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Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems. (English) Zbl 1360.65115

65F22 Ill-posedness and regularization problems in numerical linear algebra
65F10 Iterative numerical methods for linear systems
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
Full Text: DOI arXiv
[1] J. Chung, G. Easley, and D. P. O’Leary, Windowed spectral regularization of inverse problems, SIAM J. Sci. Comput., 33 (2011), pp. 3175–3200, . · Zbl 1269.65041
[2] J. Chung, J. G. Nagy, and D. P. O’Leary, A weighted GCV method for Lanczos hybrid regularization, Electron. Trans. Numer. Anal., 28 (2008), pp. 149–167, . · Zbl 1171.65029
[3] J. M. Chung, M. E. Kilmer, and D. P. O’Leary, A framework for regularization via operator approximation, SIAM J. Sci. Comput., 37 (2015), pp. B332–B359, .
[4] D. C.-L. Fong and M. Saunders, LSMR: An iterative algorithm for sparse least-squares problems, SIAM J. Sci. Comput., 33 (2011), pp. 2950–2971, . · Zbl 1232.65052
[5] G. H. Golub, M. Heath, and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), pp. 215–223, . · Zbl 0461.62059
[6] G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed., Johns Hopkins Press, Baltimore, MD, 1996. · Zbl 0865.65009
[7] K. Hämäläinen, L. Harhanen, A. Kallonen, A. Kujanpää, E. Niemi, and S. Siltanen, Tomographic X-ray Data of a Walnut, preprint, , 2015.
[8] K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, Sparse tomography, SIAM J. Sci. Comput., 35 (2013), pp. B644–B665, .
[9] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surv. Math. Ind., 3 (1993), pp. 253–315, . · Zbl 0805.65058
[10] P. C. Hansen, REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), pp. 189–194, .
[11] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998, .
[12] P. C. Hansen and T. K. Jensen, Noise propagation in regularizing iterations for image deblurring, Electron. Trans. Numer. Anal., 31 (2008), pp. 204–220, . · Zbl 1171.65032
[13] P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006, . · Zbl 1112.68127
[14] I. Hnětynková, M. Plešinger, and Z. Strakoš, The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data, BIT Numer. Math., 49 (2009), pp. 669–696, .
[15] Y. Huang and Z. Jia, Some Results on the Regularization of LSQR for Large-Scale Discrete Ill-Posed Problems, preprint, , 2015. · Zbl 1453.65080
[16] M. E. Kilmer and D. P. O’Leary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1204–1221, . · Zbl 0983.65056
[17] V. A. Morozov, On the solution of functional equations by the method of regularization, Sov. Math. Dokl., 7 (1966), pp. 414–417, . · Zbl 0187.12203
[18] J. G. Nagy, K. Palmer, and L. Perrone, Iterative methods for image deblurring: A Matlab object-oriented approach, Numer. Algorithms, 36 (2004), pp. 73–93, . · Zbl 1048.65039
[19] R. Neelamani, H. Choi, and R. Baraniuk, ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems, IEEE Trans. Signal Process., 52 (2004), pp. 418–433, . · Zbl 1369.94238
[20] C. C. Paige and M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981), pp. 398–405, . · Zbl 0471.65018
[21] C. C. Paige and M. A. Saunders, Algorithm 583: LSQR: Sparse linear equations and least squares problems, ACM Trans. Math. Software, 8 (1982), pp. 195–209, .
[22] C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), pp. 43–71, . · Zbl 0478.65016
[23] V. Paoletti, P. C. Hansen, M. F. Hansen, and M. Fedi, A computationally efficient tool for assessing the depth resolution in large-scale potential-field inversion, Geophysics, 79 (2014), pp. A33–A38, .
[24] O. Portniaguine and M. S. Zhdanov, Focusing geophysical inversion images, Geophysics, 64 (1999), pp. 874–887, .
[25] L. Reichel, F. Sgallari, and Q. Ye, Tikhonov regularization based on generalized Krylov subspace methods, Appl. Numer. Math., 62 (2012), pp. 1215–1228, . · Zbl 1246.65068
[26] R. A. Renaut, I. Hnětynková, and J. Mead, Regularization parameter estimation for large-scale Tikhonov regularization using a priori information, Comput. Stat. Data Anal., 54 (2010), pp. 3430–3445, . · Zbl 1284.62156
[27] S. Vatankhah, V. E. Ardestani, and R. A. Renaut, Automatic estimation of the regularization parameter in 2D focusing gravity inversion: Application of the method to the Safo manganese mine in the northwest of Iran, J. Geophy. Eng., 11 (2014), 045001, .
[28] S. Vatankhah, V. E. Ardestani, and R. A. Renaut, Application of the \(χ^2\) principle and unbiased predictive risk estimator for determining the regularization parameter in 3-D focusing gravity inversion, Geophys. J. Int., 200 (2015), pp. 265–277, .
[29] S. Vatankhah, R. A. Renaut, and V. E. Ardestani, Regularization parameter estimation for underdetermined problems by the \(χ^2\) principle with application to 2D focusing gravity inversion, Inverse Problems, 30 (2014), 085002, . · Zbl 1300.65019
[30] C. Vogel, Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002, . · Zbl 1008.65103
[31] Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), pp. 600–612, .
[32] B. Wohlberg and P. Rodríguez, An iteratively reweighted norm algorithm for minimization of total variation functionals, IEEE Signal Process. Lett., 14 (2007), pp. 948–951, .
[33] M. S. Zhdanov, Geophysical Inverse Theory and Regularization Problems, Methods in Geochemistry and Geophysics 36, Elsevier, Amsterdam, 2002.
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