BIT Numerical Mathematics

, Volume 27, Issue 4, pp 534–553

The truncatedSVD as a method for regularization

  • Per Christian Hansen
Part II Numerical Mathematics

Abstract

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.

AMS subject classification

65F20 65F30 

Keywords

truncated SVD regularization in standard form perturbation theory for truncated SVD numerical rank 

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References

  1. 1.
    H. C. Andrews and B. R. Hunt,Digital Image Restoration, Prentice-Hall (1977).Google Scholar
  2. 2.
    A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications, Wiley-Interscience (1974).Google Scholar
  3. 3.
    T. F. Chan and D. Foulser,Effective condition numbers for linear systems, Tech Memo 86-05, Saxpy Computer Corporation, Sunnyvale (1986).Google Scholar
  4. 4.
    T. F. Chan and P. C. Hansen,Computing truncated SVD least squares solutions by rank revealing QR-factorizations, CAM Report 86-01, Dept. of Mathematics, U.C.L.A. (1986).Google Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    L. Eldén,Algorithms for regularization of ill-conditioned least squares problems, BIT 17 (1977), 134–145.Google Scholar
  7. 7.
    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.Google Scholar
  8. 8.
    G. E. Forsythe, M. A. Malcolm and C. B. Moler,Computer Methods for Mathematical Computations, Prentice-Hall (1977).Google Scholar
  9. 9.
    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).Google Scholar
  10. 10.
    G. H. Golub and C. F. Van Loan,Matrix Computations, North Oxford Academic (1983).Google Scholar
  11. 11.
    P. C. Hansen and S. Christiansen,An SVD analysis of linear algebraic equations derived from first kind integral equations, J. Comp. Appl. Math. 12&13 (1985), 341–357.Google Scholar
  12. 12.
    R. J. Hanson,A numerical method for solving Fredholm integral equations of the first kind using singular values, SIAM J. Numer. Anal. 8 (1971), 616–622.CrossRefGoogle Scholar
  13. 13.
    C. L. Lawson and R. J. Hanson,Solving Least Squares Problems, Prentice Hall (1974).Google Scholar
  14. 14.
    A. K. Louis and F. Natterer,Mathematical problems of computerized tomography, Proc. IEEE 71 (1983), 379–389.Google Scholar
  15. 15.
    B. C. Moore and A. J. Laub,Computation of supremal (A,B)-invariant and controllability subspaces, IEEE Trans. Automat. Contr. AC-23 (1978), 783–792.CrossRefGoogle Scholar
  16. 16.
    F. Natterer,Numerical inversion of the Radon transform, Numer. Math. 30 (1978), 81–91.CrossRefGoogle Scholar
  17. 17.
    D. L. Phillips,A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9 (1962), 84–97.CrossRefGoogle Scholar
  18. 18.
    A. N. Tikhonov,Solution of incorrectly formulated problems and the regularization method, Dokl. Akad. Nauk. SSSR 151 (1963), 501–504 = Soviet Math. Dokl. 4 (1963), 1035–1038.Google Scholar
  19. 19.
    D. W. Tufts and R. Kumaresan,Singular value decomposition and improved frequency estimation using linear prediction, IEEE Trans. Acoust., Speech, Signal Processing ASSP-30 (1982), 671–675.Google Scholar
  20. 20.
    P. M. Van Dooren,The generalized eigenstructure problem in linear system theory, IEEE Trans. Automat. Contr. AC-26 (1981), 111–129.CrossRefGoogle Scholar
  21. 21.
    J. M. Varah,On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems, SIAM J. Numer. Anal. 10 (1973), 257–267.CrossRefGoogle Scholar
  22. 22.
    J. M. Varah,A practical examination of some numerical methods for linear discrete ill-posed problems, SIAM Review 21 (1979), 100–111.CrossRefGoogle Scholar
  23. 23.
    P.-Å. Wedin,Perturbation bounds in connection with the singular value decomposition, BIT 12 (1972), 99–111.CrossRefGoogle Scholar
  24. 24.
    P.-Å. Wedin,Perturbation theory for pseudo-inverses, BIT 13 (1973), 217–232.CrossRefGoogle Scholar
  25. 25.
    P.-Å. Wedin,On the almost rank deficient case of the least squares problem, BIT 13 (1973), 344–354.CrossRefGoogle Scholar

Copyright information

© BIT Foundations 1987

Authors and Affiliations

  • Per Christian Hansen
    • 1
  1. 1.Copenhagen University ObservatoryKøbenhavn KDenmark

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