Regularized total least squares: Computational aspects and error bounds

SIAM Journal on Matrix Analysis and Applications

Volumn 31, Issue 3, 2009, Pages 918-941

  Lu, Shuai ^a   Pereverzev, Sergei V ^a   Tautenhahn, Ulrich ^b  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^b UNIVERSITY OF APPLIED SCIENCES ZITTAU GÖRLITZ   (Germany)

Author keywords

Error bounds; Ill posed problems; Inverse problems; Multi parameter regularization; Noisy operator; Noisy right hand side; Regularized total least squares

Indexed keywords

ERROR BOUND; ILL POSED PROBLEM; ILL-POSED PROBLEMS; MULTIPARAMETERS; NOISY OPERATORS; RIGHT-HAND SIDES; TOTAL LEAST SQUARES;

DIFFERENTIAL EQUATIONS;

INVERSE PROBLEMS;

EID: 72449143085     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/070709086     Document Type: Article

References (24)

1. A. BECK, A. BEN-TAL, AND M. TEBOULLE, Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 425-445.
2. A. BÖTTCHER, B. HOFMANN, U. TAUTENHAHN, AND Y. YAMAMOTO, Convergence rates for Tikhonov regularization from different kinds of smoothness conditions, Appl. Anal., 85 (2006), pp. 555-578.
3. L. EldéN, Solving quadratically constrained least squares problem using a differential-geometric approach, BIT, 42 (2002), pp. 323-335.
4. W. GANDER, Least squares with a quadratic constraint, Numer. Math., 36 (1981), pp. 291-307.
5. G. H. GOLUB, P. C. HANSEN, AND D. P. O'LEARY, Tikhonov regularization and total least squares, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185-194.
6. G. H. GOLUB AND U. VON MATT, Quadratically constrained least squares and quadratic problems, Numer. Math., 59 (1991), pp. 561-580.
7. G. C. GOODWIN, M. SERON, AND J. DE DONÁ, Constrained Control and Estimation, Springer, London, 2005.
8. H. W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
9. P. C. HANSEN, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), pp. 1-35.
10. B. HOFMANN, Regularization of Applied Inverse and Ill-Posed Problems, Teubner, Leipzig, 1986.
11. S. VAN HUFFEL AND J. VANDEWALLE, The Total Least Squares Problem: Computational Aspects and Analysis, SIAM, Philadelphia, 1991.
12. V. K. IVANOV, On the approximate solution of operator equations of the first kind, USSR Comput. Math. Math. Phys., 6 (1966), pp. 1089-1094.
13. S. KREIN AND Y. I. PETUNIN, Scales of Banach spaces, Russian Math. Surveys, 21 (1966), pp. 85-159.
14. A. K. LOUIS, Inverse und Schlecht Gestellte Probleme, Teubner, Stuttgart, 1989.
15. P. MATHÉ AND U. TAUTENHAHN, Interpolation in variable Hilbert scales with application to inverse problems, Inverse Problems, 22 (2006), pp. 2271-2297.
16. V. A. MOROZOV, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), pp. 414-417.
17. M. T. NAIR, S. V. PEREVERZEV, AND U. TAUTENHAHN, Regularization in Hilbert scales under general smoothing conditions, Inverse Problems, 21 (2005), pp. 1851-1869.
18. R. A. RENAUT AND H. GUO, Efficient algorithms for solution of regularized total least squares, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 457-476.
19. D. SIMA, S. VAN HUFFEL, AND G. H. GOLUB, Regularized total least squares based on quadratic eigenvalue problem solvers, BIT, 44 (2004), pp. 793-812.
20. U. TAUTENHAHN, Error estimates for regularization methods in Hilbert scales, SIAMJ. Numer. Anal., 33 (1996), pp. 2120-2130.
21. U. TAUTENHAHN, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19 (1998), pp. 377-398.
22. A. N. TIKHONOV AND V. Y. ARSENIN, Solution of Ill-Posed Problems, Wiley, New York, 1977.
23. A. N. TIKHONOV AND V. B. GLASKO, Use of the regularization methods in non-linear problems, USSR Comput. Math. Math. Phys., 5 (1965), pp. 93-107.
24. G. M. VAINIKKO AND A. Y. VERETENNIKOV, Iteration Procedures in Ill-Posed Problems, Nauka, Moscow, 1986 (in Russian).