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

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...

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Bibliographic Details
Published in:BIT (Nordisk Tidskrift for Informationsbehandling) Vol. 49; no. 4; pp. 669 - 696
Main Authors: HNETYNKOVA, Iveta, PLESINGER, Martin, STRAKOS, Zdenek
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.12.2009
Springer
Subjects:
Computational Mathematics and Numerical Analysis
Exact sciences and technology
Mathematics
Mathematics and Statistics
Numeric Computing
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Ill-posed problems
65F10
Noise revealing
15A23
65F22
Lanczos tridiagonalization
15A06
Golub-Kahan iterative bidiagonalization
15A18
Numerical linear algebra
Iterative method
Algorithm
Regularization method
Numerical analysis
ISA06
III-posed problems
Ill posed problem
Regularization
ISSN:0006-3835, 1572-9125
Online Access:Get full text
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Summary: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.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-009-0239-7