  In the inverse Gaussian sequence space model with additional noisy
observations of the operator, we derive nonasymptotic minimax radii of testing
for ellipsoid-type alternatives simultaneously for both the signal detection
problem (testing against zero) and the goodness-of-fit testing problem (testing
against a prescribed sequence) without any regularity assumption on the null
hypothesis. The radii are the maximum of two terms, each of which only depends
on one of the noise levels. Interestingly, the term involving the noise level
of the operator explicitly depends on the null hypothesis and vanishes in the
signal detection case. The minimax radii are established by first showing a
lower bound for arbitrary null hypotheses and noise levels. For the upper bound
we consider two testing procedures, a direct test based on estimating the
energy in the image space and an indirect test. Under mild assumptions, we
prove that the testing radius of the indirect test achieves the lower bound,
which shows the minimax optimality of the radius and the test. We highlight the
assumptions under which the direct test also performs optimally. Furthermore,
we apply a classical Bonferroni method for making both the indirect and the
direct test adaptive with respect to the regularity of the alternative. The
radii of the adaptive tests are deteriorated by an additional log-factor, which
we show to be unavoidable. The results are illustrated considering Sobolev
spaces and mildly or severely ill-posed inverse problems.
