Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics

  • Frank Bauer
  • Martin GuttingEmail author
  • Mark A. LukasEmail author
Reference work entry


Many different parameter choice methods have been proposed for regularization in both deterministic and stochastic settings. The performance of a particular method in a specific setting and its comparison to other methods is sometimes hard to predict. This chapter reviews many of the existing parameter choice methods and evaluates and compares them in a large simulation study for spectral cutoff and Tikhonov regularization.

The numerical tests deal with downward continuation, i.e., an exponentially ill-posed problem, which is found in many geoscientific applications, in particular those involving satellite measurements. A wide range of scenarios for these linear inverse problems are covered with regard to both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods.


  1. Abascal J-F, Arridge SR, Bayford RH, Holder DS (2008) Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function. Physiol Meas 29:1319–1334CrossRefGoogle Scholar
  2. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Second international symposium on information theory (Tsahkadsor, 1971). Akadémiai Kiadó, Budapest, pp 267–281Google Scholar
  3. Åkesson EO, Daun KJ (2008) Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization. Appl Opt 47:407–416CrossRefGoogle Scholar
  4. Alifanov O, Rumyantsev S (1979) On the stability of iterative methods for the solution of linear ill-posed problems. Sov Math Dokl 20:1133–1136zbMATHGoogle Scholar
  5. Anderssen RS, Bloomfield P (1974) Numerical differentiation procedures for non-exact data. Numer Math 22:157–182MathSciNetCrossRefzbMATHGoogle Scholar
  6. Arcangeli R (1966) Pseudo-solution de l’équation Ax = y. C R Acad Sci Paris Sér A 263(8): 282–285MathSciNetzbMATHGoogle Scholar
  7. Bakushinskii AB (1984) Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. USSR Comput Math Math Phys 24:181–182MathSciNetCrossRefGoogle Scholar
  8. Bakushinsky A, Smirnova A (2005) On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems. Numer Funct Anal Optim 26(1):35–48MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bauer F (2007) Some considerations concerning regularization and parameter choice algorithms. Inverse Probl 23(2):837–858CrossRefMathSciNetzbMATHGoogle Scholar
  10. Bauer F, Hohage T (2005) A Lepskij-type stopping rule for regularized Newton methods. Inverse Probl 21:1975–1991MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bauer F, Kindermann S (2008) The quasi-optimality criterion for classical inverse problems. Inverse Probl 24:035002, 20MathSciNetzbMATHGoogle Scholar
  12. Bauer F, Kindermann S (2009) Recent results on the quasi-optimality principle. J Inverse Ill-Posed Probl 17(1):5–18MathSciNetCrossRefzbMATHGoogle Scholar
  13. Bauer F, Lukas MA (2011) Comparing parameter choice methods for regularization of ill-posed problems. Math Comput Simul 81(9):1795–1841MathSciNetCrossRefzbMATHGoogle Scholar
  14. Bauer F, Mathé P (2011) Parameter choice methods using minimization schemes. J Complex 27:68–85CrossRefMathSciNetzbMATHGoogle Scholar
  15. Bauer F, Munk A (2007) Optimal regularization for ill-posed problems in metric spaces. J Inverse Ill-Posed Probl 15(2):137–148MathSciNetCrossRefzbMATHGoogle Scholar
  16. Bauer F, Pereverzev S (2005) Regularization without preliminary knowledge of smoothness and error behavior. Eur J Appl Math 16(3):303–317MathSciNetCrossRefzbMATHGoogle Scholar
  17. Bauer F, Reiß M (2008) Regularization independent of the noise level: an analysis of quasi-optimality. Inverse Probl 24:055009, 16MathSciNetzbMATHGoogle Scholar
