BIT Numerical Mathematics

, Volume 59, Issue 4, pp 1031–1061 | Cite as

Unbiased predictive risk estimation of the Tikhonov regularization parameter: convergence with increasing rank approximations of the singular value decomposition

  • Rosemary A. RenautEmail author
  • Anthony W. Helmstetter
  • Saeed Vatankhah


The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between the sizes of the fit to data function and the regularization term. The unbiased predictive risk estimator is one suggested method for finding the regularization parameter when the noise in the measurements is normally distributed with known variance. In this paper we provide an algorithm using the unbiased predictive risk estimator that automatically finds both the regularization parameter and the number of terms to use from the singular value decomposition. Underlying the algorithm is a new result that proves that the regularization parameter converges with the number of terms from the singular value decomposition. For the analysis it is sufficient to assume that the discrete Picard condition is satisfied for exact data and that noise completely contaminates the measured data coefficients for a sufficiently large number of terms, dependent on both the noise level and the degree of ill-posedness of the system. A lower bound for the regularization parameter is provided leading to a computationally efficient algorithm. Supporting results are compared with those obtained using the method of generalized cross validation. Simulations for two-dimensional examples verify the theoretical analysis and the effectiveness of the algorithm for increasing noise levels, and demonstrate that the relative reconstruction errors obtained using the truncated singular value decomposition are less than those obtained using the singular value decomposition.


Inverse problems Tikhonov regularization Unbiased predictive risk estimation Regularization parameter 

