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Handbook of Geomathematics pp 1–55Cite as

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

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Abstract

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.aaaa 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.

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References

  • 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 29:1319–1334

    Google Scholar 

  • 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 267–281

    Google Scholar 

  • Åkesson EO, Daun KJ (2008) Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization. Appl Opt 47:407–416

    Google Scholar 

  • Alifanov O, Rumyantsev S (1979) On the stability of iterative methods for the solution of linear ill-posed problems. Sov Math Dokl 20:1133–1136

    MATH  Google Scholar 

  • Anderssen RS, Bloomfield P (1974) Numerical differentiation procedures for non-exact data. Numer Math 22:157–182

    MATH  MathSciNet  Google Scholar 

  • Arcangeli R (1966) Pseudo-solution de l’équation A x = y. C R Acad Sci Paris Sér A 263(8): 282–285

    MATH  MathSciNet  Google Scholar 

  • Bakushinskii AB (1984) Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. USSR Comput Math Math Phys 24:181–182

    MathSciNet  Google Scholar 

  • 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):35–48

    MATH  MathSciNet  Google Scholar 

  • Bauer F (2007) Some considerations concerning regularization and parameter choice algorithms. Inverse Probl 23(2):837–858

    MATH  Google Scholar 

  • Bauer F, Hohage T (2005) A Lepskij-type stopping rule for regularized Newton methods. Inverse Probl 21:1975–1991

    MATH  MathSciNet  Google Scholar 

  • Bauer F, Kindermann S (2008) The quasi-optimality criterion for classical inverse problems. Inverse Probl 24:035002, 20

    MathSciNet  Google Scholar 

  • Bauer F, Kindermann S (2009) Recent results on the quasi-optimality principle. J Inverse Ill-Posed Probl 17(1):5–18

    MATH  MathSciNet  Google Scholar 

  • Bauer F, Lukas MA (2011) Comparing parameter choice methods for regularization of ill-posed problems. Math Comput Simul 81(9):1795–1841

    MATH  MathSciNet  Google Scholar 

  • Bauer F, Mathé P (2011) Parameter choice methods using minimization schemes. J Complex 27:68–85

    MATH  Google Scholar 

  • Bauer F, Munk A (2007) Optimal regularization for ill-posed problems in metric spaces. J Inverse Ill-Posed Probl 15(2):137–148

    MATH  MathSciNet  Google Scholar 

  • Bauer F, Pereverzev S (2005) Regularization without preliminary knowledge of smoothness and error behavior. Eur J Appl Math 16(3):303–317

    MATH  MathSciNet  Google Scholar 

  • Bauer F, Reiß M (2008) Regularization independent of the noise level: an analysis of quasi-optimality. Inverse Probl 24:055009, 16

    Google Scholar 

  • 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):1827–1846

    MATH  MathSciNet  Google Scholar 

  • Becker SMA (2011) Regularization of statistical inverse problems and the Bakushinskii veto. Inverse Probl 27:115010, 22

    Google Scholar 

  • Blanchard G, Mathé P (2012) Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration. Inverse Probl 28:115011, 23

    Google Scholar 

  • Brezinski C, Rodriguez G, Seatzu S (2008) Error estimates for linear systems with applications to regularization. Numer Algorithms 49(1–4):85–104

    MATH  MathSciNet  Google Scholar 

  • Brezinski C, Rodriguez G, Seatzu S (2009) Error estimates for the regularization of least squares problems. Numer Algorithms 51(1):61–76

    MATH  MathSciNet  Google Scholar 

  • Brown LD, Low MG (1996) Asymptotic equivalence of nonparametric regression and white noise. Ann Stat 24(6):2384–2398

    MATH  MathSciNet  Google Scholar 

  • Calvetti D, Hansen PC, Reichel L (2002) L-curve curvature bounds via Lanczos bidiagonalization. Electron Trans Numer Anal 14:20–35

    MATH  MathSciNet  Google Scholar 

  • Candès EJ (2006) Modern statistical estimation via oracle inequalities. Acta Numer 15:257–325

    MATH  MathSciNet  Google Scholar 

  • Cavalier L (2008) Nonparametric statistical inverse problems. Inverse Probl 24(3):034004, 19

    Google Scholar 

  • Cavalier L, Golubev Y (2006) Risk hull method and regularization by projections of ill-posed inverse problems. Ann Stat 34(4):1653–1677