  18. Bauer F, Hohage T, Munk A (2009) Iteratively regularized Gauss–Newton method for nonlinear inverse problems with random noise. SIAM J Numer Anal 47(3):1827–1846MathSciNetCrossRefzbMATHGoogle Scholar
  19. Becker SMA (2011) Regularization of statistical inverse problems and the Bakushinskii veto. Inverse Probl 27:115010, 22MathSciNetzbMATHGoogle Scholar
  20. Blanchard G, Mathé P (2012) Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration. Inverse Probl 28:115011, 23MathSciNetzbMATHGoogle Scholar
  21. Brezinski C, Rodriguez G, Seatzu S (2008) Error estimates for linear systems with applications to regularization. Numer Algorithms 49(1–4):85–104MathSciNetCrossRefzbMATHGoogle Scholar
  22. Brezinski C, Rodriguez G, Seatzu S (2009) Error estimates for the regularization of least squares problems. Numer Algorithms 51(1):61–76MathSciNetCrossRefzbMATHGoogle Scholar
  23. Brown LD, Low MG (1996) Asymptotic equivalence of nonparametric regression and white noise. Ann Stat 24(6):2384–2398MathSciNetCrossRefzbMATHGoogle Scholar
  24. Calvetti D, Hansen PC, Reichel L (2002) L-curve curvature bounds via Lanczos bidiagonalization. Electron Trans Numer Anal 14:20–35MathSciNetzbMATHGoogle Scholar
  25. Candès EJ (2006) Modern statistical estimation via oracle inequalities. Acta Numer 15:257–325MathSciNetCrossRefzbMATHGoogle Scholar
  26. Cavalier L (2008) Nonparametric statistical inverse problems. Inverse Probl 24(3):034004, 19Google Scholar
  27. Cavalier L, Golubev Y (2006) Risk hull method and regularization by projections of ill-posed inverse problems. Ann Stat 34(4):1653–1677MathSciNetCrossRefzbMATHGoogle Scholar
  28. Cavalier L, Golubev GK, Picard D, Tsybakov AB (2002) Oracle inequalities for inverse problems. Ann Stat 30(3):843–874MathSciNetCrossRefzbMATHGoogle Scholar
  29. Cavalier L, Golubev Y, Lepski O, Tsybakov A (2004) Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems. Theory Probab Appl 48(3):426–446MathSciNetCrossRefzbMATHGoogle Scholar
  30. Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  31. Correia T, Gibson A, Schweiger M, Hebden J (2009) Selection of regularization parameter for optical topography. J Biomed Opt 14(3):034044, 11Google Scholar
  32. Cox DD (1988) Approximation of method of regularization estimators. Ann Stat 16(2):694–712CrossRefMathSciNetzbMATHGoogle Scholar
  33. Craven P, Wahba G (1979) Smoothing noisy data with spline functions. Numer Math 31:377–403MathSciNetCrossRefzbMATHGoogle Scholar
  34. Cummins DJ, Filloon TG, Nychka D (2001) Confidence intervals for nonparametric curve estimates: toward more uniform pointwise coverage. J Am Stat Assoc 96(453):233–246MathSciNetCrossRefzbMATHGoogle Scholar
  35. Davies AR (1982) On the maximum likelihood regularization of Fredholm convolution equations of the first kind. In: Baker C, Miller G (eds) Treatment of integral equations by numerical methods. Academic, London, pp 95–105Google Scholar
  36. Davies AR, Anderssen RS (1986) Improved estimates of statistical regularization parameters in Fourier differentiation and smoothing. Numer Math 48:671–697MathSciNetCrossRefzbMATHGoogle Scholar
  37. Ditmar P, Kusche J, Klees R (2003) Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues. J Geod 77(7–8):465–477CrossRefzbMATHGoogle Scholar
  38. Ditmar P, Klees R, Liu X (2007) Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise. J Geod 81(1):81–96CrossRefzbMATHGoogle Scholar
  39. Efron B (2001) Selection criteria for scatterplot smoothers. Ann Stat 29(2):470–504MathSciNetCrossRefzbMATHGoogle Scholar
