The recovery of signals from indirect measurements, blurred by random
noise, is considered under the assumption that prior knowledge regardi
ng the smoothness of the signal is available. For greater flexibility
the general problem is embedded in an abstract Hilbert scale. In the a
pplications Sobolev scales are used. For the construction of estimator
s we employ preconditioning along with regularized operator inversion
in the appropriate inner product, where the operator is bounded but no
t necessarily compact. A lower bound to certain minimax rates is inclu
ded? and it is shown that in generic examples the proposed estimators
attain the asymptotic minimax rate. Examples include errors-in-variabl
es (deconvolution) and indirect nonparametric regression. Special inst
ances of the latter are estimation of the source term in a differentia
l equation and the estimation of the initial state in the heat equatio
n.