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.

翻译：假设检验是数理统计学中一个研究充分的课题。近年来，该问题在反问题背景下也得到了关注，其中感兴趣的变量无法直接获取，只能在（可能）不适定算子反演后才能得到。在本研究中，我们提出了一种针对反问题中假设检验的正则化方法，即允许基础估计量（或检验统计量）存在偏差。在温和源条件型假设下，我们推导出一族具有给定显著性水平$\alpha$的检验，随后分析如何从这族检验中选择功效最大的检验。作为主要结果之一，我们证明了正则化检验始终至少与（经典）非正则化检验一样好。此外，利用凸优化工具，我们通过最大化功效泛函提供了一种自适应检验，该检验在数值模拟中比之前的非正则化检验高出数个数量级。