The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data

BIT Numerical Mathematics

Volumn 49, Issue 4, 2009, Pages 669-696

  Hnětynková, Iveta ^a,b   Plešinger, Martin ^a,c   Strakoš, Zdeněk ^a,b  

^a INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

^b CHARLES UNIVERSITY   (Czech Republic)

^c TECHNICAL UNIVERSITY OF LIBEREC   (Czech Republic)

Author keywords

Golub Kahan iterative bidiagonalization; Ill posed problems; Lanczos tridiagonalization; Noise revealing

Indexed keywords

EID: 77949654686     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-009-0239-7     Document Type: Article

References (52)

1. Björck, Å.: A bidiagonalization algorithm for solving large sparse ill-posed systems of linear equations. BIT 28, 659-670 (1988).
2. Björck, Å., Grimme, E., van Dooren, P.: An implicit shift bidiagonalization algorithm for ill-posed systems. BIT 34, 510-534 (1994).
3. Barlow, J. L., Bosner, N., Drmač, Z.: A new stable bidiagonal reduction algorithm. Linear Algebra Appl. 397, 35-84 (2005).
4. Calvetti, D., Golub, G. H., Reichel, L.: Estimation of the L-curve via Lanczos bidiagonalization. BIT 39, 603-619 (1999).
5. Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the L-curve for large discrete ill-posed problems. J. Comput. Appl. Math. 123, 423-446 (2000).
6. Calvetti, D., Reichel, L., Shuibi, A.: L-curve and curvature bounds for Tikhonov regularization. Numer. Algorithms 35, 301-314 (2004).
7. Chung, J., Nagy, J. G., O'Leary, D. P.: A weighted GCV method for Lanczos hybrid regularization. Electron. Trans. Numer. Anal. 28, 149-167 (2008).
8. Cooley, J. W., Tukey, J. W.: An algorithm for the machine computation of the complex Fourier series. Math. Comput. 19, 297-301 (1965).
9. Duhamel, P., Vetterli, M.: Fast Fourier transforms: A tutorial review and a state of the art. Signal Process. 19, 259-299 (1990).
10. Fierro, R. D., Golub, G. H., Hansen, P. C., O'Leary, D. P.: Regularization by truncated total least squares. SIAM J. Sci. Statist. Comput. 18, 1225-1241 (1997).
11. Fischer, B.: Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley-Teubner Series Advances in Numerical Mathematics. Wiley, New York (1996).
12. Fischer, B., Freund, R. W.: On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices. SIAM J. Sci. Comput. 15, 408-426 (1994).
13. Gautschi, W.: Gauss-Christoffel Quadrature Formulae. In: E. B. Christoffel, Aachen/Monschau, 1979, pp. 72-147. Birkhäuser, Basel (1981).
14. Golub, G. H., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. SIAM J. Numer. Anal. Ser. B 2, 205-224 (1965).
15. Golub, G. H., Von Matt, U.: Generalized cross-validation for large scale problems. J. Comput. Graph. Stat. 6, 1-34 (1997).
16. Hanke, M.: On Lanczos based methods for regularization of discrete ill-posed problems. BIT 41, 1008-1018 (2001).
17. Hansen, P. C.: Regularization Tools-version 3. 2 for MATLAB 6. 0, a package for analysis and solution of discrete ill-posed problems, http://www2. imm. dtu. dk/~pch/Regutools/index. html.
18. Hansen, P. C.: Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion. SIAM, Philadelphia (1998).
19. Hansen, P. C., Jensen, T. K.: Noise propagation in regularizing iterations for image deblurring. ETNA 31, 204-220 (2008).
20. Hansen, P. C., Kilmer, M. E., Kjeldsen, R.: Exploiting residual information in the parameter choice for discrete ill-posed problems. BIT 46, 41-59 (2006).
21. Hansen, P. C., Sørensen, H. O., Sükösd, Z., Poulsen, H. F.: Reconstruction of single-grain orientation distribution functions for crystalline materials. SIAM J. Image Sci. 2(2), 593-613 (2009).
22. Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409-435 (1952).
23. Hnětynková, I., Strakoš, Z.: Lanczos tridiagonalization and core problems. Linear Algebra Appl. 421, 243-251 (2007).