    MATH  MathSciNet  Google Scholar 

  • Cavalier L, Golubev GK, Picard D, Tsybakov AB (2002) Oracle inequalities for inverse problems. Ann Stat 30(3):843–874

    MATH  MathSciNet  Google Scholar 

  • 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):426–446

    MathSciNet  Google Scholar 

  • Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Correia T, Gibson A, Schweiger M, Hebden J (2009) Selection of regularization parameter for optical topography. J Biomed Opt 14(3):034044, 11

    Google Scholar 

  • Cox DD (1988) Approximation of method of regularization estimators. Ann Stat 16(2):694–712

    MATH  Google Scholar 

  • Craven P, Wahba G (1979) Smoothing noisy data with spline functions. Numer Math 31:377–403

    MATH  MathSciNet  Google Scholar 

  • Cummins DJ, Filloon TG, Nychka D (2001) Confidence intervals for nonparametric curve estimates: toward more uniform pointwise coverage. J Am Stat Assoc 96(453):233–246

    MATH  MathSciNet  Google Scholar 

  • 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 95–105

    Google Scholar 

  • Davies AR, Anderssen RS (1986) Improved estimates of statistical regularization parameters in Fourier differentiation and smoothing. Numer Math 48:671–697

    MATH  MathSciNet  Google Scholar 

  • 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):465–477

    MATH  Google Scholar 

  • 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):81–96

    MATH  Google Scholar 

  • Efron B (2001) Selection criteria for scatterplot smoothers. Ann Stat 29(2):470–504

    MATH  MathSciNet  Google Scholar 

  • 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, Heidelberg

    Google Scholar 

  • Eldén L (1984) A note on the computation of the generalized cross-validation function for ill-conditioned least squares problems. BIT 24:467–472

    MATH  MathSciNet  Google Scholar 

  • Engl HW (1993) Regularization methods for the stable solution of inverse problems. Surv Math Ind 3(2):71–143

    MATH  MathSciNet  Google Scholar 

  • Engl HW, Gfrerer H (1988) A posteriori parameter choice for general regularization methods for solving linear ill-posed problems. Appl Numer Math 4(5):395–417

    MATH  MathSciNet  Google Scholar 

  • 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 7–34

    Google Scholar 

  • Engl HW, Hanke H, Neubauer A (1996) Regularization of inverse problems. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Eubank RL (1988) Spline smoothing and nonparametric regression. Marcel Dekker, New York

    MATH  Google Scholar 

  • Evans SN, Stark PB (2002) Inverse problems as statistics. Inverse Probl 18(4):R55–R97

    MATH  MathSciNet  Google Scholar 

  • Farquharson CG, Oldenburg DW (2004) A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems. Geophys J Int 156:411–425

    Google Scholar 

  • Fitzpatrick BG (1991) Bayesian analysis in inverse problems. Inverse Probl 7(5):675–702

    MATH  MathSciNet  Google Scholar 

  • Freeden W (1999) Multiscale modelling of spaceborne geodata. Teubner, Leipzig

    MATH  Google Scholar 

  • Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, Basel

    MATH  Google Scholar 

  • Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser, Boston

    MATH  Google Scholar 

  • Freeden W, Schreiner M (2014) Satellite gravity gradiometry (SGG): from scalar to tensorial solution. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, Heidelberg

    Google Scholar 

  • Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Oxford University Press, Oxford

    MATH  Google Scholar 

  • 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):507–522

    MATH  MathSciNet  Google Scholar 

  • Girard D (1989) A fast “Monte-Carlo cross-validation” procedure for large least squares problems with noisy data. Numer Math 56(1):1–23

    MATH  MathSciNet  Google Scholar 

  • Glasko V, Kriksin Y (1984) On the quasioptimality principle for linear ill-posed problems in Hilbert space. Vychisl Math Math Fiz 24:1603–1613

    MATH  MathSciNet  Google Scholar 

  • Goldenshluger A, Pereverzev S (2000) Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probl Theory Relat Fields 118(2):169–186

    MATH  MathSciNet  Google Scholar 

  • Golub GH, von Matt U (1997) Generalized cross-validation for large scale problems. J Comput Graph Stat 6(1):1–34

    Google Scholar 

  • Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215–223

    MATH  MathSciNet  Google Scholar 

  • Grad J, Zakrajšek E (1972) LR algorithm with Laguerre shift for symmetric tridiagonal matrices. Comput J 15(3):268–270

