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

Minimax Goodness-of-Fit Testing in Ill-Posed Inverse Problems with Partially Unknown Operators