Statistical Inverse Estimation in Hilbert Scales

Abstract

The recovery of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable. For greater flexibility the general problem is embedded in an abstract Hilbert scale. In the applications Sobolev scales are used. For the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product, where the operator is bounded but not necessarily compact. A lower bound to certain minimax rates is included, and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate. Examples include errors-in-variables (deconvolution) and indirect nonparametric regression. Special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation.