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Parameter determination for Tikhonov regularization problems in general form. (English) Zbl 1391.65100
Summary: Tikhonov regularization is one of the most popular methods for computing an approximate solution of linear discrete ill-posed problems with error-contaminated data. A regularization parameter \(\lambda > 0\) balances the influence of a fidelity term, which measures how well the data are approximated, and of a regularization term, which dampens the propagation of the data error into the computed approximate solution. The value of the regularization parameter is important for the quality of the computed solution: a too large value of \(\lambda > 0\) gives an over-smoothed solution that lacks details that the desired solution may have, while a too small value yields a computed solution that is unnecessarily, and possibly severely, contaminated by propagated error. When a fairly accurate estimate of the norm of the error in the data is known, a suitable value of \(\lambda\) often can be determined with the aid of the discrepancy principle. This paper is concerned with the situation when the discrepancy principle cannot be applied. It then can be quite difficult to determine a suitable value of \(\lambda\). We consider the situation when the Tikhonov regularization problem is in general form, i.e., when the regularization term is determined by a regularization matrix different from the identity, and describe an extension of the COSE method for determining the regularization parameter \(\lambda\) in this situation. This method has previously been discussed for Tikhonov regularization in standard form, i.e., for the situation when the regularization matrix is the identity. It is well known that Tikhonov regularization in general form, with a suitably chosen regularization matrix, can give a computed solution of higher quality than Tikhonov regularization in standard form.

65F22 Ill-posedness and regularization problems in numerical linear algebra
65F10 Iterative numerical methods for linear systems
65R30 Numerical methods for ill-posed problems for integral equations
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[1] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1996), Kluwer Dordrecht · Zbl 0859.65054
[2] Dykes, L.; Reichel, L., Simplified GSVD computations for the solution of linear discrete ill-posed problems, J. Comput. Appl. Math., 255, 15-27, (2013) · Zbl 1291.65117
[3] Reichel, L.; Yu, X., Matrix decompositions for Tikhonov regularization, Electron. Trans. Numer. Anal., 43, 223-243, (2015) · Zbl 1327.65076
[4] Calvetti, D.; Lewis, B.; Reichel, L., A hybrid GMRES and TV-norm based method for image restoration, (Luk, F. T., Advanced Signal Processing Algorithms, Architectures, and Implementations XII, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), vol. 4791, (2002), The International Society for Optical Engineering Bellingham, WA), 192-200
[5] Kilmer, M. E.; Hansen, P. C.; Español, M. I., A projection-based approach to general-form Tikhonov regularization, SIAM J. Sci. Comput., 29, 315-330, (2007) · Zbl 1140.65030
[6] Bouhamidi, A.; Jbilou, K., Sylvester Tikhonov-regularization methods in image restoration, J. Comput. Appl. Math., 206, 86-98, (2007) · Zbl 1131.65036
[7] Donatelli, M.; Neuman, A.; Reichel, L., Square regularization matrices for large linear discrete ill-posed problems, Numer. Linear Algebra Appl., 19, 896-913, (2012) · Zbl 1289.65096
[8] Huang, G.; Noschese, S.; Reichel, L., Regularization matrices determined by matrix nearness problems, Linear Algebra Appl., 502, 41-57, (2016) · Zbl 1338.65296
[9] Noschese, S.; Reichel, L., Inverse problems for regularization matrices, Numer. Algorithms, 60, 531-544, (2012) · Zbl 1254.65053
[10] Reichel, L.; Ye, Q., Simple square smoothing regularization operators, Electron. Trans. Numer. Anal., 33, 63-83, (2009) · Zbl 1171.65033
[11] Hansen, P. C., Rank-deficient and discrete ill-posed problems, (1998), SIAM Philadelphia
