Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
- 65J20 Improperly posed problems

**Résumé:** We introduce and analyse numerical methods for the treatment of
inverse problems, based on an adaptive wavelet Galerkin discretization.
These methods combine the theoretical advantages of the
wavelet-vaguelette decomposition (WVD) in terms of optimally
adapting to the unknown smoothness of the solution, together with the
numerical simplicity of Galerkin methods. Two strategies
are proposed: the first one simply combines a thresholding
algorithm on the data with a Galerkin inversion on a fixed linear space,
while the second one performs the inversion through an adaptive
procedure in which a smaller space adapted to the solution
is iteratively constructed. For both methods, we recover the same
minimax rates
achieved by WVD for various function classes modeling the solution.

**Mots Clés:** *Statistical inverse problems ; Galerkin methods ; Wavelets and nonlinear methods ;
Besov spaces ; Minimax estimation
*

**Date:** 2002-07-08

**Prépublication numéro:** *PMA-749*

**Pdf file : **PMA-749.pdf