    MATH  MathSciNet  Google Scholar 

  • Groetsch CW (1984) The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston

    MATH  Google Scholar 

  • Groetsch CW, Neubauer A (1989) Regularization of ill-posed problems: optimal parameter choice in finite dimensions. J Approx Theory 58(2):184–200

    MATH  MathSciNet  Google Scholar 

  • Groetsch CW, Schock E (1984) Asymptotic convergence rate of Arcangeli’s method for ill-posed problems. Appl Anal 18:175–182

    MathSciNet  Google Scholar 

  • Gu C (2002) Smoothing spline ANOVA models. Springer, New York

    MATH  Google Scholar 

  • 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):457–480

    MATH  MathSciNet  Google Scholar 

  • Haber E, Oldenburg DW (2000) A GCV based method for nonlinear ill-posed problems. Comput Geosci 4:41–63

    MATH  MathSciNet  Google Scholar 

  • Hämarik U, Raus T (1999) On the a posteriori parameter choice in regularization methods. Proc Estonian Acad Sci Phys Math 48(2):133–145

    MATH  MathSciNet  Google Scholar 

  • 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):251–266

    MATH  MathSciNet  Google Scholar 

  • Hämarik U, Raus T (2009) About the balancing principle for choice of the regularization parameter. Numer Funct Anal Optim 30(9–10):951–970

    MATH  MathSciNet  Google Scholar 

  • Hämarik U, Tautenhahn U (2001) On the monotone error rule for parameter choice in iterative and continuous regularization methods. BIT 41(5):1029–1038

    MathSciNet  Google Scholar 

  • 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 27–55

    Google Scholar 

  • 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):924–950

    MATH  MathSciNet  Google Scholar 

  • Hämarik U, Palm R, Raus T (2011) Comparison of parameter choices in regularization algorithms in case of different information about noise level. Calcolo 48:47–59

    MATH  MathSciNet  Google Scholar 

  • 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: 2146–2157

    MATH  MathSciNet  Google Scholar 

  • Hanke M (1996) Limitations of the L-curve method in ill-posed problems. BIT 36(2):287–301

    MATH  MathSciNet  Google Scholar 

  • Hanke M, Hansen PC (1993) Regularization methods for large-scale problems. Surv Math Ind 3(4):253–315

    MATH  MathSciNet  Google Scholar 

  • Hanke M, Raus T (1996) A general heuristic for choosing the regularization parameter in ill-posed problems. SIAM J Sci Comput 17(4):956–972

    MATH  MathSciNet  Google Scholar 

  • Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34(4):561–580

    MATH  MathSciNet  Google Scholar 

  • Hansen PC (1994) Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer Algorithms 6:1–35

    MATH  MathSciNet  Google Scholar 

  • Hansen PC (1998) Rank-deficient and discrete ill-posed problems. SIAM, Philadelphia

    Google Scholar 

  • 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 119–142

    Google Scholar 

  • 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):1487–1503

    MATH  MathSciNet  Google Scholar 

  • Hansen PC, Kilmer ME, Kjeldsen RH (2006) Exploiting residual information in the parameter choice for discrete ill-posed problems. BIT 46(1):41–59

    MATH  MathSciNet  Google Scholar 

  • Hansen PC, Jensen TK, Rodriguez G (2007) An adaptive pruning algorithm for the discrete L-curve criterion. J Comput Appl Math 198(2):483–492

    MATH  MathSciNet  Google Scholar 

  • Hofinger A, Pikkarainen HK (2007) Convergence rate for the Bayesian approach to linear inverse problems. Inverse Probl 23(6):2469–2484

    MATH  MathSciNet  Google Scholar 

  • Hofmann B (1986) Regularization of applied inverse and ill-posed problems. Teubner, Leipzig

    Google Scholar 

  • Hofmann B, Mathé P (2007) Analysis of profile functions for general linear regularization methods. SIAM J Numer Anal 45(3):1122–1141

    MATH  MathSciNet  Google Scholar 

  • Hohage T (2000) Regularization of exponentially ill-posed problems. Numer Funct Anal Optim 21:439–464

    MATH  MathSciNet  Google Scholar 

  • Hutchinson M (1989) A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun Stat Simul Comput 18(3):1059–1076

    MATH  MathSciNet  Google Scholar 

  • Hutchinson MF, de Hoog FR (1985) Smoothing noisy data with spline functions. Numer Math 47:99–106