  40. Eggermont PN, LaRiccia V, Nashed MZ (2014) Noise models for ill-posed problems. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  41. Eldén L (1984) A note on the computation of the generalized cross-validation function for ill-conditioned least squares problems. BIT 24:467–472MathSciNetCrossRefzbMATHGoogle Scholar
  42. Engl HW (1993) Regularization methods for the stable solution of inverse problems. Surv Math Ind 3(2):71–143MathSciNetzbMATHGoogle Scholar
  43. Engl HW, Gfrerer H (1988) A posteriori parameter choice for general regularization methods for solving linear ill-posed problems. Appl Numer Math 4(5):395–417MathSciNetCrossRefzbMATHGoogle Scholar
  44. Engl HW, Scherzer O (2000) Convergence rate results for iterative methods for solving nonlinear ill-posed problems. In: Colton D, Engl H, Louis A, McLaughlin J, Rundell W (eds) Survey on solution methods for inverse problems. Springer, New York, pp 7–34CrossRefGoogle Scholar
  45. Engl HW, Hanke H, Neubauer A (1996) Regularization of inverse problems. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  46. Eubank RL (1988) Spline smoothing and nonparametric regression. Marcel Dekker, New YorkzbMATHGoogle Scholar
  47. Evans SN, Stark PB (2002) Inverse problems as statistics. Inverse Probl 18(4):R55–R97MathSciNetCrossRefzbMATHGoogle Scholar
  48. Farquharson CG, Oldenburg DW (2004) A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems. Geophys J Int 156:411–425CrossRefGoogle Scholar
  49. Fitzpatrick BG (1991) Bayesian analysis in inverse problems. Inverse Probl 7(5):675–702MathSciNetCrossRefzbMATHGoogle Scholar
  50. Freeden W (1999) Multiscale modelling of spaceborne geodata. Teubner, LeipzigzbMATHGoogle Scholar
  51. Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  52. Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, BaselCrossRefzbMATHGoogle Scholar
  53. Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  54. Freeden W, Schreiner M (2015) Satellite gravity gradiometry (SGG): from scalar to tensorial solution. In: Freeden W, Nashed MZ, Sonar T (Eds) Handbook of Geomathematics, 2nd Edn. SpringerGoogle Scholar
  55. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Oxford University Press, OxfordzbMATHGoogle Scholar
  56. Gfrerer H (1987) An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math Comput 49(180):507–522MathSciNetCrossRefzbMATHGoogle Scholar
  57. Girard D (1989) A fast “Monte-Carlo cross-validation” procedure for large least squares problems with noisy data. Numer Math 56(1):1–23MathSciNetCrossRefzbMATHGoogle Scholar
  58. Glasko V, Kriksin Y (1984) On the quasioptimality principle for linear ill-posed problems in Hilbert space. Vychisl Math Math Fiz 24:1603–1613MathSciNetzbMATHGoogle Scholar
  59. Goldenshluger A, Pereverzev S (2000) Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probl Theory Relat Fields 118(2):169–186MathSciNetCrossRefzbMATHGoogle Scholar
  60. Golub GH, von Matt U (1997) Generalized cross-validation for large scale problems. J Comput Graph Stat 6(1):1–34MathSciNetzbMATHGoogle Scholar
  61. Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215–223MathSciNetCrossRefzbMATHGoogle Scholar
  62. Grad J, Zakrajšek E (1972) LR algorithm with Laguerre shift for symmetric tridiagonal matrices. Comput J 15(3):268–270MathSciNetCrossRefzbMATHGoogle Scholar
  63. Groetsch CW (1984) The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, BostonzbMATHGoogle Scholar
  64. Groetsch CW, Neubauer A (1989) Regularization of ill-posed problems: optimal parameter choice in finite dimensions. J Approx Theory 58(2):184–200MathSciNetCrossRefzbMATHGoogle Scholar
  65. Groetsch CW, Schock E (1984) Asymptotic convergence rate of Arcangeli’s method for ill-posed problems. Appl Anal 18:175–182MathSciNetCrossRefzbMATHGoogle Scholar
