A Weighted GCV Method for Lanczos Hybrid Regularization

Published

January 15, 2008

Author(s)

Julianne Chung, James G. Nagy, Dianne M. O'Leary

Abstract

Lanczos-hybrid regularization methods have been proposed as effective approaches for solving large-scale ill-posed inverse problems. Lanczos methods restrict the solution to lie in a Krylov subspace, but they are hindered by semi-convergence behavior, in that the quality of the solution first increases and then decreases. Hybrid methods apply a standard regularization technique, such as Tikhonov regularization, to the projected problem at each iteration. Thus, regularization in hybrid methods is achieved both by Krylov filtering and by appropriate choice of a regularization parameter at each iteration. In this paper we describe a weighted generalized cross validation (W-GCV) method for choosing the parameter. Using this method we demonstrate that the semi-convergence behavior of the Lanczos method can be overcome, making the solution less sensitive to the number of iterations.

Citation

Electronic Transactions on Numerical Analysis

Volume

28

Pub Type

Journals

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Keywords

generalized cross validation, ill-posed problems, iterative methods, Lanczos bidiagonalization, LSQR, regularization, Tikhonov

Citation

Chung, J. , Nagy, J. and O'Leary, D. (2008), A Weighted GCV Method for Lanczos Hybrid Regularization, Electronic Transactions on Numerical Analysis, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=51142 (Accessed October 18, 2025)