  We consider a Gaussian sequence model that contains ill-posed inverse
problems as special cases. We assume that the associated operator is partially
unknown in the sense that its singular functions are known and the
corresponding singular values are unknown but observed with Gaussian noise. For
the considered model, we study the minimax goodness-of-fit testing problem.
Working with certain ellipsoids in the space of squared-summable sequences of
real numbers, with a ball of positive radius removed, we obtain lower and upper
bounds for the minimax separation radius in the non-asymptotic framework, i.e.,
for fixed values of the involved noise levels. Examples of mildly and severely
ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth
sequences are examined in detail and minimax rates of goodness-of-fit testing
are obtained for illustrative purposes.