  66. Gu C (2002) Smoothing spline ANOVA models. Springer, New YorkCrossRefzbMATHGoogle Scholar
  67. Gu C, Bates DM, Chen Z, Wahba G (1989) The computation of generalized cross-validation functions through Householder tridiagonalization with applications to the fitting of interaction spline models. SIAM J Matrix Anal Appl 10(4):457–480MathSciNetCrossRefzbMATHGoogle Scholar
  68. Haber E, Oldenburg DW (2000) A GCV based method for nonlinear ill-posed problems. Comput Geosci 4:41–63MathSciNetCrossRefzbMATHGoogle Scholar
  69. Hämarik U, Raus T (1999) On the a posteriori parameter choice in regularization methods. Proc Estonian Acad Sci Phys Math 48(2):133–145MathSciNetzbMATHGoogle Scholar
  70. Hämarik U, Raus T (2006) On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data. J Inverse Ill-Posed Probl 14(3):251–266MathSciNetCrossRefzbMATHGoogle Scholar
  71. Hämarik U, Raus T (2009) About the balancing principle for choice of the regularization parameter. Numer Funct Anal Optim 30(9–10):951–970MathSciNetCrossRefzbMATHGoogle Scholar
  72. Hämarik U, Tautenhahn U (2001) On the monotone error rule for parameter choice in iterative and continuous regularization methods. BIT 41(5):1029–1038MathSciNetCrossRefGoogle Scholar
  73. Hämarik U, Tautenhahn U (2003) On the monotone error rule for choosing the regularization parameter in ill-posed problems. In: Lavrent’ev MM et al (ed) Ill-posed and non-classical problems of mathematical physics and analysis. VSP, Utrecht, pp 27–55Google Scholar
  74. Hämarik U, Palm R, Raus T (2009) On minimization strategies for choice of the regularization parameter in ill-posed problems. Numer Funct Anal Optim 30(9–10):924–950MathSciNetCrossRefzbMATHGoogle Scholar
  75. Hämarik U, Palm R, Raus T (2011) Comparison of parameter choices in regularization algorithms in case of different information about noise level. Calcolo 48:47–59MathSciNetCrossRefzbMATHGoogle Scholar
  76. Hämarik U, Palm R, Raus T (2012) A family of rules for parameter choices in Tikhonov regularization of ill-posed problems with inexact noise level. J Comput Appl Math 236: 2146–2157MathSciNetCrossRefzbMATHGoogle Scholar
  77. Hanke M (1996) Limitations of the L-curve method in ill-posed problems. BIT 36(2):287–301MathSciNetCrossRefzbMATHGoogle Scholar
  78. Hanke M, Hansen PC (1993) Regularization methods for large-scale problems. Surv Math Ind 3(4):253–315MathSciNetzbMATHGoogle Scholar
  79. Hanke M, Raus T (1996) A general heuristic for choosing the regularization parameter in ill-posed problems. SIAM J Sci Comput 17(4):956–972MathSciNetCrossRefzbMATHGoogle Scholar
  80. Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34(4):561–580MathSciNetCrossRefzbMATHGoogle Scholar
  81. Hansen PC (1994) Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer Algorithms 6:1–35MathSciNetCrossRefzbMATHGoogle Scholar
  82. Hansen PC (1998) Rank-deficient and discrete ill-posed problems. SIAM, PhiladelphiaCrossRefGoogle Scholar
  83. Hansen PC (2001) The L-curve and its use in the numerical treatment of inverse problems. In: Johnstone PR (ed) Computational inverse problems in electrocardiography. WIT, Southampton, pp 119–142Google Scholar
  84. Hansen PC, O’Leary DP (1993) The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 14(6):1487–1503MathSciNetCrossRefzbMATHGoogle Scholar
  85. Hansen PC, Kilmer ME, Kjeldsen RH (2006) Exploiting residual information in the parameter choice for discrete ill-posed problems. BIT 46(1):41–59MathSciNetCrossRefzbMATHGoogle Scholar
  86. Hansen PC, Jensen TK, Rodriguez G (2007) An adaptive pruning algorithm for the discrete L-curve criterion. J Comput Appl Math 198(2):483–492MathSciNetCrossRefzbMATHGoogle Scholar
