On a generalization of Regin ´ ska’s parameter choice rule and its numerical realization in large-scale multi-parameter Tikhonov regularization Fermín S. Viloche Bazán a,⇑,1 , Leonardo S. Borges b,2 , Juliano B. Francisco a,3 a Department of Mathematics, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil b Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, CP 6065, 13081-970 Campinas SP, Brazil article info Keywords: Parameter choice rules Multi-parameter Tikhonov regularization Large-scale discrete ill-posed problems abstract A crucial problem concerning Tikhonov regularization is the proper choice of the regular- ization parameter. This paper deals with a generalization of a parameter choice rule due to Regin ´ ska (1996) [31], analyzed and algorithmically realized through a fast fixed-point method in Bazán (2008) [3], which results in a fixed-point method for multi-parameter Tikhonov regularization called MFP. Like the single-parameter case, the algorithm does not require any information on the noise level. Further, combining projection over the Krylov subspace generated by the Golub–Kahan bidiagonalization (GKB) algorithm and the MFP method at each iteration, we derive a new algorithm for large-scale multi- parameter Tikhonov regularization problems. The performance of MFP when applied to well known discrete ill-posed problems is evaluated and compared with results obtained by the discrepancy principle. The results indicate that MFP is efficient and competitive. The efficiency of the new algorithm on a super-resolution problem is also illustrated. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Linear least squares problems of the form min kg  Af k 2 2 ; A 2 R mn ; m P n; g 2 R m ; f 2 R n ð1:1Þ with A large and ill-conditioned arise in a number of areas in science and engineering. They are commonly referred to as discrete ill-posed problems and arise, for example, when discretizing first kind integral equations with smooth kernel as in signal processing and image restoration, or when seeking to determine the internal structure of a system by external mea- surements, e.g., computerized tomography. In practical applications g represents data obtained experimentally and it is of the form g ¼ g exact þ e, where e represents noise, g exact denotes the unknown error-free data and Af exact ¼ g exact . Note that due to the noise and the ill-conditioning of A, the naive least squares solution of (1.1), f LS ¼ A y g (where A y denotes the Moore–Penrose pseudoinverse of A) is dominated by noise and thus some form of regularization is needed in order to obtain a useful approximation to f exact . The earliest and most known and well established regularization method is that due to Tik- honov [35] where f exact is approximated by regularized solutions defined as 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.08.054 ⇑ Corresponding author. E-mail addresses: fermin@mtm.ufsc.br (F.S. Viloche Bazán), lsbplsb@yahoo.com (L.S. Borges), juliano@mtm.ufsc.br (J.B. Francisco). 1 The work of this author is supported by CNPq, Brazil, Grant 308154/2008-8. 2 This research is supported by FAPESP, Brazil, Grant 2009/52193-1. 3 The work of this author is supported by CNPq, Brazil, Grant 479729/2011-5. Applied Mathematics and Computation 219 (2012) 2100–2113 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc