# zbMATH — the first resource for mathematics

Generalized singular value decomposition with iterated Tikhonov regularization. (English) Zbl 07193275
Summary: Linear discrete ill-posed problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods try to reduce the sensitivity by replacing the given problem by a nearby one, whose solution is less affected by perturbations. This paper describes how generalized singular value decomposition can be combined with iterated Tikhonov regularization and illustrates that the method so obtained determines approximate solutions of higher quality than the more commonly used approach of pairing generalized singular value decomposition with (standard) Tikhonov regularization. The regularization parameter is determined with the aid of the discrepancy principle. This requires the application of a zero-finder. Several zero-finders are compared.

##### MSC:
 65 Numerical analysis 39 Difference and functional equations
##### Keywords:
ill-posed problem; iterated Tikhonov; GSVD; zero-finder
##### Software:
Regularization tools
Full Text:
##### References:
 [1] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer: Kluwer Dordrecht · Zbl 0859.65054 [2] Hansen, P. C., Rank-Deficient and Discrete Ill-Posed Problems (1998), SIAM: SIAM Philadelphia [3] Fenu, C.; Reichel, L.; Rodriguez, G., GCV for tikhonov regularization via global Golub-Kahan decomposition, Numer. Linear Algebra Appl., 23, 467-484 (2016) · Zbl 1374.65064 [4] Kindermann, S., Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems, Electron. Trans. Numer. Anal., 38, 233-257 (2011) · Zbl 1287.65043 [5] Reichel, L.; Rodriguez, G., Old and new parameter choice rules for discrete ill-posed problems, Numer. Algorithms, 63, 65-87 (2013) · Zbl 1267.65045 [6] Buccini, A., Regularizing preconditioners by non-stationary iterated Tikhonov with general penalty term, Appl. Numer. Math., 116, 64-81 (2017) · Zbl 1372.65161 [7] Buccini, A.; Donatelli, M.; Reichel, L., Iterated Tikhonov regularization with a general penalty term, Numer. Linear Algebra Appl., 24, Article e2089 pp. (2017), 12 pages · Zbl 1424.65045 [8] Huang, G.; Reichel, L.; Yin, F., Projected nonstationary iterated Tikhonov regularization, BIT, 56, 467-487 (2016) · Zbl 1341.65017 [9] Brill, M.; Schock, E., Iterative solution of ill-posed problems –a survey, (Model Optimization in Exploration Geophysics, Vol. 1 (1987)), 17-37 [10] Golub, G. H.; Van Loan, C. F., Matrix Computations (2013), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 1268.65037 [11] Bai, Z., The CSD, GSVD, their Applications and Computation (1992), Institute for Mathematics and Its Applications, University of Minnesota: Institute for Mathematics and Its Applications, University of Minnesota Minneapolis, IMA Preprint 958 [12] Bai, Z.; Demmel, J. W., Computing the generalized singular value decomposition, SIAM J. Sci. Comput., 14, 1464-1486 (1993) · Zbl 0789.65024 [13] Dykes, L.; Noschese, S.; Reichel, L., Rescaling the GSVD with application to ill-posed problems, Numer. Algorithms, 68, 531-545 (2015) · Zbl 1314.65058 [14] Dykes, L.; Reichel, L., Simplified GSVD computations for the solution of linear discrete ill-posed problems, J. Comput. Appl. Math., 225, 15-27 (2013) · Zbl 1291.65117 [15] Reichel, L.; Shyshkov, A., A new zero-finder for Tikhonov regularization, BIT, 48, 627-643 (2004) · Zbl 1161.65029 [16] Reinsch, C., Smoothing by spline functions. II, Numer. Math., 16, 451-454 (1971) · Zbl 1248.65020 [17] Hansen, P. C., Regularization tools: A MATLAB package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6, 1-35 (1994) · Zbl 0789.65029 [18] Baart, M. L., The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems, IMA J. Numer. Anal., 2, 241-247 (1982) · Zbl 0484.65021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.