  87. Hofinger A, Pikkarainen HK (2007) Convergence rate for the Bayesian approach to linear inverse problems. Inverse Probl 23(6):2469–2484MathSciNetCrossRefzbMATHGoogle Scholar
  88. Hofmann B (1986) Regularization of applied inverse and ill-posed problems. Teubner, LeipzigCrossRefzbMATHGoogle Scholar
  89. Hofmann B, Mathé P (2007) Analysis of profile functions for general linear regularization methods. SIAM J Numer Anal 45(3):1122–1141MathSciNetCrossRefzbMATHGoogle Scholar
  90. Hohage T (2000) Regularization of exponentially ill-posed problems. Numer Funct Anal Optim 21:439–464MathSciNetCrossRefzbMATHGoogle Scholar
  91. Hutchinson M (1989) A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun Stat Simul Comput 18(3):1059–1076MathSciNetCrossRefzbMATHGoogle Scholar
  92. Hutchinson MF, de Hoog FR (1985) Smoothing noisy data with spline functions. Numer Math 47:99–106MathSciNetCrossRefzbMATHGoogle Scholar
  93. Jansen M, Malfait M, Bultheel A (1997) Generalized cross validation for wavelet thresholding. Signal Process 56(1):33–44CrossRefzbMATHGoogle Scholar
  94. Jin Q-N (2000) On the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed problems. Math Comput 69(232):1603–1623CrossRefMathSciNetzbMATHGoogle Scholar
  95. Jin Q-N, Hou Z-Y (1999) On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems. Numer Math 83(1):139–159MathSciNetCrossRefzbMATHGoogle Scholar
  96. Jin Q, Tautenhahn U (2009) On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer Math 111(4):509–558MathSciNetCrossRefzbMATHGoogle Scholar
  97. Johnstone PR, Gulrajani RM (2000) Selecting the corner in the L-curve approach to Tikhonov regularization. IEEE Trans Biomed Eng 47(9):1293–1296CrossRefGoogle Scholar
  98. Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, New YorkzbMATHGoogle Scholar
  99. Kaltenbacher B, Neubauer A, Scherzer O (2008) Iterative regularization methods for nonlinear ill-posed problems. Walter de Gruyter, BerlinCrossRefzbMATHGoogle Scholar
  100. Kilmer ME, O’Leary DP (2001) Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J Matrix Anal Appl 22(4):1204–1221MathSciNetCrossRefzbMATHGoogle Scholar
  101. Kindermann S (2011) Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron Trans Numer Anal 38:233–257MathSciNetzbMATHGoogle Scholar
  102. Kindermann S, Neubauer A (2008) On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Probl Imaging 2(2):291–299MathSciNetCrossRefzbMATHGoogle Scholar
  103. Kohn R, Ansley CF, Tharm D (1991) The performance of cross-validation and maximum likelihood estimators of spline smoothing parameters. J Am Stat Assoc 86:1042–1050MathSciNetCrossRefGoogle Scholar
  104. Kou SC, Efron B (2002) Smoothers and the \(C_{p}\), generalized maximum likelihood, and extended exponential criteria: a geometric approach. J Am Stat Assoc 97(459):766–782MathSciNetCrossRefzbMATHGoogle Scholar
  105. Larkin FM (1972) Gaussian measure in Hilbert space and applications in numerical analysis. Rocky Mt J Math 2:379–421MathSciNetCrossRefzbMATHGoogle Scholar
  106. Lawson CL, Hanson RJ (1974) Solving least squares problems. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  107. Leonov A (1979) Justification of the choice of regularization parameter according to quasi-optimality and quotient criteria. USSR Comput Math Math Phys 18(6):1–15CrossRefzbMATHGoogle Scholar
  108. Lepskij O (1990) On a problem of adaptive estimation in Gaussian white noise. Theory Probab Appl 35(3):454–466MathSciNetCrossRefzbMATHGoogle Scholar
