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A new Tikhonov regularization method

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Abstract

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size, which allow evaluation of the singular value decomposition of the matrix defining the problem, are the truncated singular value decomposition and Tikhonov regularization. The present paper proposes a novel choice of regularization matrix for Tikhonov regularization that bridges the gap between Tikhonov regularization and truncated singular value decomposition. Computed examples illustrate the benefit of the proposed method.

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References

  1. Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Extrapolation techniques for ill-conditioned linear systems. Numer. Math. 81, 1–29 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with applications to regularization. Numer. Algor. 49, 85–104 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algor. 51, 61–76 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Donatelli, M., Serra-Capizzano, S.: Filter factor analysis of an iterative multilevel regularization method. Electron. Trans. Numer. Anal. 29, 163–177 (2008)

    MathSciNet  Google Scholar 

  5. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  6. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984)

    MATH  Google Scholar 

  7. Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  8. Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algor. 46, 189–194 (2007)

    Article  MATH  Google Scholar 

  9. Klann, E., Ramlau, R.: Regularization by fractional filter methods and data smoothing. Inverse Probl. 24, 025018 (2008)

    Article  MathSciNet  Google Scholar 

  10. Morigi, S., Reichel, L., Sgallari, F.: A truncated projected SVD method for linear discrete ill-posed problems. Numer. Algor. 43, 197–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. 33, 63–83 (2009)

    MathSciNet  Google Scholar 

  12. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

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

    Martin Fuhry & Lothar Reichel

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  1. Martin Fuhry
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  2. Lothar Reichel
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Correspondence to Lothar Reichel.

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Fuhry, M., Reichel, L. A new Tikhonov regularization method. Numer Algor 59, 433–445 (2012). https://doi.org/10.1007/s11075-011-9498-x

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  • DOI: https://doi.org/10.1007/s11075-011-9498-x

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