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Journal of Scientific Computing

, Volume 63, Issue 1, pp 163–184 | Cite as

Simple and Efficient Determination of the Tikhonov Regularization Parameter Chosen by the Generalized Discrepancy Principle for Discrete Ill-Posed Problems

  • Fermín S. Viloche Bazán
Article

Abstract

Discrete ill-posed problems where both the coefficient matrix and the right hand side are contaminated by noise appear in a variety of engineering applications. In this paper we consider Tikhonov regularized solutions where the regularization parameter is chosen by the generalized discrepancy principle (GDP). In contrast to Newton-based methods often used to compute such parameter, we propose a new algorithm referred to as GDP-FP, where derivatives are not required and where the regularization parameter is calculated efficiently by a fixed-point iteration procedure. The algorithm is globally and monotonically convergent. Additionally, a specialized version of GDP-FP based on a projection method, that is well-suited for large-scale Tikhonov problems, is also proposed and analyzed in detail. Numerical examples are presented to illustrate the effectiveness of the proposed algorithms on test problems from the literature.

Keywords

Discrete ill-posed problems Tikhonov regularization  Projection method Generalized discrepancy principle Noisy operator Noisy right hand side 

References

  1. 1.
    Ang, D.D., Gorenflo, R., Le, V.K., Trong, D.D.: Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction, Lect. Notes Math. 1792. Springer, Berlin (2002)Google Scholar
  2. 2.
    Bazán, F.S.V.: Fixed-point iterations in determining the Tikhonov regularization parameter. Inverse Probl. 24, 1–15 (2008)CrossRefGoogle Scholar
  3. 3.
    Bazán, F.S.V., Francisco, J.B., Leem, K.H., Pelekanos, G.: A maximum product criterion as a Tikhonov parameter choice rule for Kirsch’s factorization method. J. Comput. Appl. Math. 236, 4264–4275 (2012)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bazán, F.S.V., Borges, L.S.: GKB-FP:an algorithm for large-scale discrete ill-posed problems. BIT 50(3), 481–507 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bazán, F.S.V., Cunha, M.C., Borges, L.S.: Extension of GKB-FP to large-scale general-form Tikhonov regularization. Numer. Linear Algebra Appl. 21, 316–339 (2014)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Björck, Å.: A bidiagonalization algorithm for solving ill-posed systems of linear equations. BIT 28, 659–670 (1988)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chung, J., Nagy, J.G., O’Leary, D.P.: A weighted-GCV method for Lanczos-hybrid regularization. Electron. Trans. Numer. Anal. 28, 149–167 (2008)MathSciNetGoogle Scholar
  8. 8.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, New York (1992)CrossRefMATHGoogle Scholar
  9. 9.
    Colton, D., Piana, M., Potthast, R.: A simple method using Morozov’s discrepancy principle for solving inverse scattering problems. Inverse Probl. 13, 1477–1493 (1999)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least square problems. BIT 22, 487–502 (1982)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems (Mathematics and its Applications), vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)CrossRefGoogle Scholar
  12. 12.
    Golub, G.H., Hansen, P.C., O’Leary, D.P.: Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21, 185–194 (1999)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  14. 14.
    Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discreet Ill-posed problems. Numer. Algorithms 6, 1–35 (1994)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  16. 16.
    Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equ. Appl. 22, 463–480 (2010)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kirsch, A.: Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Probl. 14, 1489–1512 (1998)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Lampe, J., Voss, H.: Large-scale Tikhonov regularization of total least squares. J. Comput. Appl. Math. 238, 95–108 (2013)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Leem, K.H., Pelekanos, G., Bazán, F.S.V.: Fixed-point iterations in determining a Tikhonov regularization parameter in Kirsch’s factorization method. Appl. Math. Comput. 216, 3747–3753 (2010)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Lu, S., Pereverzev, S.V., Tautenhahn, U.: Regularized total least squares: computational aspects and error bounds. SIAM J. Matrix Anal. Appl. 31, 918–941 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Lu, S., Pereverzev, S.V., Shao, Y., Tautenhahn, U.: On the generalized discrepancy principle for Tikhonov regularization in Hilbert Scales. J. Integral Equ. Appl. 22(3), 483–517 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Markovsky, I., Van Huffel, S.: Overview of total least squares methods. Signal Process. 87, 2283–2302 (2007)CrossRefMATHGoogle Scholar
  23. 23.
    Morozov, V.A.: Regularization Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)CrossRefGoogle Scholar
  24. 24.
    Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8(1), 43–71 (1982)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Paige, C.C., Saunders, M.A.: Algorithm 583. LSQR: sparse linear equations and least squares problems. ACM Trans. Math. Softw. 8(2), 195–209 (1982)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. 33, 63–83 (2009)MathSciNetGoogle Scholar
  27. 27.
    Renaut, R.A., Guo, G.: Efficient algorithms for solution of regularized total least squares. SIAM J. Matrix Anal. Appl. 21, 457–476 (2005)MathSciNetGoogle Scholar
  28. 28.
    Sima, D., Van Huffel, S., Golub, G.H.: Regularized total least squares based on quadratic eigenvalue problems solvers. BIT 44, 793–812 (2004)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Tautenhahn, U.: Regularization of linear ill-posed problems with noisy right hand side and noisy operator. J. Inverse Ill Posed Probl. 16, 507–523 (2008)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Tikhonov, A.N., Arsenin, V.Y.: Solution of Ill-Posed Problems. Wiley, New York (1977)Google Scholar
  31. 31.
    Vinokurov, V.A.: Two notes on the choice of regularization parameter. USSR Comput. Math. Math. Phys. 12(2), 249–253 (1972)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsFederal University of Santa CatarinaFlorianópolisBrazil

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