Penalized estimators for non linear inverse problems
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 173-191.

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.

DOI : https://doi.org/10.1051/ps:2008024
Classification : 60G17,  62G07
Mots clés : ill-posed inverse problems, Tikhonov estimator, projection estimator, penalized estimation, model selection
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     author = {Loubes, Jean-Michel and Lude\~na, Carenne},
     title = {Penalized estimators for non linear inverse problems},
     journal = {ESAIM: Probability and Statistics},
     pages = {173--191},
     publisher = {EDP-Sciences},
     volume = {14},
     year = {2010},
     doi = {10.1051/ps:2008024},
     mrnumber = {2741964},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2008024/}
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Loubes, Jean-Michel; Ludeña, Carenne. Penalized estimators for non linear inverse problems. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 173-191. doi : 10.1051/ps:2008024. http://www.numdam.org/articles/10.1051/ps:2008024/

[1] Y. Baraud, Model selection for regression on a fixed design. Probab. Theory Relat. Fields 117 (2000) 467-493. | Zbl 0997.62027

[2] L. Birgé and P. Massart, Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields. 138 (2007) 33-73. | Zbl 1112.62082

[3] 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. | Zbl 1077.65060

[4] N. Bissantz, T. Hohage, A. Munk and F. Ruymgaart, Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 (2007) 2610-2636.

[5] O. Bousquet, Concentration inequalities for sub-additive functions using the entropy method. Stoch. Inequalities Appl. 56 (2003) 213-247. | Zbl 1037.60015

[6] L. Cavalier, G.K. Golubev, D. Picard and A.B. Tsybakov, Oracle inequalities for inverse problems. Ann. Statist. 30 (2002) 843-874. Dedicated to the memory of Lucien Le Cam. | Zbl 1029.62032

[7] P. Chow and R. Khasminskii, Statistical approach to dynamical inverse problems. Commun. Math. Phys. 189 (1997) 521-531. | Zbl 1090.62523

[8] D. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (1995) 101-126. | Zbl 0826.65117

[9] H. Engl, Regularization methods for solving inverse problems, in ICIAM 99 (Edinburgh), pp. 47-62. Oxford Univ. Press, Oxford (2000). | Zbl 0992.65058

[10] H. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems. Math. Appl. 375. Kluwer Academic Publishers Group, Dordrecht (1996). | Zbl 0859.65054

[11] F. Gamboa, New Bayesian methods for ill posed problems. Statist. Decisions 17 (1999) 315-337. | Zbl 0952.62027

[12] Q. Jin and U. Amato, A discrete scheme of Landweber iteration for solving nonlinear ill-posed problems. J. Math. Anal. Appl. 253 (2001) 187-203. | Zbl 0992.47032

[13] J. Kalifa and S. Mallat, Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 (2003) 58-109. | Zbl 1102.62318

[14] B. Kaltenbacher, Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems. Inv. Prob. 16 (2000) 1523-1539. | Zbl 0978.65045

[15] J.-M. Loubes and C. Ludena, Adaptive complexity regularization for inverse problems. Electron. J. Statist. 2 (2008) 661-677.

[16] B. Mair and F. Ruymgaart, Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 (1996) 1424-1444. | Zbl 0864.62020

[17] A. Neubauer, Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales. Appl. Anal. 46 (1992) 59-72. | Zbl 0788.47052

[18] F. O'Sullivan, Convergence characteristics of methods of regularization estimators for nonlinear operator equations. SIAM J. Numer. Anal. 27 (1990) 1635-1649. | Zbl 0724.65136

[19] R. Snieder, An extension of Backus-Gilbert theory to nonlinear inverse problems. Inv. Prob. 7 (1991) 409-433. | Zbl 0737.35155

[20] U. Tautenhahn and Qi-nian Jin, Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems. Inv. Prob. 19 (2003) 1-21. | Zbl 1030.65061

[21] 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. | Zbl 0920.65038

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