Tikhonov regularization of large linear problems

BIT Numerical Mathematics

Volumn 43, Issue 2, 2003, Pages 263-283

  Calvetti, Daniela ^a   Reichel, Lothar ^b  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

^b KENT STATE UNIVERSITY   (United States)

Author keywords

Discrepancy principle; Gauss quadrature; Ill posed problem; Lanczos bidiagonalization; Regularization

Indexed keywords

EID: 4043103309     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1026083619097     Document Type: Article

References (22)

1. M. L. Baart, The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems, IMA J. Numer. Anal., 2 (1982), pp. 241-247.
2. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994.
3. Å. Björck, A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT, 18 (1988), pp. 659-670.
4. Å. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
5. D. Calvetti, G. H. Golub, and L. Reichel, Estimation of the L-curve via Lanczos bidiagonalization, BIT, 39 (1999), pp. 603-619.
6. D. Calvetti, P. C. Hansen, and L. Reichel, L-curve curvature bounds via Lanczos bidiagonalization, Elec. Trans. Numer. Anal., 14 (2002), pp. 20-35.
7. D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, Tikhonov regularization and the L-curve for large, discrete ill-posed problems, J. Comput. Appl. Math., 123 (2000), pp. 423-446.
8. D. Calvetti, L. Reichel, F. Sgallari, and G. Spaletta, A regularizing Lanczos iteration method for underdetermined linear systems, J. Comput. Appl. Math., 115 (2000), pp. 101-120.
9. D. Calvetti, L. Reichel, and Q. Zhang, Iterative solution methods for large linear discrete ill-posed problems, Applied and Computational Control, Signals and Circuits, 1 (1999), pp. 313-367.
10. L. Eldén, Algorithms for the regularization of ill-conditioned least squares problems, BIT, 17 (1977), pp. 134-145.
11. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
12. A. Frommer and P. Maass, Fast CG-based methods for Tikhonov-Phillips regularization, SIAM J. Sci. Comput., 20 (1999), pp. 1831-1850.
13. G. H. Golub and G. Meurant, Matrices, moments and quadrature, in Numerical Analysis 1993, eds. D. F. Griffiths and G. A. Watson, Longman, Essex, England, 1994, pp. 105-156.
14. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, 1996.
15. G. H. Golub and U. von Matt, Tikhonov regularization for large scale problems, in Workshop on Scientific Computing, eds. G. H. Golub, S. H. Lui, F. Luk, and R. Plemmons, Springer, New York, 1997, pp. 3-26.
16. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
17. P. C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algor., 6 (1994), pp. 1-35. Software is available in Netlib at http://www.netlib.org.
18. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
19. 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.
20. 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.
21. D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM, 9 (1962), pp. 84-97.
22. C. B. Shaw, Jr., Improvements of the resolution of an instrument by numerical solution of an integral equation, J. Math. Anal. Appl., 37 (1972), pp. 83-112.