorsten Hohage - Institut für Numerische und Angewandte Mathematik, Lotzestr. 16-18 D-37083 Göttingen, Germany (email)
Mihaela Pricop - Institut für Numerische und Angewandte Mathematik, Lotzestr. 16-18 D-37083 Göttingen, Germany (email)

Abstract: We consider the inverse problem to identify coefficient functions in boundary value problems from noisy measurements of the solutions. Our estimators are defined as minimizers of a Tikhonov functional, which is the sum of a nonlinear data misfit term and a quadratic penalty term involving a Hilbert scale norm. In this abstract framework we derive estimates of the expected squared error under certain assumptions on the forward operator. These assumptions are shown to be satisfied for two classes of inverse elliptic boundary value problems. The theoretical results are confirmed by Monte Carlo simulations.

Keywords:  statistical inverse problem, nonlinear Tikhonov regularization, Hilbert scales, parameter identification problems, inverse boundary value problems.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: April 2007;      Revised: December 2007;      Available Online: April 2008.

 References