24. Hnětynková, I., Plešinger, M., Strakoš, Z.: Lanczos tridiagonalization, Golub-Kahan bidiagonalization and core problem. PAMM 6, 717-718 (2006).
25. Jensen, T. K., Hansen, P. C.: Iterative regularization with minimum-residual methods. BIT 47, 103-120 (2007).
26. Karlin, S., Shapley, L. S.: Geometry of Moment Spaces. American Mathematical Society, Providence (1953).
27. Kilmer, M. E., O'Leary, D. P.: Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J. Matrix Anal. Appl. 22, 1204-1221 (2001).
28. Kilmer, M. E., Hansen, P. C., Español, M. I.: A projection-based approach to general form Tikhonov regularization. SIAM J. Sci. Comput. 29, 315-330 (2006).
29. Lanczos, C.: Linear Differential Operators. van Nostrand, London (1961).
30. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. 45, 255-282 (1950).
31. Meurant, G., Strakoš, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numer. 15, 471-542 (2006).
32. Morozov, V. A.: On the solution of functional equations by the method of regularization (in Russian). Sov. Math. Dokl. 7, 414-417 (1966).
33. Morozov, V. A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984).
34. Nguyen, N., Milanfar, P., Golub, G. H.: Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement. IEEE Trans. Image Proces. 10, 1299-1308 (2001).
35. O'Leary, D. P.: Near-optimal parameters for Tikhonov and other regularization methods. SIAM J. Sci. Comput. 23, 1161-1171 (2001).
36. O'Leary, D. P., Simmons, J. A.: A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems. SIAM J. Sci. Stat. Comput. 2, 474-489 (1981).
37. O'Leary, D. P., Strakoš, Z., Tichý, P.: On sensitivity of Gauss-Christoffel quadrature. Numer. Math. 107, 147-174 (2007).
38. Paige, C. C.: Bidiagonalization of matrices and solution of linear equations. SIAM J. Numer. Anal. 11, 197-209 (1974).
39. Paige, C. C.: Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem. Linear Algebra Appl. 34, 235-258 (1980).
40. Paige, C. C.: A useful form of unitary matrix obtained from any sequence of unit 2-norm n-vectors. SIAM J. Matrix Anal. Appl. 31, 565-583 (2009).
41. Paige, C. C., Saunders, M. A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43-71 (1982).
42. Paige, C. C., Saunders, M. A.: ALGORITHM 583 LSQR: Sparse Linear equations and least squares problems. ACM Trans. Math. Softw. 8, 195-209 (1982).
43. Paige, C. C., Strakoš, Z.: Scaled total least squares fundamentals. Numer. Math. 91, 117-146 (2002).
44. Paige, C. C., Strakoš, Z.: Unifying least squares, total least squares and data least squares. In: van Huffel, S., Lemmerling, P. (eds.) Total Least Squares and Errors-in-Variables Modeling, pp. 25-34. Kluwer Academic, Dordrecht (2002).
45. Paige, C. C., Strakoš, Z.: Core problem in linear algebraic systems. SIAM J. Matrix Anal. Appl. 27, 861-875 (2006).
46. Rust, B. W.: Parameter selection for constrained solutions to ill-posed problems. Comput. Sci. Stat. 32, 333-347 (2000).
47. Rust, B. W., O'Leary, D. P.: Residual periodograms for choosing regularization parameters for ill-posed problems. Inverse Probl. 24 (2008). doi: 10. 1088/0266-5611/24/3/034005.
48. Saunders, M. A.: Computing projections with LSQR. BIT 37, 96-104 (1997).
49. Sima, D. M., van Huffel, S.: Using core formulations for ill-posed linear systems. PAMM 5, 795-796 (2005).
50. Sima, D. M.: Regularization techniques in model fitting and parameter estimation. Ph. D. thesis, Dept. of Electrical Engineering, Katholieke Universiteit Leuven (2006).
51. Simon, H. D., Zha, H.: Low-rank matrix approximation using the Lanczos bidiagonalization process with applications. SIAM J. Sci. Stat. Comput. 21, 2257-2274 (2000).
52. Strakoš, Z.: Model reduction using the Vorobyev moment problem. Numer. Algorithms 51, 363-376 (2009).