×

Subspace-restricted singular value decompositions for linear discrete ill-posed problems. (English) Zbl 1204.65038

The authors solve noisy underdetermined ill-posed linear systems, \(Ax=b+e\), of small dimensions by means of truncated singular value decomposition (SVD), truncated generalized SVD and truncated subspace-restricted SVD that allow the solution subspace or the range of \(A\) to contain specified subspaces. Theoretical comparisons and computed examples are presented.

MSC:

65F22 Ill-posedness and regularization problems in numerical linear algebra
65F20 Numerical solutions to overdetermined systems, pseudoinverses
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0865.65009
[2] Hansen, P. C., Truncated SVD solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Stat. Comput., 11, 503-518 (1990) · Zbl 0699.65029
[3] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer: Kluwer Dordrecht · Zbl 0859.65054
[4] Calvetti, D.; Lewis, B.; Reichel, L., GMRES, \(L\)-curves and discrete ill-posed problems, BIT, 42, 44-65 (2002) · Zbl 1003.65040
[5] Eldén, L., Partial least squares vs. Lanczos bidiagonalization I: analysis of a projection method for multiple regression, Comput. Statist. Data Anal., 46, 11-31 (2004) · Zbl 1429.62288
[6] Hansen, P. C.; Sekii, T.; Shibahashi, H., The modified truncated SVD method for regularization in general form, SIAM J. Sci. Stat. Comput., 13, 1142-1150 (1992) · Zbl 0760.65044
[7] 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
[8] Hansen, P. C., Regularization, GSVD and truncated GSVD, BIT, 29, 491-504 (1989) · Zbl 0682.65021
[9] Bultheel, A., Laurent Series and their Padé Approximations (1987), Birkhäuser: Birkhäuser Basel · Zbl 0624.30005
[10] Bultheel, A.; Van Barel, M., (Linear Algebra. Linear Algebra, Rational Approximation and Orthogonal Polynomials (1997), Elsevier: Elsevier Amsterdam) · Zbl 0890.65024
[11] Bultheel, A.; Van Barel, M.; Van gucht, P., Orthogonal bases in discrete least squares rational approximation, J. Comput. Appl. Math., 164-165, 175-194 (2004) · Zbl 1046.93010
[12] Bultheel, A.; Van Barel, M.; Rolain, Y.; Pintelon, R., Numerically robust transfer function modelling from noisy frequency domain data, IEEE Trans. Automat. Control, 50, 1835-1839 (2005) · Zbl 1365.93156
[13] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0729.15001
[14] Hansen, P. C., Rank Deficient and Discrete Ill-Posed Problems (1998), SIAM: SIAM Philadelphia
[15] Hansen, P. C., Regularization tools version 4.0 for MATLAB 7.3, Numer. Algorithms, 46, 189-194 (2007) · Zbl 1128.65029
[16] 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
[17] Hansen, P. C.; Nagy, J. G.; O’Leary, D. P., Deblurring Images: Matrices, Spectra, and Filtering (2006), SIAM: SIAM Philadelphia · Zbl 1112.68127
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.