Testing inverse problems: A direct or an indirect problem?

In this paper, we consider ill-posed inverse problem models Y = Tf + ε ξ where T denotes a compact operator, ε a noise level, ξ a Gaussian white noise and f the function of interest. Recently, minimax rates of testing in such models have been obtained in various situations, both from asymptotic and non-asymptotic point of views. Nevertheless, it seems necessary to propose test strategies attaining these rates, being easy to implement and robust with respect to the characteristics of the operator. In particular, we prove that the inversion of the operator is not always necessary. This result provides interesting perspectives, for instance in the specific cases where the operator is difficult to handle.