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ill-posed inverse problems

ESAIM: Probability and Statistics (1)

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Penalized estimators for non linear inverse problems
Jean-Michel Loubes, Carenne Ludeña
Journal:

ESAIM: Probability and Statistics / Volume 14 / 2010

Published online by Cambridge University Press:

29 July 2010, pp. 173-191

Print publication:

2010
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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.