Nonlinear Tikhonov regularization in Hilbert scales with balancing principle tuning parameter in statistical inverse problems

References

• Engl HW , Hanke M , Neubauer A . Regularization of inverse problems. Vol. 375, Mathematics and its applications. Dordrecht: Kluwer Academic Publishers Group; 1996.
   Google Scholar
• Sudakov VN , Halfin LA . A statistical approach to the correctness of the problems of mathematical physics. Dokl Akad Nauk SSSR. 1964;157:1058–1060.
   Google Scholar
• Bissantz N , Hohage T , Munk A , et al . Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J Numer Anal. 2007;45(6):2610–2636.
   Web of Science ®Google Scholar
• Mathé P , Pereverzev S . Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods. SIAM J Numer Anal. 2001;38:1999–2021.
   Web of Science ®Google Scholar
• O’Sullivan F . Convergence characteristics of methods of regularization estimators for nonlinear operator equations. SIAM J Numer Anal. 1990;27(6):1635–1649.
   Web of Science ®Google Scholar
• Lukas MA . Robust generalized cross-validation for choosing the regularization parameter. Inverse Probl. 2006;22(5):1883–1902.
   Web of Science ®Google Scholar
• Wahba G . Spline models for observational data. Vol. 59, CBMS-NSF regional conference series in applied mathematics. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 1990.
   Google Scholar
• Bissantz N , Hohage T , Munk A . Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inverse Probl. 2004;20(6):1773–1789.
   Web of Science ®Google Scholar
• Klann E , Ramlau R . Regularization by fractional filter methods and data smoothing. Inverse Probl. 2008;24(2):025018, 26.
   Web of Science ®Google Scholar
• Klann E , Maaß P , Ramlau R . Two-step regularization methods for linear inverse problems. J Inverse Ill-Posed Probl. 2006;14(6):583–607.
   Google Scholar
• Tsybakov AB . Introduction to nonparametric estimation. Springer series in statistics. New York (NY): Springer; 2009. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats.
   Google Scholar
• Grama I , Nussbaum M . Asymptotic equivalence for nonparametric regression. Math Methods Statist. 2002;11(1):1–36.
   Google Scholar
• Meister A , Reiß M . Asymptotic equivalence for nonparametric regression with non-regular errors. Probab Theory Rel Fields. 2013;155(1–2):201–229.
   Web of Science ®Google Scholar
• Werner F , Hohage T . Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data. Inverse Probl. 2012;28(10):104004, 15.
   Web of Science ®Google Scholar
• Hohage T , Pricop M . Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Probl Imag. 2008;2(2):271–290.
   Web of Science ®Google Scholar
• Natterer F . Error bounds for Tikhonov regularization in Hilbert scales. Appl Anal. 1984;18(1–2):29–37.
   Web of Science ®Google Scholar
• Neubauer A . Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales. Appl Anal. 1992;46(1–2):59–72.
   Google Scholar
• Mair BA , Ruymgaart FH . Statistical inverse estimation in Hilbert scales. SIAM J Appl Math. 1996;56(5):1424–1444.
   Web of Science ®Google Scholar
• Nussbaum M , Pereverzev S . The degree of ill-posedness in stochastic and deterministic noise models. Berlin: WIAS; 1999. Technical Report.
   Google Scholar
• Goldenshluger A , Pereverzev S . On adaptive inverse estimation of linear functionals in Hilbert scales. Bernoulli. 2003;9:783–807.
   Web of Science ®Google Scholar
• Tautenhahn U . Error estimates for regularized solutions of nonlinear ill-posed problems. Inverse Probl. 1994;10(2):485–500.
   Web of Science ®Google Scholar
• Polzehl J , Spokoiny V . Error estimates for regularization methods in Hilbert scales. SIAM J Numer Anal. 1996;33(6):2120–2130.
   Web of Science ®Google Scholar
• Lepskiĭ OV . A problem of adaptive estimation in Gaussian white noise. Teor Veroyatnost i Primenen. 1990;35(3):459–470.
   Web of Science ®Google Scholar
• Mathé P , Pereverzev S . Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Probl. 2003;19(3):789–803.
   Web of Science ®Google Scholar
• Lu S , Pereverzev S , Ramlau R . An analysis of Tikhonov regularization for nonlinear ill-posed problems under general smoothness assumptions. Inverse Probl. 2007;23(1):217–230.
   Web of Science ®Google Scholar
• Bauer F , Hohage T . A Lepskij-type stopping rule for regularized Newton methods. Inverse Probl. 2005;21(6):1975–1991.
