Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Geomathematics: Its Role, Its Aim, and Its Potential Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

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

verfasst von : Frank Bauer, Martin Gutting, Mark A. Lukas

Erschienen in: Handbook of Geomathematics

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.
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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Multiparameter Regularization in Downward Continuation of Satellite Data
Nächstes Kapitel Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment
Literatur
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 29:1319–1334CrossRef
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
Åkesson EO, Daun KJ (2008) Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization. Appl Opt 47:407–416CrossRef
Alifanov O, Rumyantsev S (1979) On the stability of iterative methods for the solution of linear ill-posed problems. Sov Math Dokl 20:1133–1136MATH
Arcangeli R (1966) Pseudo-solution de l’équation Ax = y. C R Acad Sci Paris Sér A 263(8): 282–285MathSciNetMATH
Bakushinskii AB (1984) Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion. USSR Comput Math Math Phys 24:181–182MathSciNetCrossRef
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):35–48MathSciNetCrossRefMATH
Bauer F, Kindermann S (2008) The quasi-optimality criterion for classical inverse problems. Inverse Probl 24:035002, 20MathSciNetMATH
Bauer F, Lukas MA (2011) Comparing parameter choice methods for regularization of ill-posed problems. Math Comput Simul 81(9):1795–1841MathSciNetCrossRefMATH
Bauer F, Munk A (2007) Optimal regularization for ill-posed problems in metric spaces. J Inverse Ill-Posed Probl 15(2):137–148MathSciNetCrossRefMATH
Bauer F, Pereverzev S (2005) Regularization without preliminary knowledge of smoothness and error behavior. Eur J Appl Math 16(3):303–317MathSciNetCrossRefMATH
Bauer F, Reiß M (2008) Regularization independent of the noise level: an analysis of quasi-optimality. Inverse Probl 24:055009, 16MathSciNetMATH
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):1827–1846MathSciNetCrossRefMATH
Becker SMA (2011) Regularization of statistical inverse problems and the Bakushinskii veto. Inverse Probl 27:115010, 22MathSciNetMATH
Blanchard G, Mathé P (2012) Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration. Inverse Probl 28:115011, 23MathSciNetMATH
Brezinski C, Rodriguez G, Seatzu S (2008) Error estimates for linear systems with applications to regularization. Numer Algorithms 49(1–4):85–104MathSciNetCrossRefMATH
Brezinski C, Rodriguez G, Seatzu S (2009) Error estimates for the regularization of least squares problems. Numer Algorithms 51(1):61–76MathSciNetCrossRefMATH
Calvetti D, Hansen PC, Reichel L (2002) L-curve curvature bounds via Lanczos bidiagonalization. Electron Trans Numer Anal 14:20–35MathSciNetMATH
Cavalier L (2008) Nonparametric statistical inverse problems. Inverse Probl 24(3):034004, 19
Cavalier L, Golubev Y (2006) Risk hull method and regularization by projections of ill-posed inverse problems. Ann Stat 34(4):1653–1677MathSciNetCrossRefMATH
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):426–446MathSciNetCrossRefMATH
Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory, 2nd edn. Springer, BerlinCrossRefMATH
Correia T, Gibson A, Schweiger M, Hebden J (2009) Selection of regularization parameter for optical topography. J Biomed Opt 14(3):034044, 11
Cummins DJ, Filloon TG, Nychka D (2001) Confidence intervals for nonparametric curve estimates: toward more uniform pointwise coverage. J Am Stat Assoc 96(453):233–246MathSciNetCrossRefMATH
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
Davies AR, Anderssen RS (1986) Improved estimates of statistical regularization parameters in Fourier differentiation and smoothing. Numer Math 48:671–697MathSciNetCrossRefMATH
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):465–477CrossRefMATH
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):81–96CrossRefMATH
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
Eldén L (1984) A note on the computation of the generalized cross-validation function for ill-conditioned least squares problems. BIT 24:467–472MathSciNetCrossRefMATH
Engl HW (1993) Regularization methods for the stable solution of inverse problems. Surv Math Ind 3(2):71–143MathSciNetMATH
Engl HW, Gfrerer H (1988) A posteriori parameter choice for general regularization methods for solving linear ill-posed problems. Appl Numer Math 4(5):395–417MathSciNetCrossRefMATH
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 7–34CrossRef
Engl HW, Hanke H, Neubauer A (1996) Regularization of inverse problems. Kluwer, DordrechtCrossRefMATH
Eubank RL (1988) Spline smoothing and nonparametric regression. Marcel Dekker, New YorkMATH
Farquharson CG, Oldenburg DW (2004) A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems. Geophys J Int 156:411–425CrossRef
Freeden W (1999) Multiscale modelling of spaceborne geodata. Teubner, LeipzigMATH
Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Chapman & Hall/CRC, Boca RatonMATH
Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, BaselCrossRefMATH
Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser, BostonCrossRefMATH
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. Springer
Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Oxford University Press, OxfordMATH
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):507–522MathSciNetCrossRefMATH
Girard D (1989) A fast “Monte-Carlo cross-validation” procedure for large least squares problems with noisy data. Numer Math 56(1):1–23MathSciNetCrossRefMATH
Glasko V, Kriksin Y (1984) On the quasioptimality principle for linear ill-posed problems in Hilbert space. Vychisl Math Math Fiz 24:1603–1613MathSciNetMATH
Goldenshluger A, Pereverzev S (2000) Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probl Theory Relat Fields 118(2):169–186MathSciNetCrossRefMATH
Golub GH, von Matt U (1997) Generalized cross-validation for large scale problems. J Comput Graph Stat 6(1):1–34MathSciNetMATH
Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215–223MathSciNetCrossRefMATH
Groetsch CW (1984) The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, BostonMATH
Groetsch CW, Neubauer A (1989) Regularization of ill-posed problems: optimal parameter choice in finite dimensions. J Approx Theory 58(2):184–200MathSciNetCrossRefMATH
Groetsch CW, Schock E (1984) Asymptotic convergence rate of Arcangeli’s method for ill-posed problems. Appl Anal 18:175–182MathSciNetCrossRefMATH
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):457–480MathSciNetCrossRefMATH
Hämarik U, Raus T (1999) On the a posteriori parameter choice in regularization methods. Proc Estonian Acad Sci Phys Math 48(2):133–145MathSciNetMATH
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):251–266MathSciNetCrossRefMATH
Hämarik U, Raus T (2009) About the balancing principle for choice of the regularization parameter. Numer Funct Anal Optim 30(9–10):951–970MathSciNetCrossRefMATH
Hämarik U, Tautenhahn U (2001) On the monotone error rule for parameter choice in iterative and continuous regularization methods. BIT 41(5):1029–1038MathSciNetCrossRef
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
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):924–950MathSciNetCrossRefMATH
Hämarik U, Palm R, Raus T (2011) Comparison of parameter choices in regularization algorithms in case of different information about noise level. Calcolo 48:47–59MathSciNetCrossRefMATH
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: 2146–2157MathSciNetCrossRefMATH
Hanke M, Hansen PC (1993) Regularization methods for large-scale problems. Surv Math Ind 3(4):253–315MathSciNetMATH
Hanke M, Raus T (1996) A general heuristic for choosing the regularization parameter in ill-posed problems. SIAM J Sci Comput 17(4):956–972MathSciNetCrossRefMATH
Hansen PC (1994) Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer Algorithms 6:1–35MathSciNetCrossRefMATH
Hansen PC (1998) Rank-deficient and discrete ill-posed problems. SIAM, PhiladelphiaCrossRef
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
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):1487–1503MathSciNetCrossRefMATH
Hansen PC, Kilmer ME, Kjeldsen RH (2006) Exploiting residual information in the parameter choice for discrete ill-posed problems. BIT 46(1):41–59MathSciNetCrossRefMATH
Hansen PC, Jensen TK, Rodriguez G (2007) An adaptive pruning algorithm for the discrete L-curve criterion. J Comput Appl Math 198(2):483–492MathSciNetCrossRefMATH
Hofinger A, Pikkarainen HK (2007) Convergence rate for the Bayesian approach to linear inverse problems. Inverse Probl 23(6):2469–2484MathSciNetCrossRefMATH
Hofmann B (1986) Regularization of applied inverse and ill-posed problems. Teubner, LeipzigCrossRefMATH
Hofmann B, Mathé P (2007) Analysis of profile functions for general linear regularization methods. SIAM J Numer Anal 45(3):1122–1141MathSciNetCrossRefMATH
Hutchinson M (1989) A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun Stat Simul Comput 18(3):1059–1076MathSciNetCrossRefMATH
Jansen M, Malfait M, Bultheel A (1997) Generalized cross validation for wavelet thresholding. Signal Process 56(1):33–44CrossRefMATH
Jin Q-N (2000) On the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed problems. Math Comput 69(232):1603–1623CrossRefMathSciNetMATH
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):139–159MathSciNetCrossRefMATH
Jin Q, Tautenhahn U (2009) On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer Math 111(4):509–558MathSciNetCrossRefMATH
Johnstone PR, Gulrajani RM (2000) Selecting the corner in the L-curve approach to Tikhonov regularization. IEEE Trans Biomed Eng 47(9):1293–1296CrossRef
Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, New YorkMATH
Kaltenbacher B, Neubauer A, Scherzer O (2008) Iterative regularization methods for nonlinear ill-posed problems. Walter de Gruyter, BerlinCrossRefMATH
Kilmer ME, O’Leary DP (2001) Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J Matrix Anal Appl 22(4):1204–1221MathSciNetCrossRefMATH
Kindermann S (2011) Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron Trans Numer Anal 38:233–257MathSciNetMATH
Kindermann S, Neubauer A (2008) On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Probl Imaging 2(2):291–299MathSciNetCrossRefMATH
Kohn R, Ansley CF, Tharm D (1991) The performance of cross-validation and maximum likelihood estimators of spline smoothing parameters. J Am Stat Assoc 86:1042–1050MathSciNetCrossRef
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):766–782MathSciNetCrossRefMATH
Lawson CL, Hanson RJ (1974) Solving least squares problems. Prentice-Hall, Englewood CliffsMATH
Leonov A (1979) Justification of the choice of regularization parameter according to quasi-optimality and quotient criteria. USSR Comput Math Math Phys 18(6):1–15CrossRefMATH
Li K-C (1986) Asymptotic optimality of \(C_{L}\) and generalized cross-validation in ridge regression with application to spline smoothing. Ann Stat 14:1101–1112CrossRefMATH
Li K-C (1987) Asymptotic optimality for \(C_{p}\), \(C_{L}\), cross-validation and generalized cross-validation: discret index set. Ann Stat 15:958–975CrossRefMATH
Lu S, Mathé P (2013) Heuristic parameter selection based on functional minimization: optimality and model function approach. Math Comput 82(283):1609–1630CrossRefMathSciNetMATH
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
Lukas MA (1993) Asymptotic optimality of generalized cross-validation for choosing the regularization parameter. Numer Math 66(1):41–66MathSciNetCrossRefMATH
Lukas MA (1995) On the discrepancy principle and generalised maximum likelihood for regularisation. Bull Aust Math Soc 52(3):399–424MathSciNetCrossRefMATH
Lukas MA (1998a) Asymptotic behaviour of the minimum bound method for choosing the regularization parameter. Inverse Probl 14(1):149–159MathSciNetCrossRefMATH
Lukas MA (1998b) Comparisons of parameter choice methods for regularization with discrete noisy data. Inverse Probl 14(1):161–184MathSciNetCrossRefMATH
Lukas MA (2006) Robust generalized cross-validation for choosing the regularization parameter. Inverse Probl 22(5):1883–1902MathSciNetCrossRefMATH
Lukas MA (2008) Strong robust generalized cross-validation for choosing the regularization parameter. Inverse Probl 24:034006, 16MathSciNetMATH
Mathé P, Pereverzev SV (2003) Geometry of linear ill-posed problems in variable Hilbert spaces. Inverse Probl 19(3):789–803CrossRefMATH
Mathé P, Pereverzev SV (2006) Regularization of some linear ill-posed problems with discretized random noisy data. Math Comput 75(256):1913–1929CrossRefMathSciNetMATH
Michel V (2014) Tomography: problems and multiscale solutions. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, Heidelberg
Morozov VA (1966) On the solution of functional equations by the method of regularization. Soviet Math Dokl 7:414–417MathSciNetMATH
Morozov VA (1984) Methods for solving incorrectly posed problems. Springer, New YorkCrossRef
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):281–294MathSciNetCrossRefMATH
Nair MT, Schock E, Tautenhahn U (2003) Morozov’s discrepancy principle under general source conditions. Z Anal Anwend 22:199–214MathSciNetCrossRefMATH
Neubauer A (1988) An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error. Appl Numer Math 4(6):507–519MathSciNetCrossRefMATH
