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BIT Numerical Mathematics

, Volume 54, Issue 1, pp 129–145 | Cite as

The structure of iterative methods for symmetric linear discrete ill-posed problems

  • L. Dykes
  • F. Marcellán
  • L. Reichel
Article

Abstract

The iterative solution of large linear discrete ill-posed problems with an error contaminated data vector requires the use of specially designed methods in order to avoid severe error propagation. Range restricted minimal residual methods have been found to be well suited for the solution of many such problems. This paper discusses the structure of matrices that arise in a range restricted minimal residual method for the solution of large linear discrete ill-posed problems with a symmetric matrix. The exploitation of the structure results in a method that is competitive with respect to computer storage, number of iterations, and accuracy.

Keywords

Ill-posed problem Iterative method Truncated iteration 

Mathematics Subject Classification (2000)

65F10 65F22 

Notes

Acknowledgments

We would like to thank the referees for comments. The work of F. M. was supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under grant MTM2012-36732-C03-01. Work of L. R. was supported by Universidad Carlos III de Madrid in the Department of Mathematics during the academic year 2010-2011 within the framework of the Chair of Excellence Program and by NSF grant DMS-1115385.

References

  1. 1.
    Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94, 203–228 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Brezinski, C., Redivo-Zaglia, M., Sadok, H.: New look-ahead Lanczos-type algorithms for linear systems. Numer. Math. 83, 53–85 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Buhmann, M.D., Iserles, A.: On orthogonal polynomials transformed by the QR algorithm. J. Comput. Appl. Math. 43, 117–134 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Calvetti, D., Lewis, B., Reichel, L.: On the choice of subspace for iterative methods for linear discrete ill-posed problems. Int. J. Appl. Math. Comput. Sci. 11, 1069–1092 (2001)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Calvetti, D., Reichel, L., Zhang, Q.: Conjugate gradient algorithms for symmetric inconsistent linear systems. In: Brown, J.D., Chu, M.T., Ellison, D.C., Plemmons, R. J. (ed.) Proceedings of the Cornelius Lanczos International Centenary Conference, pp. 267–272. SIAM, Philadelphia (1994)Google Scholar
  6. 6.
    Dykes, L., Reichel, L.: A family of range restricted iterative methods for linear discrete ill-posed problems. Dolomites Res Notes Approx 6, 27–36 (2013)CrossRefGoogle Scholar
  7. 7.
    Gautschi, W.: Orthogonal polynomials: computation and approximation. Oxford University Press, Oxford (2004)Google Scholar
  8. 8.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)zbMATHGoogle Scholar
  9. 9.
    Hanke, M.: Conjugate gradient type methods for Ill-posed problems. Longman, Harlow (1995)zbMATHGoogle Scholar
  10. 10.
    Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189–194 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hansen, P.C., Jensen, T.K.: Noise propagation in regularizing iterations for image deblurring. Electron. Trans. Numer. Anal. 31, 204–220 (2008)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kautsky, J., Golub, G.H.: On the calculation of Jacobi matrices. Linear Algebra Appl. 52–53, 439–455 (1983)Google Scholar
  13. 13.
    Kindermann, S.: Convergence analysis of minimization-based noise level free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233–257 (2011)MathSciNetGoogle Scholar
  14. 14.
    Morigi, S., Reichel, L., Sgallari, F., Zama, F.: Iterative methods for ill-posed problems and semiconvergent sequences. J. Comput. Appl. Math. 193, 157–167 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Neuman, A., Reichel, L., Sadok, H.: Implementations of range restricted iterative methods for linear discrete ill-posed problems. Linear Algebra Appl. 436, 3974–3990 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Neuman, A., Reichel, L., Sadok, H.: Algorithms for range restricted iterative methods for linear discrete ill-posed problems. Numer. Algorithms 59, 325–331 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9, 84–97 (1962)CrossRefzbMATHGoogle Scholar
  19. 19.
    Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63, 65–87 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  21. 21.
    Shaw Jr, C.B.: Improvements of the resolution of an instrument by numerical solution of an integral equation. J. Math. Anal. Appl. 37, 83–112 (1972)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA
  2. 2.University SchoolHunting ValleyUSA
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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