   Web of Science ®Google Scholar
• Polzehl J , Spokoiny VG . Adaptive weights smoothing with applications to image restoration. J R Stat Soc Ser B Stat Methodol. 2000;62(2):335–354.
   Google Scholar
• Kreĭn SG , Petunin JI . Scales of Banach spaces. Uspehi Mat Nauk. 1966;21(2 (128)):89–168.
   Google Scholar
• Pereverzev S , Schock E . On the adaptive selection of the parameter in regularization of ill-posed problems. SIAM J Numer Anal. 2005;43(5):2060–2076. electronic.
   Web of Science ®Google Scholar
• Mathé P . The Lepskiĭ principle revisited. Inverse Probl. 2006;22(3):L11–L15.
   Web of Science ®Google Scholar
• Bauer F , Pereverzev S . Regularization without preliminary knowledge of smoothness and error behaviour. Eur J Appl Math. 2005;16(3):303–317.
   Web of Science ®Google Scholar
• Bauer F . An alternative approach to the oblique derivative problem in potential theory [PhD thesis]. Aachen: Universität Kaiserslautern; 2004.
   Google Scholar
• Neubauer A . On Landweber iteration for nonlinear ill-posed problems in Hilbert scales. Numer Math. 2000;85(2):309–328.
   Web of Science ®Google Scholar
• Egger H , Neubauer A . Preconditioning Landweber iteration in Hilbert scales. Numer Math. 2005;101(4):643–662.
   Web of Science ®Google Scholar
• Pricop M . Tikhonov regularization in Hilbert scales for nonlinear statistical inverse problems. Uelvesbüll: Der Andere Verlag; 2007.
   Google Scholar
• Renardy M , Rogers RC . An introduction to partial differential equations. 2nd ed. Vol. 13, Texts in applied mathematics. New York (NY): Springer-Verlag; 2004.
   Google Scholar
• Goldenshluger A , Nemirovski A . On spatially adaptive estimation of nonparametric regression. Math Methods Statist. 1997;6(2):135–170.
   Google Scholar
• Avery M . Literature review for local polynomial regression, unpublished manuscript; 2013. Available from: http://www4.ncsu.edu/mravery/AveryReview2.pdf
   Google Scholar
• Adams RA , Fournier J . Sobolev spaces. 2nd ed. Vol. 140, Pure and applied mathematics. Oxford: Elsevier Science; 2003.
   Google Scholar
• Carbery A , Wright J . Distributional and lq norm inequalities for polynomials over convex bodies in R n . Math Res Lett. 2001;8(3):233–248.
   Web of Science ®Google Scholar
• Janson S . Gaussian Hilbert spaces. Vol. 129, Cambridge tracts in mathematics. Cambridge: Cambridge University Press; 1997.
   Google Scholar
• Bagnoli M , Bergstrom T . Log-concave probability and its applications. Econom Theory. 2005;26(2):445–469.
   Web of Science ®Google Scholar
• Xia Y , Li WK . Asymptotic behavior of bandwidth selected by the cross-validation method for local polynomial fitting. J Multivariate Anal. 2002;83(2):265–287.
   Web of Science ®Google Scholar
• Li Q , Racine J . Cross-validated local linear nonparametric regression. Statist Sinica. 2004;14(2):485–512.
   Web of Science ®Google Scholar
• Fan J , Gijbels I . Local polynomial modelling and its applications. Vol. 66, Monographs on statistics and applied probability. London: Chapman & Hall; 1996.
   Google Scholar
• Lepskiĭ OV . Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor Veroyatnost i Primenen. 1991;36(4):645–659.
   Web of Science ®Google Scholar
• Lepskiĭ OV . Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation. Adaptive estimates. Teor Veroyatnost i Primenen. 1992;37(3):468–481.
   Web of Science ®Google Scholar
• Hall PG , Racine JS . Infinite order cross-validated local polynomial regression. Department of Economics Working Papers McMaster University;2013.
   Google Scholar
• Chua SK , Wheeden RL . A note on sharp 1-dimensional Poincaré inequalities. Proc Amer Math Soc. 2006;134(8):2309–2316. electronic.
   Web of Science ®Google Scholar
• Talenti G . Best constant in Sobolev inequality. Ann Mat Pura Appl (4). 1976;110:353–372.
   Google Scholar
• Taylor ME . Partial differential equations. I. Vol. 115, Applied mathematical sciences. New York (NY): Springer-Verlag; 1996.
   Google Scholar
• Payne LE , Weinberger HF . An optimal Poincaré inequality for convex domains. Arch Rat Mech Anal. 1960;1960(5):286–292.
   Google Scholar