The truncatedSVD as a method for regularization
- Per Christian Hansen
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The truncated singular value decomposition (SVD) is considered as a method for regularization of ill-posed linear least squares problems. In particular, the truncated SVD solution is compared with the usual regularized solution. Necessary conditions are defined in which the two methods will yield similar results. This investigation suggests the truncated SVD as a favorable alternative to standard-form regularization in cases of ill-conditioned matrices with well-determined numerical rank.
- H. C. Andrews and B. R. Hunt,Digital Image Restoration, Prentice-Hall (1977).
- A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications, Wiley-Interscience (1974).
- Chan, T. F., Foulser, D. (1986) Effective condition numbers for linear systems. Saxpy Computer Corporation, Sunnyvale
- Chan, T. F., Hansen, P. C. (1986) Computing truncated SVD least squares solutions by rank revealing QR-factorizations. Dept. of Mathematics, U.C.L.A.
- U. Eckhardt and K. Mika,Numerical treatment of incorrectly posed problems — a case study; in J. Albrecht and L. Collatz (Eds.),Numerical Treatment of Integral Equations, Workshop on numerical treatment of integral equations, Oberwolfach. November 18–24, 1979, Birkhäuser Verlag (1980), pp. 92–101.
- Eldén, L. (1977) Algorithms for regularization of ill-conditioned least squares problems. BIT 17: pp. 134-145
- L. Eldén,The numerical solution of a non-characteristic Cauchy problem for a parabolic equation; in P. Deuflhard and E. Hairer (Eds.),Numerical Treatment of Inverse Problems in Differential and Integral Equations, Birkhäuser Verlag (1983), pp. 246–268.
- G. E. Forsythe, M. A. Malcolm and C. B. Moler,Computer Methods for Mathematical Computations, Prentice-Hall (1977).
- G. H. Golub, V. Klema and G. W. Stewart,Rank degeneracy and least squares problems, Technical Report TR-456, Computer Science Department, University of Maryland (1976).
- G. H. Golub and C. F. Van Loan,Matrix Computations, North Oxford Academic (1983).
- Hansen, P. C., Christiansen, S. (1985) An SVD analysis of linear algebraic equations derived from first kind integral equations. J. Comp. Appl. Math. 12&13: pp. 341-357
- Hanson, R. J. (1971) A numerical method for solving Fredholm integral equations of the first kind using singular values. SIAM J. Numer. Anal. 8: pp. 616-622 CrossRef
- C. L. Lawson and R. J. Hanson,Solving Least Squares Problems, Prentice Hall (1974).
- Louis, A. K., Natterer, F. (1983) Mathematical problems of computerized tomography. Proc. IEEE 71: pp. 379-389
- Moore, B. C., Laub, A. J. (1978) Computation of supremal (A,B)-invariant and controllability subspaces. IEEE Trans. Automat. Contr. AC-23: pp. 783-792 CrossRef
- Natterer, F. (1978) Numerical inversion of the Radon transform. Numer. Math. 30: pp. 81-91 CrossRef
- Phillips, D. L. (1962) A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9: pp. 84-97 CrossRef
- Tikhonov, A. N. (1963) Solution of incorrectly formulated problems and the regularization method. Dokl. Akad. Nauk. SSSR 151: pp. 501-504
- Tufts, D. W., Kumaresan, R. (1982) Singular value decomposition and improved frequency estimation using linear prediction. IEEE Trans. Acoust., Speech, Signal Processing ASSP-30: pp. 671-675
- Dooren, P. M. (1981) The generalized eigenstructure problem in linear system theory. IEEE Trans. Automat. Contr. AC-26: pp. 111-129 CrossRef
- Varah, J. M. (1973) On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems. SIAM J. Numer. Anal. 10: pp. 257-267 CrossRef
- Varah, J. M. (1979) A practical examination of some numerical methods for linear discrete ill-posed problems. SIAM Review 21: pp. 100-111 CrossRef
- Wedin, P.-Å. (1972) Perturbation bounds in connection with the singular value decomposition. BIT 12: pp. 99-111 CrossRef
- Wedin, P.-Å. (1973) Perturbation theory for pseudo-inverses. BIT 13: pp. 217-232 CrossRef
- Wedin, P.-Å. (1973) On the almost rank deficient case of the least squares problem. BIT 13: pp. 344-354 CrossRef
- The truncatedSVD as a method for regularization
BIT Numerical Mathematics
Volume 27, Issue 4 , pp 534-553
- Cover Date
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- Online ISSN
- Kluwer Academic Publishers
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- truncated SVD
- regularization in standard form
- perturbation theory for truncated SVD
- numerical rank
- Industry Sectors
- Author Affiliations
- 1. Copenhagen University Observatory, Øster Voldgade 3, DK-1350, København K, Denmark