Regularization by truncated total least squares

SIAM Journal on Scientific Computing

Volumn 18, Issue 4, 1997, Pages 1223-1241

  Fierro, R D ^a   Golub, G H ^b   Hansen, P C ^c   O'Leary, D P ^d  

^a CALIFORNIA STATE UNIVERSITY   (United States)

^b Stanford University   (United States)

^c TECHNICAL UNIVERSITY OF DENMARK   (Denmark)

^d University of Maryland ^*  (United States)

Author keywords

Bidiagonalization; Discrete ill posed problems; Regularization; Total least squares

Indexed keywords

BIDIAGONALIZATION; COEFFICIENT MATRIX; DISCRETE ILL POSED PROBLEMS; REGULARIZATION; TRUNCATED TOTAL LEAST SQUARES;

ALGORITHMS; BOUNDARY VALUE PROBLEMS; COMPUTATIONAL METHODS; ITERATIVE METHODS; MATRIX ALGEBRA; PROBLEM SOLVING;

LEAST SQUARES APPROXIMATIONS;

EID: 0031173551     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827594263837     Document Type: Article

References (30)

1. J. R. BUNCH AND C. P. NIELSEN, Updating the singular value decomposition, Numer. Math., 31 (1978), pp. 111-129.
2. T. F. CHAN AND P. C. HANSEN, Some applications of the rank revealing QR factorization, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 727-741.
3. J. K. CULLUM AND R. WILLOUGHBY, Lanczos Algorithms for Large Symmetric Eigenvalue Computations; Vol. I, Theory, Birkhauser Boston, Cambridge, MA, 1985.
4. L. ELDÉN, A weighted pseudoinverse, generalized singular values, and constrained least squares problems, BIT, 22 (1982), pp. 487-502.
5. R. D. FIERRO AND J. R. BUNCH, Collinearity and total least squares, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1167-1181.
6. R. D. FIERRO AND J. R. BUNCH, Perturbation theory for orthogonal projection methods with application to least squares and total least squares, Linear Algebra Appl., 234 (1996), pp. 71-96.
7. F. R. GANTMACHER, The Theory of Matrices; Vol. II, Chelsea, New York, 1959.
8. G. H. GOLUB, M. T. HEATH, AND G. WAHBA, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), pp. 215-223.
9. G. H. GOLUB AND W. KAHAN, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2 (Series B) (1965), pp. 205-224.
10. G. H. GOLUB AND C. F. VAN LOAN, An analysis of the total least squares problem, SIAM J. Numer. Anal., 17 (1980), pp. 883-893.
11. G. GOLUB AND J. WELSCH, Calculation of Gauss quadrature rules, Math. Comp., 23 (1969), pp. 221-230.
12. C. W. GROETSCH, Inverse Problems in the Mathematical Sciences, Vieweg, Wiesbaden, 1993.
13. M. HANKE AND P. C. HANSEN, Regularization methods for large-scale problems, Surveys Math. Indust., 3 (1993), pp. 253-315.
14. P. C. HANSEN, Regularization, GSVD and truncated GSVD, BIT, 29 (1989), pp. 491-504.
15. P. C. HANSEN, Truncated SVD solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 503-518.
16. P. C. HANSEN, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), pp. 561-580.
17. P. C. HANSEN, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), pp. 1-35.
18. P. C. HANSEN AND D. P. O'LEARY, The. use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), pp. 1487-1503.
19. M. R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bureau Standards, 49 (1952), pp. 409-436.
20. D. P. O'LEARY AND J. A. SIMMONS, A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 474-489.
21. 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.
22. G. SZEGÖ, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.
23. G. W. STEWART, On the invariance of perturbed null vectors under column scaling, Numer. Math., 44 (1984), pp. 61-65.
24. S. VAN HUFFEL AND J. VANDEWALLE, Algebraic relationships between classical regression and total least-squares estimation, Linear Algebra Appl., 93 (1987), pp. 149-162.
25. S. VAN HUFFEL AND J. VANDEWALLE, Analysis and solution of the nongeneric total least squares problem, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 327-348.
26. S. VAN HUFFEL AND J. VANDEWALLE, The Total Least Squares Problem - Computational Aspects and Analysis, SIAM, Philadelphia, PA, 1991.
27. S. VAN HUFFEL, J. VANDEWALLE, AND A. HAEGEMANS, An efficient and reliable algorithm for computing the singular subspace of a matrix, associated with its smallest singular values, J. Comput. Appl. Math., 19 (1987), pp. 313-330.
28. S. VAN HUFFEL, and H. ZHA, An efficient total least squares algorithm based on a rank-revealing two-sided orthogonal decomposition, Numer. Algorithms, 4 (1993), pp. 101-133.
29. M. WEI, The analysis for the total least squares problem with more than one solution, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 746-763.
30. M. WEI, Algebraic relations between the total least squares and least squares problems with more than one solution, Numer. Math., 62 (1992), pp. 123-148.