Mathematics Subject Classification




  1. 1.
    Abascal, J.-F.P.J., Arridge, S.R., Bayford, R.H., Holder, D.S.: Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function. Physiol. Meas. 29, 1319–1334 (2008)CrossRefGoogle Scholar
  2. 2.
    Aster, R.C., Borchers, B., Thurber, C.H.: Parameter Estimation and Inverse Problems, 2nd edn. Elsevier, Amsterdam (2013)zbMATHGoogle Scholar
  3. 3.
    Bakushinskii, A.B.: Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. USSR Comput. Math. Math. Phys. 24(4), 181182 (1984)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problems. Math. Comput. Simul. 81, 1795–1841 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Björck, A.: Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia (1996)CrossRefGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Drineas, P., Mahoney, M.W.: RandNLA: randomized numerical linear algebra. Commun. ACM 59, 80–90 (2016)CrossRefGoogle Scholar
  8. 8.
    Drineas, P., Mahoney, M.W., Muthukrishnan, S., Sarlós, T.: Faster least squares approximation. Numer. Math. 117, 219–249 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fenu, C., Reichel, L., Rodrigues, G., Sadok, H.: GCV for Tikhonov regularization by partial SVD. BIT Numer. Math. 57, 1019–1039 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gazzola, S., Hansen, P.C., Nagy, J.G.: IR tools: a MATLAB package of iterative regularization methods and large-scale test problems. Numer. Algorithms. (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215–223 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Press, Baltimore (1996)zbMATHGoogle Scholar
  13. 13.
    Gower, R.M., Richtárik, P.: Randomized iterative methods for linear systems. SIAM J. Matrix Anal. Appl. 36, 1660–1690 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hämäläinen, K., Harhanen, L., Kallonen, A., Kujanpää, A., Niemi, E., Siltanen, S.: Tomographic X-ray data of a walnut. arXiv:1502.04064 (2015)
  15. 15.
    Hämarik, U., Palm, R., Raus, T.: On minimizationstrategies for choice of the regularization parameter in ill-posed problems. Numer. Funct. Anal. Optim. 30, 924–950 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hämarik, U., Palm, R., Raus, T.: A family of rules for parameter choice in Tikhonov regularization of ill-posed problems with inexact noise level. J. Comput. Appl. Math. 236, 2146–2157 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253–315 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hansen, J.K., Hogue, J.D., Sander, G.K., Renaut, R.A., Popat, S.C.: Non-negatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy data. J. Comput. Appl. Math. 278, 52–74 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hansen, P.C.: The discrete Picard condition for discrete ill-posed problems. BIT Numer. Math. 30, 658–672 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hansen, P.C.: Regularization tools—a Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 46, 189–194 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics, Philadelphia (1998)CrossRefGoogle Scholar
  22. 22.
    Hansen, P.C.: Discrete Inverse Problems. Society for Industrial and Applied Mathematics, Philadelphia (2010)CrossRefGoogle Scholar
  23. 23.
    Hofmann, B.: Regularization for Applied Inverse and Ill-posed Problems: A Numerical Approach, Teubner-Texte zur Mathematik. Teubner, B.G., Berlin (1986)CrossRefGoogle Scholar
  24. 24.
    Levin, E., Meltzer, A.Y.: Estimation of the regularization parameter in linear discrete ill-posed problems using the Picard parameter. SIAM J. Sci. Comput. 39, A2741–A2762 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lin, Y., Wohlberg, B., Guo, H.: UPRE method for total variation parameter selection. Signal Process. 90, 2546–2551 (2010)CrossRefGoogle Scholar
  26. 26.
    Mahoney, M.W.: Randomized algorithms for matrices and data. Found. Trends® Mach. Learn. 3, 123–224 (2011)zbMATHGoogle Scholar
  27. 27.
    Mead, J.L., Renaut, R.A.: A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems. Inverse Probl. 25, 025002 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Meng, X., Saunders, M.A., Mahoney, M.W.: LSRN: a parallel iterative solver for strongly over- or underdetermined systems. SIAM J. Sci. Comput. 36, C95–C118 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Morozov, V.A.: On the solution of functional equations by the method of regularization. Sov. Math. Dokl. 7, 414–417 (1966)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Renaut, R.A., Horst, M., Wang, Y., Cochran, D., Hansen, J.: Efficient estimation of regularization parameters via downsampling and the singular value expansion. BIT Numer. Math. 57, 499–529 (2017)CrossRefGoogle Scholar
  31. 31.
    Renaut, R.A., Vatankhah, S., Ardestani, V.E.: Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems. SIAM J. Sci. Comput. 39, B221–B243 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rokhlin, V., Tygert, M.: A fast randomized algorithm for overdetermined linear least-squares regression. Proc. Natl. Acad. Sci. 105, 13212–13217 (2008)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Stein, C.M.: Estimation of the mean of a multivariate normal distribution. Ann. Stat. 9, 1135–1151 (1981)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Taroudaki, V., O’Leary, D.P.: Near-optimal spectral filtering and error estimation for solving ill-posed problems. SIAM J. Sci. Comput. 37, A2947–A2968 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Toma, A., Sixou, B., Peyrin, F.: Iterative choice of the optimal regularization parameter in tv image restoration. Inverse Probl. Imaging 9, 1171 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Vatankhah, S., Ardestani, V.E., Renaut, R.A.: Automatic estimation of the regularization parameter in 2D focusing gravity inversion: application of the method to the Safo manganese mine in the northwest of Iran. J. Geophys. Eng. 11, 045001 (2014)CrossRefGoogle Scholar
  37. 37.
    Vatankhah, S., Ardestani, V.E., Renaut, R.A.: Application of the \(\chi ^2\) principle and unbiased predictive risk estimator for determining the regularization parameter in 3-D focusing gravity inversion. Geophys. J. Int. 200, 265–277 (2015)CrossRefGoogle Scholar
  38. 38.
    Vatankhah, S., Renaut, R.A., Ardestani, V.E.: 3-D projected \(\ell _1\) inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation. Geophys. J. Int. 210, 1872–1887 (2017)CrossRefGoogle Scholar
  39. 39.
    Vatankhah, S., Renaut, R.A., Ardestani, V.E.: A fast algorithm for regularized focused 3-D inversion of gravity data using the randomized SVD. Geophysics (2018)Google Scholar
  40. 40.
    Vatankhah, S., Renaut, R.A., Ardestani, V.E.: Total variation regularization of the 3-D gravity inverse problem using a randomized generalized singular value decomposition. Geophys. J. Int. 213, 695–705 (2018)CrossRefGoogle Scholar
  41. 41.
    Vogel, C.: Computational Methods for Inverse Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Institute of GeophysicsUniversity of TehranTehranIran

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