    MATH  MathSciNet  Google Scholar 

  • Jansen M, Malfait M, Bultheel A (1997) Generalized cross validation for wavelet thresholding. Signal Process 56(1):33–44

    MATH  Google Scholar 

  • Jin Q-N (2000) On the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed problems. Math Comput 69(232):1603–1623

    MATH  Google Scholar 

  • 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):139–159

    MATH  MathSciNet  Google Scholar 

  • Jin Q, Tautenhahn U (2009) On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer Math 111(4):509–558

    MATH  MathSciNet  Google Scholar 

  • Johnstone PR, Gulrajani RM (2000) Selecting the corner in the L-curve approach to Tikhonov regularization. IEEE Trans Biomed Eng 47(9):1293–1296

    Google Scholar 

  • Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, New York

    MATH  Google Scholar 

  • Kaltenbacher B, Neubauer A, Scherzer O (2008) Iterative regularization methods for nonlinear ill-posed problems. Walter de Gruyter, Berlin

    MATH  Google Scholar 

  • Kilmer ME, O’Leary DP (2001) Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J Matrix Anal Appl 22(4):1204–1221

    MATH  MathSciNet  Google Scholar 

  • Kindermann S (2011) Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron Trans Numer Anal 38:233–257

    MATH  MathSciNet  Google Scholar 

  • Kindermann S, Neubauer A (2008) On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Probl Imaging 2(2):291–299

    MATH  MathSciNet  Google Scholar 

  • Kohn R, Ansley CF, Tharm D (1991) The performance of cross-validation and maximum likelihood estimators of spline smoothing parameters. J Am Stat Assoc 86:1042–1050

    MathSciNet  Google Scholar 

  • 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):766–782

    MATH  MathSciNet  Google Scholar 

  • Larkin FM (1972) Gaussian measure in Hilbert space and applications in numerical analysis. Rocky Mt J Math 2:379–421

    MATH  MathSciNet  Google Scholar 

  • Lawson CL, Hanson RJ (1974) Solving least squares problems. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Leonov A (1979) Justification of the choice of regularization parameter according to quasi-optimality and quotient criteria. USSR Comput Math Math Phys 18(6):1–15

    MATH  Google Scholar 

  • Lepskij O (1990) On a problem of adaptive estimation in Gaussian white noise. Theory Probab Appl 35(3):454–466

    MATH  MathSciNet  Google Scholar 

  • Li K-C (1986) Asymptotic optimality of C L and generalized cross-validation in ridge regression with application to spline smoothing. Ann Stat 14:1101–1112

    MATH  Google Scholar 

  • Li K-C (1987) Asymptotic optimality for C p , C L , cross-validation and generalized cross-validation: discret index set. Ann Stat 15:958–975

    MATH  Google Scholar 

  • Lu S, Mathé P (2013) Heuristic parameter selection based on functional minimization: optimality and model function approach. Math Comput 82(283):1609–1630

    MATH  Google Scholar 

  • Lu S, Mathé P (2014) Discrepancy based model selection in statistical inverse problems. J Complex 30(3):290-308

    MATH  Google Scholar 

  • 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, Heidelberg

    Google Scholar 

  • Lukas MA (1988) Convergence rates for regularized solutions. Math Comput 51(183):107–131

    MATH  MathSciNet  Google Scholar 

  • Lukas MA (1993) Asymptotic optimality of generalized cross-validation for choosing the regularization parameter. Numer Math 66(1):41–66

    MATH  MathSciNet  Google Scholar 

  • Lukas MA (1995) On the discrepancy principle and generalised maximum likelihood for regularisation. Bull Aust Math Soc 52(3):399–424

    MATH  MathSciNet  Google Scholar 

  • Lukas MA (1998a) Asymptotic behaviour of the minimum bound method for choosing the regularization parameter. Inverse Probl 14(1):149–159

    MATH  MathSciNet  Google Scholar 

  • Lukas MA (1998b) Comparisons of parameter choice methods for regularization with discrete noisy data. Inverse Probl 14(1):161–184

    MATH  MathSciNet  Google Scholar 

  • Lukas MA (2006) Robust generalized cross-validation for choosing the regularization parameter. Inverse Probl 22(5):1883–1902

    MATH  MathSciNet  Google Scholar 

  • Lukas MA (2008) Strong robust generalized cross-validation for choosing the regularization parameter. Inverse Probl 24:034006, 16