  109. Li K-C (1986) Asymptotic optimality of \(C_{L}\) and generalized cross-validation in ridge regression with application to spline smoothing. Ann Stat 14:1101–1112CrossRefzbMATHGoogle Scholar
  110. Li K-C (1987) Asymptotic optimality for \(C_{p}\), \(C_{L}\), cross-validation and generalized cross-validation: discret index set. Ann Stat 15:958–975CrossRefzbMATHGoogle Scholar
  111. Lu S, Mathé P (2013) Heuristic parameter selection based on functional minimization: optimality and model function approach. Math Comput 82(283):1609–1630CrossRefMathSciNetzbMATHGoogle Scholar
  112. Lu S, Mathé P (2014) Discrepancy based model selection in statistical inverse problems. J Complex 30(3):290-308CrossRefMathSciNetzbMATHGoogle Scholar
  113. Lu S, Pereverzev S (2014) Multiparameter regularization in downward continuation of satellite data. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  114. Lukas MA (1988) Convergence rates for regularized solutions. Math Comput 51(183):107–131MathSciNetCrossRefzbMATHGoogle Scholar
  115. Lukas MA (1993) Asymptotic optimality of generalized cross-validation for choosing the regularization parameter. Numer Math 66(1):41–66MathSciNetCrossRefzbMATHGoogle Scholar
  116. Lukas MA (1995) On the discrepancy principle and generalised maximum likelihood for regularisation. Bull Aust Math Soc 52(3):399–424MathSciNetCrossRefzbMATHGoogle Scholar
  117. Lukas MA (1998a) Asymptotic behaviour of the minimum bound method for choosing the regularization parameter. Inverse Probl 14(1):149–159MathSciNetCrossRefzbMATHGoogle Scholar
  118. Lukas MA (1998b) Comparisons of parameter choice methods for regularization with discrete noisy data. Inverse Probl 14(1):161–184MathSciNetCrossRefzbMATHGoogle Scholar
  119. Lukas MA (2006) Robust generalized cross-validation for choosing the regularization parameter. Inverse Probl 22(5):1883–1902MathSciNetCrossRefzbMATHGoogle Scholar
  120. Lukas MA (2008) Strong robust generalized cross-validation for choosing the regularization parameter. Inverse Probl 24:034006, 16MathSciNetzbMATHGoogle Scholar
  121. Lukas MA (2010) Robust GCV choice of the regularization parameter for correlated data. J Integral Equ Appl 22(3):519–547MathSciNetCrossRefzbMATHGoogle Scholar
  122. Mair BA (1994) Tikhonov regularization for finitely and infinitely smoothing operators. SIAM J Math Anal 25:135–147MathSciNetCrossRefzbMATHGoogle Scholar
  123. Mair BA, Ruymgaart FH (1996) Statistical inverse estimation in Hilbert scales. SIAM J Appl Math 56(5):1424–1444MathSciNetCrossRefzbMATHGoogle Scholar
  124. Mathé P (2006) What do we learn from the discrepancy principle? Z Anal Anwend 25(4):411–420MathSciNetCrossRefzbMATHGoogle Scholar
  125. Mathé P, Pereverzev SV (2003) Geometry of linear ill-posed problems in variable Hilbert spaces. Inverse Probl 19(3):789–803CrossRefzbMATHGoogle Scholar
  126. Mathé P, Pereverzev SV (2006) Regularization of some linear ill-posed problems with discretized random noisy data. Math Comput 75(256):1913–1929CrossRefMathSciNetzbMATHGoogle Scholar
  127. Michel V (2014) Tomography: problems and multiscale solutions. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, HeidelbergGoogle Scholar
  128. Morozov VA (1966) On the solution of functional equations by the method of regularization. Soviet Math Dokl 7:414–417MathSciNetzbMATHGoogle Scholar
  129. Morozov VA (1984) Methods for solving incorrectly posed problems. Springer, New YorkCrossRefGoogle Scholar
  130. Nair MT, Rajan MP (2002) Generalized Arcangeli’s discrepancy principles for a class of regularization methods for solving ill-posed problems. J Inverse Ill-Posed Probl 10(3):281–294MathSciNetCrossRefzbMATHGoogle Scholar
