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An iterative Lavrentiev regularization method

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Abstract

This paper presents an iterative method for the computation of approximate solutions of large linear discrete ill-posed problems by Lavrentiev regularization. The method exploits the connection between Lanczos tridiagonalization and Gauss quadrature to determine inexpensively computable lower and upper bounds for certain functionals. This approach to bound functionals was first described in a paper by Dahlquist, Eisenstat, and Golub. A suitable value of the regularization parameter is determined by a modification of the discrepancy principle.

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References

  1. A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications, Kluwer, Dordrecht, 1994.

  2. D. Calvetti, P. C. Hansen, and L. Reichel, L-curve curvature bounds via Lanczos bidiagonalization, Electron. Trans. Numer. Anal., 14 (2002), pp. 20–35.

    MATH  MathSciNet  Google Scholar 

  3. D. Calvetti, B. Lewis, L. Reichel, and F. Sgallari, Tikhonov regularization with nonnegativity constraint, Electron. Trans. Numer. Anal., 18 (2004), pp. 153–173.

    MATH  MathSciNet  Google Scholar 

  4. D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, Tikhonov regularization and the L-curve for large discrete ill-posed problems, J. Comput. Appl. Math., 123 (2000), pp. 423–446.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, An iterative method with error estimators, J. Comput. Appl. Math., 127 (2001), pp. 93–119.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Calvetti and L. Reichel, Tikhonov regularization of large linear problems, BIT, 43 (2003), pp. 263–283.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Calvetti and L. Reichel, Tikhonov regularization with a solution constraint, SIAM J. Sci. Comput., 26 (2004), pp. 224–239.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Calvetti, L. Reichel, and D. C. Sorensen, An implicitly restarted Lanczos method for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal., 2 (1994), pp. 1–21.

    MATH  MathSciNet  Google Scholar 

  9. G. Dahlquist, S. C. Eisenstat, and G. H. Golub, Bounds for the error of linear systems of equations using the theory of moments, J. Math. Anal. Appl., 37 (1972), pp. 151–166.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Dahlquist, G. H. Golub, and S. G. Nash, Bounds for the error in linear systems, in Semi-Infinite Programming, R. Hettich (ed.), Lecture Notes in Control and Computer Science # 15, Springer, Berlin, 1979, pp. 154–172.

  11. W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004.

  12. F. Girosi, M. Jones, and T. Poggio, Regularization theory and neural network architecture, Neural Comput., 7 (1995), pp. 219–269.

    Article  Google Scholar 

  13. G. H. Golub and G. Meurant, Matrices, moments and quadrature, in Numerical Analysis 1993, D. F. Griffiths and G. A. Watson (eds.), Longman, Essex, 1994, pp. 105–156.

  14. G. H. Golub and G. Meurant, Matrices, moments and quadrature II: How to compute the norm of the error in iterative methods, BIT, 37 (1997), pp. 687–705.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd edn., Johns Hopkins University Press, Baltimore, 1996.

  16. G. H. Golub and U. von Matt, Quadratically constrained least squares and quadratic problems, Numer. Math., 59 (1991), pp. 561–580.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. H. Golub and U. von Matt, Tikhonov regularization for large scale problems, in Workshop on Scientific Computing, G. H. Golub, S. H. Lui, F. Luk, and R. Plemmons (eds.), Springer, New York, 1997, pp. 3–26.

  18. C. W. Groetsch and J. Guacamene, Arcangeli’s method for Fredholm equations of the first kind, Proc. Am. Math. Soc., 99 (1987), pp. 256–260.

    Article  MATH  Google Scholar 

  19. M. Hanke, A note on Tikhonov regularization of large linear problems, BIT, 43 (2003), pp. 449–451.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), pp. 1–35. Software is available in Netlib at http://www.netlib.org.

    Article  MATH  MathSciNet  Google Scholar 

  21. Z. P. Liang and P. C. Lauterbur, An efficient method for dynamic magnetic resonance imaging, IEEE Trans. Med. Imag., 13 (1994), pp. 677–686.

    Article  Google Scholar 

  22. D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM, 9 (1962), pp. 84–97.

    Article  MATH  Google Scholar 

  23. D. C. Sorensen, Numerical methods for large eigenvalue problems, Acta Numer., 11 (2002), pp. 519–584.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. Szegő, Orthogonal Polynomials, 4th edn., Amer. Math. Soc., Providence, 1975.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universitá degli Studi di Bologna, Piazza Porta S. Donato 5, 40127, Bologna, Italy

    S. Morigi

  2. Department of Mathematical Sciences, Kent State University, Kent, 44242, OH, USA

    L. Reichel

  3. Dipartimento di Matematica CIRAM, Universitá degli Studi di Bologna, Via Saragozza 8, 40123, Bologna, Italy

    F. Sgallari

Authors
  1. S. Morigi
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  2. L. Reichel
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  3. F. Sgallari
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Corresponding author

Correspondence to L. Reichel.

Additional information

In memory of Germund Dahlquist (1925–2005).

AMS subject classification (2000)

65R30, 65R32, 65F10

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Morigi, S., Reichel, L. & Sgallari, F. An iterative Lavrentiev regularization method . Bit Numer Math 46, 589–606 (2006). https://doi.org/10.1007/s10543-006-0070-3

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  • DOI: https://doi.org/10.1007/s10543-006-0070-3

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