    MathSciNet  Google Scholar 

  • Lukas MA (2010) Robust GCV choice of the regularization parameter for correlated data. J Integral Equ Appl 22(3):519–547

    MATH  MathSciNet  Google Scholar 

  • Mair BA (1994) Tikhonov regularization for finitely and infinitely smoothing operators. SIAM J Math Anal 25:135–147

    MATH  MathSciNet  Google Scholar 

  • Mair BA, Ruymgaart FH (1996) Statistical inverse estimation in Hilbert scales. SIAM J Appl Math 56(5):1424–1444

    MATH  MathSciNet  Google Scholar 

  • Mathé P (2006) What do we learn from the discrepancy principle? Z Anal Anwend 25(4):411–420

    MATH  MathSciNet  Google Scholar 

  • Mathé P, Pereverzev SV (2003) Geometry of linear ill-posed problems in variable Hilbert spaces. Inverse Probl 19(3):789–803

    MATH  Google Scholar 

  • Mathé P, Pereverzev SV (2006) Regularization of some linear ill-posed problems with discretized random noisy data. Math Comput 75(256):1913–1929

    MATH  Google Scholar 

  • Michel V (2014) Tomography: problems and multiscale solutions. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, Heidelberg

    Google Scholar 

  • Morozov VA (1966) On the solution of functional equations by the method of regularization. Soviet Math Dokl 7:414–417

    MATH  MathSciNet  Google Scholar 

  • Morozov VA (1984) Methods for solving incorrectly posed problems. Springer, New York

    Google Scholar 

  • 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):281–294

    MATH  MathSciNet  Google Scholar 

  • Nair MT, Schock E, Tautenhahn U (2003) Morozov’s discrepancy principle under general source conditions. Z Anal Anwend 22:199–214

    MATH  MathSciNet  Google Scholar 

  • Neubauer A (1988) An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error. Appl Numer Math 4(6):507–519

    MATH  MathSciNet  Google Scholar 

  • Neubauer A (2008) The convergence of a new heuristic parameter selection criterion for general regularization methods. Inverse Probl 24(5):055005, 10

    Google Scholar 

  • Opsomer J, Wang Y, Yang Y (2010) Nonparametric regression with correlated errors. Stat Sci 16:134–153

    MathSciNet  Google Scholar 

  • Palm R (2010) Numerical comparison of regularization algorithms for solving ill-posed problems. PhD thesis, University of Tartu, Estonia

    Google Scholar 

  • Pensky M, Sapatinas T (2010) On convergence rates equivalency and sampling strategies in functional deconvolution models. Ann Stat 38(3):1793–1844

    MATH  MathSciNet  Google Scholar 

  • 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: 901–916

    MATH  MathSciNet  Google Scholar 

  • Phillips D (1962) A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 9:84–97

    MATH  MathSciNet  Google Scholar 

  • 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):157–174

    Google Scholar 

  • Raus T (1984) On the discrepancy principle for the solution of ill-posed problems. Uch Zap Tartu Gos Univ 672:16–26

    MathSciNet  Google Scholar 

  • Raus T (1985) The principle of the residual in the solution of ill-posed problems with nonselfadjoint operator. Tartu Riikl Ül Toimetised 715:12–20

    MathSciNet  Google Scholar 

  • 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 73–87

    Google Scholar 

  • 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 77–89

    Google Scholar 

  • Raus T, Hämarik U (2007) On the quasioptimal regularization parameter choices for solving ill-posed problems. J Inverse Ill-Posed Probl 15(4):419–439

    MATH  MathSciNet  Google Scholar 

  • Raus T, Hämarik U (2009) New rule for choice of the regularization parameter in (iterated) Tikhonov method. Math Model Anal 14:187–198

    MATH  MathSciNet  Google Scholar 

  • Reginska T (1996) A regularization parameter in discrete ill-posed problems. SIAM J Sci Comput 17(3):740–749

    MATH  MathSciNet  Google Scholar 

  • Reichel L, Rodriguez G (2013) Old and new parameter choice rules for discrete ill-posed problems. Numer Algorithms 63:65–87

    MATH  MathSciNet  Google Scholar 

  • Robinson T, Moyeed R (1989) Making robust the cross-validatory choice of smoothing parameter in spline smoothing regression. Commun Stat Theory Methods 18(2):523–539