[12] Bauer, F.; Lukas, M. A., Comparing parameter choice methods for regularization of ill-posed problem, Math. Comput. Simulation, 81, 1795-1841, (2011) · Zbl 1220.65063
[13] Brezinski, C.; Rodriguez, G.; Seatzu, S., Error estimates for the regularization of least squares problems, Numer. Algorithms, 51, 61-76, (2009) · Zbl 1166.65331
[14] Calvetti, D.; Hansen, P. C.; Reichel, L., L-curve curvature bounds via Lanczos bidiagonalization, Electron. Trans. Numer. Anal., 14, 20-35, (2002) · Zbl 1029.65041
[15] Castellanos, J. L.; Gómez, S.; Guerra, V., The triangle method for finding the corner of the L-curve, Appl. Numer. Math., 43, 359-373, (2002) · Zbl 1014.65022
[16] 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
[17] Fenu, C.; Reichel, L.; Rodriguez, G.; Sadok, H., GCV for Tikhonov regularization by partial SVD, BIT, 57, 1019-1039, (2017) · Zbl 1386.65118
[18] Hansen, P. C.; Jensen, T. K.; Rodriguez, G., An adaptive pruning algorithm for the discrete L-curve criterion, J. Comput. Appl. Math., 198, 483-492, (2006) · Zbl 1101.65044
[19] Hochstenbach, M. E.; Reichel, L.; Rodriguez, G., Regularization parameter determination for discrete ill-posed problems, J. Comput. Appl. Math., 273, 132-149, (2015) · Zbl 1295.65046
[20] Kindermann, S., Convergence analysis of minimization-based noise-level-free parameter choice rules for ill-posed problems, Electron. Trans. Numer. Anal., 38, 233-257, (2011) · Zbl 1287.65043
[21] Kindermann, S., Discretization independent convergence rates for noise level-free parameter choice rules for the regularization of ill-conditioned problems, Electron. Trans. Numer. Anal., 40, 58-81, (2013) · Zbl 1288.65085
[22] Reichel, L.; Rodriguez, G., Old and new parameter choice rules for discrete ill-posed problems, Numer. Algorithms, 63, 65-87, (2013) · Zbl 1267.65045
[23] Reichel, L.; Rodriguez, G.; Seatzu, S., Error estimates for large-scale ill-posed problems, Numer. Algorithms, 51, 341-361, (2009) · Zbl 1166.65332
[24] Hansen, P. C., Regularization, GSVD and truncated GSVD, BIT, 29, 491-504, (1989) · Zbl 0682.65021
[25] Gazzola, S.; Novati, P.; Russo, M. R., On Krylov projection methods and Tikhonov regularization, Electron. Trans. Numer. Anal., 44, 83-123, (2015) · Zbl 1312.65065
[26] Hochstenbach, M. E.; Reichel, L., An iterative method for Tikhonov regularization with a general linear regularization operator, J. Integral Equations Appl., 22, 463-480, (2010)
[27] Lampe, J.; Reichel, L.; Voss, H., Large-scale Tikhonov regularization via reduction by orthogonal projection, Linear Algebra Appl., 436, 2845-2865, (2012) · Zbl 1241.65044
[28] Z. Bai, The CSD, GSVD, their applications and computation, IMA preprint 958, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, 1992.
[29] Golub, G. H.; Van Loan, C. F., Matrix computations, (2013), Johns Hopkins University Press Baltimore · Zbl 1268.65037
[30] Dykes, L.; Noschese, S.; Reichel, L., Rescaling the GSVD with application to ill-posed problems, Numer. Algorithms, 68, 531-545, (2015) · Zbl 1314.65058
[31] Daniel, J. W.; Gragg, W. B.; Kaufman, L.; Stewart, G. W., Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp., 30, 772-795, (1976) · Zbl 0345.65021
[32] Baglama, J.; Reichel, L., Decomposition methods for large linear discrete ill-posed problems, J. Comput. Appl. Math., 198, 332-342, (2007) · Zbl 1106.65035
[33] Morigi, S.; Reichel, L.; Sgallari, F., A truncated projected SVD method for linear discrete ill-posed problems, Numer. Algorithms, 43, 197-213, (2006) · Zbl 1114.65039
[34] Hansen, P. C., Regularization tools version 4.0 for MATLAB 7.3, Numer. Algorithms, 46, 189-194, (2007) · Zbl 1128.65029
[35] Reichel, L.; Sadok, H., A new L-curve for ill-posed problems, J. Comput. Appl. Math., 219, 493-508, (2008) · Zbl 1145.65035
[36] Regińska, T., A regularization parameter in discrete ill-posed problems, SIAM J. Sci. Comput., 17, 740-749, (1996) · Zbl 0865.65023
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