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 ≈ 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 b1,..., bd are available, i.e., the T-TLS method is applied to the problem AX ≈ B, where B = [b1,..., bd] 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.
truncated total least squares multiple right-hand sides eigenvalues of rank-d update ill-posed problem regularization filter factors
15A18 65F20 65F22 65F30
This is a preview of subscription content, log in to check access.
G. H. Golub, C. F. Van Loan: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, 2013.Google Scholar
J. Hadamard: Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann, Paris, 1932.zbMATHGoogle Scholar
P. C. Hansen: 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.CrossRefGoogle Scholar
P. C. Hansen: Discrete Inverse Problems. Insight and Algorithms. Fundamentals of Algorithms 7, Society for Industrial and Applied Mathematics, Philadelphia, 2010.CrossRefGoogle Scholar
P. C. Hansen, J. G. Nagy, D. P. O’Leary: Deblurring Images. Matrices, Spectra, and Filtering. Fundamentals of Algorithms 3, Society for Industrial and Applied Mathematics, Philadelphia, 2006.CrossRefzbMATHGoogle Scholar
P. C. Hansen, V. Pereyra, G. Scherer: Least Squares Data Fitting with Applications, Johns Hopkins University Press, Baltimore. 2013.zbMATHGoogle Scholar
I. Hnětynková, M. Plešinger, D. M. Sima: Solvability of the core problem with multiple right-hand sides in the TLS sense. SIAM J. Matrix Anal. Appl. 37 (2016), 861–876.MathSciNetCrossRefzbMATHGoogle Scholar
I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, S. Van Huffel: The total least squares problem in AX B: a new classification with the relationship to the classical works. SIAM J. Matrix Anal. Appl. 32 (2011), 748–770.MathSciNetCrossRefzbMATHGoogle Scholar
I. Hnětynková, M. Plešinger, Z. Strakoš: The core problem within a linear approximation problem AX B with multiple right-hand sides. SIAM J. Matrix Anal. Appl. 34 (2013), 917–931.MathSciNetCrossRefzbMATHGoogle Scholar
I. Hnětynková, M. Plešinger, Z. Strakoš: Band generalization of the Golub-Kahan bidiagonalization, generalized Jacobi matrices, and the core problem. SIAM J. Matrix Anal. Appl. 36 (2015), 417–434.MathSciNetCrossRefzbMATHGoogle Scholar
F. Natterer: The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Chichester. 1986.Google Scholar
S. Van Huffel, J. Vandewalle: The Total Least Squares Problem: Computational Aspects and Analysis. Frontiers in Applied Mathematics 9, Society for Industrial and Applied Mathematics, Philadelphia, 1991.CrossRefzbMATHGoogle Scholar