Volume 44, pp. 83-123, 2015.

On Krylov projection methods and Tikhonov regularization

Silvia Gazzola, Paolo Novati, and Maria Rosaria Russo

Abstract

In the framework of large-scale linear discrete ill-posed problems, Krylov projection methods represent an essential tool since their development, which dates back to the early 1950's. In recent years, the use of these methods in a hybrid fashion or to solve Tikhonov regularized problems has received great attention especially for problems involving the restoration of digital images. In this paper we review the fundamental Krylov-Tikhonov techniques based on Lanczos bidiagonalization and the Arnoldi algorithms. Moreover, we study the use of the unsymmetric Lanczos process that, to the best of our knowledge, has just marginally been considered in this setting. Many numerical experiments and comparisons of different methods are presented.

Full Text (PDF) [1.5 MB]

Key words

discrete ill-posed problems, Krylov projection methods, Tikhonov regularization, Lanczos bidiagonalization, nonsymmetric Lanczos process, Arnoldi algorithm, discrepancy principle, generalized cross validation, L-curve criterion, Regińska criterion, image deblurring

AMS subject classifications

65F10, 65F22, 65R32.

Links to the cited ETNA articles

[17]Vol. 28 (2007-2008), pp. 149-167 Julianne Chung, James G. Nagy, and Dianne P. O'Leary: A weighted-GCV method for Lanczos-hybrid regularization
[26]Vol. 40 (2013), pp. 452-475 Silvia Gazzola and Paolo Novati: Multi-parameter Arnoldi-Tikhonov methods
[38]Vol. 31 (2008), pp. 204-220 Per Christian Hansen and Toke Koldborg Jensen: Noise propagation in regularizing iterations for image deblurring
[65]Vol. 33 (2008-2009), pp. 63-83 Lothar Reichel and Qiang Ye: Simple square smoothing regularization operators

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