  Ill-posed inverse problems arise in various scientific fields. We consider
the signal detection problem for mildly, severely and extremely ill-posed
inverse problems with $l^q$-ellipsoids (bodies), $q\in(0,2]$, for Sobolev,
analytic and generalized analytic classes of functions under the Gaussian white
noise model. We study both rate and sharp asymptotics for the error
probabilities in the minimax setup. By construction, the derived tests are,
often, nonadaptive. Minimax rate-optimal adaptive tests of rather simple
structure are also constructed.
