Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T16:53:03.411Z Has data issue: false hasContentIssue false
  • English
  • Français

Penalized estimators for non linear inverse problems

Published online by Cambridge University Press:  29 July 2010

Jean-Michel Loubes  and
Carenne Ludeña
Jean-Michel Loubes
Affiliation:
Institut de Mathématiques, Équipe de Statistique et Probabilités, Université Paul Sabatier, 31000 Toulouse, France
Carenne Ludeña
Affiliation:
Departamento de Matemáticas, IVIC, Venezuela
Get access

Abstract

In this article we tackle the problem of inverse non linear ill-posed problems from a statistical point of view. We discuss the problem of estimating an indirectly observed function, without prior knowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show that their rate of convergence is optimal for the given estimation procedure.

Keywords

Ill-posed Inverse ProblemsTikhonov estimatorprojection estimatorpenalized estimationmodel selection
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 173 - 191
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baraud, Y., Model selection for regression on a fixed design. Probab. Theory Relat. Fields 117 (2000) 467–493. CrossRef
L. Birgé and P. Massart, Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields. 138 (2007) 33–73. CrossRef
N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inv. Prob. 20 (2004) 1773–1789.
Bissantz, N., Hohage, T., Munk, A. and Ruymgaart, F., Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 (2007) 2610–2636. CrossRef
Bousquet, O., Concentration inequalities for sub-additive functions using the entropy method. Stoch. Inequalities Appl. 56 (2003) 213–247. CrossRef
Cavalier, L., Golubev, G.K., Picard, D. and Tsybakov, A.B., Oracle inequalities for inverse problems. Ann. Statist. 30 (2002) 843–874. Dedicated to the memory of Lucien Le Cam.
Chow, P. and Khasminskii, R., Statistical approach to dynamical inverse problems. Commun. Math. Phys. 189 (1997) 521–531. CrossRef
Donoho, D., Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (1995) 101–126. CrossRef
H. Engl, Regularization methods for solving inverse problems, in ICIAM 99 (Edinburgh), pp. 47–62. Oxford Univ. Press, Oxford (2000).
H. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems. Math. Appl. 375. Kluwer Academic Publishers Group, Dordrecht (1996).
Gamboa, F., New Bayesian methods for ill posed problems. Statist. Decisions 17 (1999) 315–337.
Jin, Q. and Amato, U., A discrete scheme of Landweber iteration for solving nonlinear ill-posed problems. J. Math. Anal. Appl. 253 (2001) 187–203. CrossRef
Kalifa, J. and Mallat, S., Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 (2003) 58–109.
Kaltenbacher, B., Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems. Inv. Prob. 16 (2000) 1523–1539. CrossRef
Loubes, J.-M. and Ludena, C., Adaptive complexity regularization for inverse problems. Electron. J. Statist. 2 (2008) 661–677. CrossRef
Mair, B. and Ruymgaart, F., Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 (1996) 1424–1444. CrossRef
Neubauer, A., Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales. Appl. Anal. 46 (1992) 59–72. CrossRef
O'Sullivan, F., Convergence characteristics of methods of regularization estimators for nonlinear operator equations. SIAM J. Numer. Anal. 27 (1990) 1635–1649. CrossRef
Snieder, R., An extension of Backus-Gilbert theory to nonlinear inverse problems. Inv. Prob. 7 (1991) 409–433. CrossRef
Tautenhahn, U. and Qi-nian Jin, Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems. Inv. Prob. 19 (2003) 1–21. CrossRef
A.N. Tikhonov, A.S. Leonov and A.G. Yagola, Nonlinear ill-posed problems, volumes 1 and 2. Appl. Math. Math. Comput. 14. Chapman & Hall, London (1998). Translated from the Russian.