  131. Nair MT, Schock E, Tautenhahn U (2003) Morozov’s discrepancy principle under general source conditions. Z Anal Anwend 22:199–214MathSciNetCrossRefzbMATHGoogle Scholar
  132. Neubauer A (1988) An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error. Appl Numer Math 4(6):507–519MathSciNetCrossRefzbMATHGoogle Scholar
  133. Neubauer A (2008) The convergence of a new heuristic parameter selection criterion for general regularization methods. Inverse Probl 24(5):055005, 10Google Scholar
  134. Opsomer J, Wang Y, Yang Y (2010) Nonparametric regression with correlated errors. Stat Sci 16:134–153MathSciNetzbMATHGoogle Scholar
  135. Palm R (2010) Numerical comparison of regularization algorithms for solving ill-posed problems. PhD thesis, University of Tartu, EstoniaGoogle Scholar
  136. Pensky M, Sapatinas T (2010) On convergence rates equivalency and sampling strategies in functional deconvolution models. Ann Stat 38(3):1793–1844MathSciNetCrossRefzbMATHGoogle Scholar
  137. Pereverzev S, Schock E (2000) Morozov’s discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite dimensional subspaces. Numer Funct Anal Optim 21: 901–916MathSciNetCrossRefzbMATHGoogle Scholar
  138. Phillips D (1962) A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 9:84–97MathSciNetCrossRefzbMATHGoogle Scholar
  139. Pohl S, Hofmann B, Neubert R, Otto T, Radehaus C (2001) A regularization approach for the determination of remission curves. Inverse Probl Eng 9(2):157–174CrossRefGoogle Scholar
  140. Raus T (1984) On the discrepancy principle for the solution of ill-posed problems. Uch Zap Tartu Gos Univ 672:16–26MathSciNetzbMATHGoogle Scholar
  141. Raus T (1985) The principle of the residual in the solution of ill-posed problems with nonselfadjoint operator. Tartu Riikl Ül Toimetised 715:12–20MathSciNetzbMATHGoogle Scholar
  142. Raus T (1990) An a posteriori choice of the regularization parameter in case of approximately given error bound of data. In: Pedas A (ed) Collocation and projection methods for integral equations and boundary value problems. Tartu University, Tartu, pp 73–87Google Scholar
  143. Raus T (1992) About regularization parameter choice in case of approximately given error bounds of data. In: Vainikko G (ed) Methods for solution of integral equations and ill-posed problems. Tartu University, Tartu, pp 77–89Google Scholar
  144. Raus T, Hämarik U (2007) On the quasioptimal regularization parameter choices for solving ill-posed problems. J Inverse Ill-Posed Probl 15(4):419–439MathSciNetCrossRefzbMATHGoogle Scholar
  145. Raus T, Hämarik U (2009) New rule for choice of the regularization parameter in (iterated) Tikhonov method. Math Model Anal 14:187–198MathSciNetCrossRefzbMATHGoogle Scholar
  146. Reginska T (1996) A regularization parameter in discrete ill-posed problems. SIAM J Sci Comput 17(3):740–749MathSciNetCrossRefzbMATHGoogle Scholar
  147. Reichel L, Rodriguez G (2013) Old and new parameter choice rules for discrete ill-posed problems. Numer Algorithms 63:65–87MathSciNetCrossRefzbMATHGoogle Scholar
  148. Robinson T, Moyeed R (1989) Making robust the cross-validatory choice of smoothing parameter in spline smoothing regression. Commun Stat Theory Methods 18(2):523–539MathSciNetCrossRefzbMATHGoogle Scholar
  149. Rust BW (2000) Parameter selection for constrained solutions to ill-posed problems. Comput Sci Stat 32:333–347Google Scholar
  150. Rust BW, O’Leary DP (2008) Residual periodograms for choosing regularization parameters for ill-posed problems. Inverse Probl 24(3):034005, 30Google Scholar
  151. Santos R, De Pierro A (2003) A cheaper way to compute generalized cross-validation as a stopping rule for linear stationary iterative methods. J Comput Graph Stat 12(2):417–433CrossRefMathSciNetGoogle Scholar
