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BIT Numerical Mathematics

, Volume 49, Issue 4, pp 669–696 | Cite as

The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data

  • Iveta Hnětynková
  • Martin Plešinger
  • Zdeněk Strakoš
Article

Abstract

Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem.

In this paper we consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems.

Keywords

Ill-posed problems Golub-Kahan iterative bidiagonalization Lanczos tridiagonalization Noise revealing 

Mathematics Subject Classification (2000)

15A06 15A18 15A23 65F10 65F22 

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Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  • Iveta Hnětynková
    • 1
    • 2
  • Martin Plešinger
    • 1
    • 3
  • Zdeněk Strakoš
    • 1
    • 2
  1. 1.Institute of Computer ScienceAcademy of SciencesPrague 8Czech Republic
  2. 2.Faculty of Mathematics and PhysicsCharles University in PraguePrague 8Czech Republic
  3. 3.Faculty of MechatronicsTechnical University of LiberecLiberecCzech Republic

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