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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
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Abstract

In this article we tackle the problem of inverse non linear ill-posedproblems from a statistical point of view. We discuss the problemof estimating an indirectly observed function, without priorknowledge 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 estimatorprojectionestimatorpenalized estimationmodel selection

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 173 - 191
Copyright
© EDP Sciences, SMAI, 2010

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