Journal of Engineering Mathematics

, Volume 93, Issue 1, pp 99–112 | Cite as

Vector extrapolation applied to truncated singular value decomposition and truncated iteration

  • A. Bouhamidi
  • K. Jbilou
  • L. Reichel
  • H. Sadok
  • Z. Wang


This paper is concerned with the computation of accurate approximate solutions of linear systems of equations and linear least-squares problems with a very ill-conditioned matrix and error-contaminated data. The solution of these kinds of problems requires regularization. Common regularization methods include truncated singular value decomposition (TSVD) and truncated iteration with a Krylov subspace method. It can be difficult to determine when to truncate. Recently, it has been demonstrated that extrapolation of approximate solutions determined by TSVD gives a new sequence of approximate solutions that is less sensitive to errors in data than the original approximate solutions. The present paper describes a novel approach to determining a suitable truncation index by comparing the original and extrapolated approximate solutions. Applications to TSVD and the LSQR iterative method are presented.


Discrete ill-posed problem LSQR Truncation criterion Truncated iteration  Truncated singular value decomposition Vector extrapolation 



This research was supported in part by National Science Foundation Grant DMS-1115385. The research was also supported by the Jiangsu Oversea Research & Training Program for Prominent Young & Middle-Aged University Teachers and Presidents and the Fundamental Research Funds for Central Universities Grant NZ2012307.


  1. 1.
    Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  2. 2.
    Hansen PC (1998) Rank-deficient and discrete ill-posed problems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  3. 3.
    Bouhamidi A, Jbilou K, Reichel L, Sadok H (2011) An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure. Linear Algebra Appl 434:1677–1688CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Jbilou K, Reichel L, Sadok H (2009) Vector extrapolation enhanced TSVD for linear discrete ill-posed problems. Numer Algorithms 51:195–208CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. 5.
    Kindermann S (2011) Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron Trans Numer Anal 38:233–257zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bauer F, Lukas MA (2011) Comparing parameter choice methods for regularization of ill-posed problem. Math Comput Simul 81:1795–1841CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Reichel L, Rodriguez G (2013) Old and new parameter choice rules for discrete ill-posed problems. Numer Algorithms 63:65–87CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brezinski C, Redivo Zaglia M, Rodriguez G, Seatzu S (1998) Extrapolation techniques for ill-conditioned linear systems. Numer Math 81:1–29CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Brezinski C, Rodriguez G, Seatzu S (2008) Error estimates for linear systems with applications to regularization. Numer Algorithms 49:85–104CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    Brezinski C, Rodriguez G, Seatzu S (2009) Error estimates for the regularization of least squares problems. Numer Algorithms 51:61–76CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Reichel L, Rodriguez G, Seatzu S (2009) Error estimates for large-scale ill-posed problems. Numer Algorithms 51:341–361CrossRefADSzbMATHMathSciNetGoogle Scholar
  12. 12.
    Brezinski C, Redivo Zaglia M (1991) Extrapolation methods: theory and practice. North-Holland, AmsterdamzbMATHGoogle Scholar
  13. 13.
    Duminil S, Sadok H (2011) Reduced rank extrapolation applied to electronic structure computations. Electron Trans Numer Anal 38:347–362zbMATHMathSciNetGoogle Scholar
  14. 14.
    Jbilou K, Sadok H (2000) Vector extrapolation methods. Applications and numerical comparison. J Comput Appl Math 122:149–165CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    Paige CC, Saunders MA (1982) LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans Math Softw 8:43–71CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  17. 17.
    Eddy RP (1979) Extrapolation to the limit of a vector sequence. In: Wang PCC (ed) Information linkage between applied mathematics and industry. Academic Press, New York, pp 387–396Google Scholar
  18. 18.
    Mesina M (1977) Convergence acceleration for the iterative solution of \(x=Ax+f\). Comput Method Appl Mech Eng 10:165–173CrossRefADSzbMATHMathSciNetGoogle Scholar
  19. 19.
    Ford WD, Sidi A (1988) Recursive algorithms for vector extrapolation methods. Appl Numer Math 4:477–489CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hansen PC (2007) Regularization tools version 4.0 for MATLAB 7.3. Numer Algorithms 46:189–194CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    Baart ML (1982) The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems. IMA J Numer Anal 2:241–247CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Wing GM (1991) A primer on integral equations of the first kind. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  23. 23.
    Baker CTH (1977) The numerical treatment of integral equations. Clarendon Press, OxfordzbMATHGoogle Scholar
  24. 24.
    Shaw CB Jr (1972) Improvements of the resolution of an instrument by numerical solution of an integral equation. J Math Anal Appl 37:83–112CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  26. 26.
    Carasso AS (1982) Determining surface temperatures from interior observations. SIAM J Appl Math 42:558–574CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • A. Bouhamidi
    • 1
  • K. Jbilou
    • 1
  • L. Reichel
    • 2
  • H. Sadok
    • 1
  • Z. Wang
    • 3
  1. 1.Laboratoire de Mathématiques Pures et Appliquées, Centre Universtaire de la Mi-VoixUniversité du Littoral Calais CedexFrance
  2. 2.Department of Mathematical SciencesKent State UniversityKentUSA
  3. 3.Department of Mathematics, College of ScienceNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China

Personalised recommendations