    MATH  MathSciNet  Google Scholar 

  • Rust BW (2000) Parameter selection for constrained solutions to ill-posed problems. Comput Sci Stat 32:333–347

    Google Scholar 

  • Rust BW, O’Leary DP (2008) Residual periodograms for choosing regularization parameters for ill-posed problems. Inverse Probl 24(3):034005, 30

    Google Scholar 

  • 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):417–433

    Google Scholar 

  • 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):1796–1838

    MATH  MathSciNet  Google Scholar 

  • Spokoiny V, Vial C (2009) Parameter tuning in pointwise adaptation using a propagation approach. Ann Stat 37:2783–2807

    MATH  MathSciNet  Google Scholar 

  • Tarantola A (1987) Inverse problem theory: methods for data fitting and model parameter estimation. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Tautenhahn U, Hämarik U (1999) The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Probl 15(6):1487–1505

    MATH  Google Scholar 

  • Thompson AM, Kay JW, Titterington DM (1989) A cautionary note about crossvalidatory choice. J Stat Comput Simul 33:199–216

    MATH  MathSciNet  Google Scholar 

  • 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 13:3326–3339

    Google Scholar 

  • Tikhonov A, Arsenin V (1977) Solutions of ill-posed problems. Wiley, New York

    MATH  Google Scholar 

  • Tikhonov A, Glasko V (1965) Use of the regularization method in non-linear problems. USSR Comput Math Math Phys 5(3):93–107

    Google Scholar 

  • Tsybakov A (2000) On the best rate of adaptive estimation in some inverse problems. C R Acad Sci Paris Sér I Math 330(9):835–840

    MATH  MathSciNet  Google Scholar 

  • Vio R, Ma P, Zhong W, Nagy J, Tenorio L, Wamsteker W (2004) Estimation of regularization parameters in multiple-image deblurring. Astron Astrophys 423:1179–1186

    Google Scholar 

  • 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):109–117

    MATH  MathSciNet  Google Scholar 

  • Vogel CR (1996) Non-convergence of the L-curve regularization parameter selection method. Inverse Probl 12(4):535–547

    MATH  Google Scholar 

  • Vogel CR (2002) Computational methods for inverse problems. SIAM, Philadelphia

    MATH  Google Scholar 

  • Wahba G (1977) Practical approximate solutions to linear operator equations when the data are noisy. SIAM J Numer Anal 14(4):651–667

    MATH  MathSciNet  Google Scholar 

  • Wahba G (1985) A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann Stat 13:1378–1402

    MATH  MathSciNet  Google Scholar 

  • Wahba G (1990) Spline models for observational data. SIAM, Philadelphia

    MATH  Google Scholar 

  • 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):1685–1700

    MATH  MathSciNet  Google Scholar 

  • Wang Y (1998) Smoothing spline models with correlated random errors. J Am Stat Assoc 93: 341–348

    MATH  Google Scholar 

  • Wecker WE, Ansley CF (1983) The signal extraction approach to nonlinear regression and spline smoothing. J Am Stat Assoc 78(381):81–89

    MATH  MathSciNet  Google Scholar 

  • Yagola AG, Leonov AS, Titarenko VN (2002) Data errors and an error estimation for ill-posed problems. Inverse Probl Eng 10(2):117–129

    Google Scholar 

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Authors and Affiliations

  1. DZ Bank AG, Kapitalmärkte Handel, Quantitative Modelle, F/KHSQ, Frankfurt, Germany

    Frank Bauer

  2. Geomathematics Group, University of Siegen, Siegen, Germany

    Martin Gutting

  3. Mathematics and Statistics, School of Engineering and Information Technology, Murdoch University, Murdoch, WA, Australia

    Mark A. Lukas

Authors
  1. Frank Bauer
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  2. Martin Gutting
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  3. Mark A. Lukas
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Corresponding author

Correspondence to Frank Bauer .

Editor information

Editors and Affiliations

  1. Geomathematische Gruppe, TU Kaiserslautern FB Mathematik, Kaiserslautern, Germany

    Willi Freeden

  2. University of Central Florida Dept. Mathematics, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Inst. Analysis, TU Braunschweig FB 1 Mathematik und Informatik, Braunschweig, Germany

    Thomas Sonar

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Bauer, F., Gutting, M., Lukas, M.A. (2013). Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27793-1_99-1

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  • DOI: https://doi.org/10.1007/978-3-642-27793-1_99-1

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  • Online ISBN: 978-3-642-27793-1

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