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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Adaptive wavelet Galerkin methods for linear inverse problems


Code(s) de Classification MSC:

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