关于反问题的不稳定性机制
On instability mechanisms for inverse problems
AI总结：
本文提出三种基于压缩性质的反问题不稳定性机制，应用于唯一延拓、向后热方程及Calderón型问题，并估计可控性成本。
摘要：
本文提出了线性和非线性反问题的三种稳健的不稳定性机制。所有这些机制都基于强压缩性质（在奇异值或熵数界意义上），我们分别通过相应正算子的强全局光滑性、仅弱全局光滑性或微局部光滑性推导出这些性质。作为应用，我们例如为唯一延拓、向后热方程以及一般几何中（可能具有粗糙系数）的线性和非线性Calderón型问题提供了新的不稳定性论证。我们的不稳定性机制在控制理论背景下也可能引起关注，为相当一般设置下的（近似）可控性成本提供估计。这是同一作者的文章“关于反问题的不稳定性机制”Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp的修订版。
英文摘要：
In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderón type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings. This is a revised version of the article ``On instability mechanisms for inverse problems'' Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp by the same authors.
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