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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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1569-3945
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Volume 19, Issue 1 (Jan 2011)

Issues

Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess

Michael V. Klibanov / Anatoly B. Bakushinsky
  • Institute for System Analysis of The Russian Academy of Science, Prospect 60 letya Oktyabrya 9, 117312, Moscow, Russia.
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/ Larisa Beilina
  • Department of Mathematical Sciences, Chalmers University and Gothenburg University, SE-42196, Gothenburg, Sweden.
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Published Online: 2011-05-02 | DOI: https://doi.org/10.1515/jiip.2011.024

Abstract

Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented.

Keywords.: Uniqueness theorem; Tikhonov functional; a single value of the level of error

About the article

Received: 2010-09-28

Published Online: 2011-05-02

Published in Print: 2011-05-01


Citation Information: Journal of Inverse and Ill-posed Problems, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip.2011.024.

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