Adaptive estimation for inverse problems with noisy operators

Consider an inverse problem with random noise where we want to estimate a function f. Moreover, suppose that the operator A that we need to invert is not completely known: we know its eigenfunctions and observe its singular values but with some noise. To construct our estimator θ, we minimize a modification of an unbiased risk estimator. We obtain some non-asymptotic exact oracle inequalities. Considering smooth functions in some standard classes of functions, we prove that θ is asymptotically minimax among a given class of estimators.