  This work deals with the ill-posed inverse problem of reconstructing a
function $f$ given implicitly as the solution of $g = Af$, where $A$ is a
compact linear operator with unknown singular values and known eigenfunctions.
We observe the function $g$ and the singular values of the operator subject to
Gaussian white noise with respective noise levels $\varepsilon$ and $\sigma$.
  We develop a minimax theory in terms of both noise levels and propose an
orthogonal series estimator attaining the minimax rates. This estimator
requires the optimal choice of a dimension parameter depending on certain
characteristics of $f$ and $A$. This work addresses the fully data-driven
choice of the dimension parameter combining model selection with Lepski's
method. We show that the fully data-driven estimator preserves minimax
optimality over a wide range of classes for $f$ and $A$ and noise levels
$\varepsilon$ and $\sigma$. The results are illustrated considering Sobolev
spaces and mildly and severely ill-posed inverse problems.
