Nonlinear estimation for linear inverse problems with error in the operator.pdf

a r X i v : 0 8 0 3 .1 9 5 6 v 1 [ m a t h .S T ] 1 3 M a r 2 0 0 8 The Annals of Statistics 2008, Vol. 36, No. 1, 310–336 DOI: 10.1214/009053607000000721 c? Institute of Mathematical Statistics, 2008 NONLINEAR ESTIMATION FOR LINEAR INVERSE PROBLEMS WITH ERROR IN THE OPERATOR1 By Marc Hoffmann and Markus Reiss University of Marne-la-Valle?e and University of Heidelberg We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their per- formances depend on the underlying structure of the operator, quan- tified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov classes. 1. Introduction. Linear inverse problems with error in the operator. We want to recover f ∈L2(D), where D is a domain in Rd, from data gε =Kf + εW? ,(1.1) where K is an unknown linear operator K :L2(D)→ L2(Q), Q is a domain in Rq, W? is Gaussian white noise and ε 0 is the noise level. We do not know K exactly, but we have access to Kδ =K + δB?.(1.2) The process Kδ is a blurred version of K, polluted by a Gaussian opera- tor white noise B? with a noise level δ 0. The operator K acting on f is unknown and treated as a nuisance parameter. However, preliminary statis- tical inference about K is possible, with an accuracy governed by δ. Another equivalent approach is to consider that for experimental reasons we never have access to K in practice, but rather to Kδ . The error level δ can be linked to the accuracy of supplementary experiments; see Efromovich and Koltchinskii [11] and the examples below. In most interesting cases K?1 Received December 2004; revised April 2007. 1Supported in part by the European research training network Dynstoch. AMS 2000 subject classifications. 65J20, 62G07. Key words and phrases. Statistical inverse problem, Galerkin projection method, wavelet thresholding, minimax rate, degree of ill-posedness, mat