Neubauer A (2008) The convergence of a new heuristic parameter selection criterion for general regularization methods. Inverse Probl 24(5):055005, 10
Opsomer J, Wang Y, Yang Y (2010) Nonparametric regression with correlated errors. Stat Sci 16:134–153MathSciNetMATH
Palm R (2010) Numerical comparison of regularization algorithms for solving ill-posed problems. PhD thesis, University of Tartu, Estonia
Pensky M, Sapatinas T (2010) On convergence rates equivalency and sampling strategies in functional deconvolution models. Ann Stat 38(3):1793–1844MathSciNetCrossRefMATH
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: 901–916MathSciNetCrossRefMATH
Phillips D (1962) A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 9:84–97MathSciNetCrossRefMATH
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):157–174CrossRef
Raus T (1984) On the discrepancy principle for the solution of ill-posed problems. Uch Zap Tartu Gos Univ 672:16–26MathSciNetMATH
Raus T (1985) The principle of the residual in the solution of ill-posed problems with nonselfadjoint operator. Tartu Riikl Ül Toimetised 715:12–20MathSciNetMATH
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
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
Raus T, Hämarik U (2007) On the quasioptimal regularization parameter choices for solving ill-posed problems. J Inverse Ill-Posed Probl 15(4):419–439MathSciNetCrossRefMATH
Raus T, Hämarik U (2009) New rule for choice of the regularization parameter in (iterated) Tikhonov method. Math Model Anal 14:187–198MathSciNetCrossRefMATH
Reichel L, Rodriguez G (2013) Old and new parameter choice rules for discrete ill-posed problems. Numer Algorithms 63:65–87MathSciNetCrossRefMATH
Robinson T, Moyeed R (1989) Making robust the cross-validatory choice of smoothing parameter in spline smoothing regression. Commun Stat Theory Methods 18(2):523–539MathSciNetCrossRefMATH
Rust BW (2000) Parameter selection for constrained solutions to ill-posed problems. Comput Sci Stat 32:333–347
Rust BW, O’Leary DP (2008) Residual periodograms for choosing regularization parameters for ill-posed problems. Inverse Probl 24(3):034005, 30
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):417–433CrossRefMathSciNet
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):1796–1838MathSciNetCrossRefMATH
Tarantola A (1987) Inverse problem theory: methods for data fitting and model parameter estimation. Elsevier, AmsterdamMATH
Tautenhahn U, Hämarik U (1999) The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Probl 15(6):1487–1505CrossRefMathSciNetMATH
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 13:3326–3339CrossRef
Tikhonov A, Arsenin V (1977) Solutions of ill-posed problems. Wiley, New YorkMATH
Tikhonov A, Glasko V (1965) Use of the regularization method in non-linear problems. USSR Comput Math Math Phys 5(3):93–107CrossRefMATH
Tsybakov A (2000) On the best rate of adaptive estimation in some inverse problems. C R Acad Sci Paris Sér I Math 330(9):835–840MathSciNetCrossRefMATH
Vio R, Ma P, Zhong W, Nagy J, Tenorio L, Wamsteker W (2004) Estimation of regularization parameters in multiple-image deblurring. Astron Astrophys 423:1179–1186CrossRef
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):109–117MathSciNetCrossRefMATH
Wahba G (1977) Practical approximate solutions to linear operator equations when the data are noisy. SIAM J Numer Anal 14(4):651–667MathSciNetCrossRefMATH
Wahba G (1985) A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann Stat 13:1378–1402MathSciNetCrossRefMATH
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):1685–1700MathSciNetCrossRefMATH
Wang Y (1998) Smoothing spline models with correlated random errors. J Am Stat Assoc 93: 341–348CrossRefMATH
Wecker WE, Ansley CF (1983) The signal extraction approach to nonlinear regression and spline smoothing. J Am Stat Assoc 78(381):81–89MathSciNetCrossRefMATH
Yagola AG, Leonov AS, Titarenko VN (2002) Data errors and an error estimation for ill-posed problems. Inverse Probl Eng 10(2):117–129CrossRef
Metadaten
Titel
Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics
verfasst von
Frank Bauer
Martin Gutting
Mark A. Lukas
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Buch
Handbook of Geomathematics

Print ISBN: 978-3-642-54550-4

Electronic ISBN: 978-3-642-54551-1

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-642-54551-1_99

Premium Partner

    Bildnachweise