  Testing of hypotheses is a well studied topic in mathematical statistics.
Recently, this issue has also been addressed in the context of Inverse
Problems, where the quantity of interest is not directly accessible but only
after the inversion of a (potentially) ill-posed operator. In this study, we
propose a regularized approach to hypothesis testing in Inverse Problems in the
sense that the underlying estimators (or test statistics) are allowed to be
biased. Under mild source-condition type assumptions we derive a family of
tests with prescribed level $\alpha$ and subsequently analyze how to choose the
test with maximal power out of this family. As one major result we prove that
regularized testing is always at least as good as (classical) unregularized
testing. Furthermore, using tools from convex optimization, we provide an
adaptive test by maximizing the power functional, which then outperforms
previous unregularized tests in numerical simulations by several orders of
magnitude.
