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Keywords:
truncated total least squares; multiple right-hand sides; eigenvalues of rank-$d$ update; ill-posed problem; regularization; filter factors
truncated total least squares; multiple right-hand sides; eigenvalues of rank-$d$ update; ill-posed problem; regularization; filter factors
Summary:
The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems $Ax\approx b$ were analyzed by Fierro, Golub, Hansen, and O'Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of $A$ applied to $b$. This paper focuses on the situation when multiple observations $b_1,\ldots ,b_d$ are available, i.e., the T-TLS method is applied to the problem $AX\approx B$, where $B=[b_1,\ldots ,b_d]$ is a matrix. It is proved that the filtering representation of the T-TLS solution can be generalized to this case. The corresponding filter factors are explicitly derived.
References:
[1] Björck, Å.: Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia (1996). DOI 10.1137/1.9781611971484 | MR 1386889 | Zbl 0847.65023
[2] Bunch, J. R., Nielsen, C. P.: Updating the singular value decomposition. Numer. Math. 31 (1978), 111-129. DOI 10.1007/BF01397471 | MR 0509670 | Zbl 0421.65028
[3] Bunch, J. R., Nielsen, C. P., Sorensen, D. C.: Rank-one modification of the symmetric eigenproblem. Numer. Math. 31 (1978), 31-48. DOI 10.1007/BF01396012 | MR 0508586 | Zbl 0369.65007
[4] Fierro, R. D., Golub, G. H., Hansen, P. C., O'Leary, D. P.: Regularization by truncated total least squares. SIAM J. Sci. Comput. 18 (1997), 1223-1241. DOI 10.1137/S1064827594263837 | MR 1453566 | Zbl 0891.65040
[5] Golub, G. H.: Some modified matrix eigenvalue problems. SIAM Rev. 15 (1973), 318-334. DOI 10.1137/1015032 | MR 0329227 | Zbl 0254.65027
[6] Golub, G. H., Loan, C. F. Van: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17 (1980), 883-893. DOI 10.1137/0717073 | MR 0595451 | Zbl 0468.65011
[7] Golub, G. H., Loan, C. F. Van: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore (2013). MR 3024913 | Zbl 1268.65037
[8] Hadamard, J.: Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann, Paris (1932). Zbl 0006.20501
[9] Hansen, P. C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM Monographs on Mathematical Modeling and Computation 4, Society for Industrial and Applied Mathematics, Philadelphia (1998). DOI 10.1137/1.9780898719697 | MR 1486577 | Zbl 0890.65037
[10] Hansen, P. C.: Discrete Inverse Problems. Insight and Algorithms. Fundamentals of Algorithms 7, Society for Industrial and Applied Mathematics, Philadelphia (2010). DOI 10.1137/1.9780898718836 | MR 2584074 | Zbl 1197.65054
[11] Hansen, P. C., Nagy, J. G., O'Leary, D. P.: Deblurring Images. Matrices, Spectra, and Filtering. Fundamentals of Algorithms 3, Society for Industrial and Applied Mathematics, Philadelphia (2006). DOI 10.1137/1.9780898718874 | MR 2271138 | Zbl 1112.68127
[12] Hansen, P. C., Pereyra, V., Scherer, G.: Least Squares Data Fitting with Applications, Johns Hopkins University Press, Baltimore. (2013). MR 3012616 | Zbl 1270.65008
[13] Hnětynková, I., Plešinger, M., Sima, D. M.: Solvability of the core problem with multiple right-hand sides in the TLS sense. SIAM J. Matrix Anal. Appl. 37 (2016), 861-876. DOI 10.1137/15M1028339 | MR 3523076 | Zbl 1343.15002
[14] Hnětynková, I., Plešinger, M., Sima, D. M., Strakoš, Z., Huffel, S. Van: The total least squares problem in $AX\approx B$: a new classification with the relationship to the classical works. SIAM J. Matrix Anal. Appl. 32 (2011), 748-770. DOI 10.1137/100813348 | MR 2825323 | Zbl 1235.15016
[15] Hnětynková, I., Plešinger, M., Strakoš, Z.: The core problem within a linear approximation problem $AX\approx B$ with multiple right-hand sides. SIAM J. Matrix Anal. Appl. 34 (2013), 917-931. DOI 10.1137/120884237 | MR 3073354 | Zbl 1279.65041
[16] Hnětynková, I., Plešinger, M., Strakoš, Z.: Band generalization of the Golub-Kahan bidiagonalization, generalized Jacobi matrices, and the core problem. SIAM J. Matrix Anal. Appl. 36 (2015), 417-434. DOI 10.1137/140968914 | MR 3335497 | Zbl 1320.65057
[17] Natterer, F.: The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Chichester. (1986). DOI 10.1007/978-3-663-01409-6 | MR 0856916 | Zbl 0617.92001
[18] Huffel, S. Van, Vandewalle, J.: The Total Least Squares Problem: Computational Aspects and Analysis. Frontiers in Applied Mathematics 9, Society for Industrial and Applied Mathematics, Philadelphia (1991). DOI 10.1137/1.9781611971002 | MR 1118607 | Zbl 0789.62054
[19] Wang, X.-F.: Total least squares problem with the arbitrary unitarily invariant norms. Linear Multilinear Algebra 65 (2017), 438-456. DOI 10.1080/03081087.2016.1189493 | MR 3589611 | Zbl 06687266
[20] Wei, M.: Algebraic relations between the total least squares and least squares problems with more than one solution. Numer. Math. 62 (1992), 123-148. DOI 10.1007/BF01396223 | MR 1159048 | Zbl 0761.65030
[21] Wei, M.: The analysis for the total least squares problem with more than one solution. SIAM J. Matrix Anal. Appl. 13 (1992), 746-763. DOI 10.1137/0613047 | MR 1168020 | Zbl 0758.65039
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