  152. Scherzer O, Engl H, Kunisch K (1993) Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J Numer Anal 30(6):1796–1838MathSciNetCrossRefzbMATHGoogle Scholar
  153. Spokoiny V, Vial C (2009) Parameter tuning in pointwise adaptation using a propagation approach. Ann Stat 37:2783–2807MathSciNetCrossRefzbMATHGoogle Scholar
  154. Tarantola A (1987) Inverse problem theory: methods for data fitting and model parameter estimation. Elsevier, AmsterdamzbMATHGoogle Scholar
  155. Tautenhahn U, Hämarik U (1999) The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Probl 15(6):1487–1505CrossRefMathSciNetzbMATHGoogle Scholar
  156. Thompson AM, Kay JW, Titterington DM (1989) A cautionary note about crossvalidatory choice. J Stat Comput Simul 33:199–216MathSciNetCrossRefzbMATHGoogle Scholar
  157. Thompson AM, Brown JC, Kay JW, Titterington DM (1991) A study of methods for choosing the smoothing parameter in image restoration by regularization. IEEE Trans Pattern Anal Machine Intell 13:3326–3339CrossRefGoogle Scholar
  158. Tikhonov A, Arsenin V (1977) Solutions of ill-posed problems. Wiley, New YorkzbMATHGoogle Scholar
  159. Tikhonov A, Glasko V (1965) Use of the regularization method in non-linear problems. USSR Comput Math Math Phys 5(3):93–107CrossRefzbMATHGoogle Scholar
  160. Tsybakov A (2000) On the best rate of adaptive estimation in some inverse problems. C R Acad Sci Paris Sér I Math 330(9):835–840MathSciNetCrossRefzbMATHGoogle Scholar
  161. Vio R, Ma P, Zhong W, Nagy J, Tenorio L, Wamsteker W (2004) Estimation of regularization parameters in multiple-image deblurring. Astron Astrophys 423:1179–1186CrossRefGoogle Scholar
  162. Vogel CR (1986) Optimal choice of a truncation level for the truncated SVD solution of linear first kind integral equations when data are noisy. SIAM J Numer Anal 23(1):109–117MathSciNetCrossRefzbMATHGoogle Scholar
  163. Vogel CR (1996) Non-convergence of the L-curve regularization parameter selection method. Inverse Probl 12(4):535–547CrossRefMathSciNetzbMATHGoogle Scholar
  164. Vogel CR (2002) Computational methods for inverse problems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  165. Wahba G (1977) Practical approximate solutions to linear operator equations when the data are noisy. SIAM J Numer Anal 14(4):651–667MathSciNetCrossRefzbMATHGoogle Scholar
  166. Wahba G (1985) A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann Stat 13:1378–1402MathSciNetCrossRefzbMATHGoogle Scholar
  167. Wahba G (1990) Spline models for observational data. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  168. Wahba G, Wang YH (1990) When is the optimal regularization parameter insensitive to the choice of the loss function? Commun Stat Theory Methods 19(5):1685–1700MathSciNetCrossRefzbMATHGoogle Scholar
  169. Wang Y (1998) Smoothing spline models with correlated random errors. J Am Stat Assoc 93: 341–348CrossRefzbMATHGoogle Scholar
  170. Wecker WE, Ansley CF (1983) The signal extraction approach to nonlinear regression and spline smoothing. J Am Stat Assoc 78(381):81–89MathSciNetCrossRefzbMATHGoogle Scholar
  171. Yagola AG, Leonov AS, Titarenko VN (2002) Data errors and an error estimation for ill-posed problems. Inverse Probl Eng 10(2):117–129CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.DZ Bank AG, Kapitalmärkte Handel, Quantitative Modelle, F/KHSQFrankfurtGermany
  2. 2.Geomathematics GroupUniversity of SiegenSiegenGermany
  3. 3.Mathematics and Statistics, School of Engineering and Information TechnologyMurdoch UniversityMurdochAustralia

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