Uncertainty principles for inverse source and inverse scattering problems (C06)

逆源和逆散射问题是不适定逆问题，其目标是从相关辐射或散射波的远程观测中恢复有关源或散射体的信息。观测数据中包含的结构信息可以用来建立唯一性结果，设计新的重构算法，提高算法的稳定性。在这个项目中，我们研究了两种特殊类型的结构信息：不确定性原理和单调关系。我们将重点关注(1)宽带多频观测源问题的不确定性原理、稳定性结果和重建算法；(2)结合远场分裂和相位恢复；(3)基于单调性的形状表征的稳定数值实现；(4)基于时间相关逆散射问题的单调性原理和基于单调性的形状表征。

Inverse source and inverse scattering problems are ill-posed inverse problems, where the goal is to recover information about sources or scatterers from remote observations of associated radiated or scattered waves. Structural information contained in the observed data can be used to establish uniqueness results, devise new types of reconstruction algorithms, and improve their stability. In this project, we study two particular types of such structural information: uncertainty principles and monotonicity relations. We will focus on (1) uncertainty principles, stability results, and reconstruction algorithms for source problems with broadband multifrequency observations, (2) combining far field splitting and phase retrieval, (3) a stable numerical implementation of monotonicity-based shape characterizations, and (4) monotonicity principles and monotonicity-based shape characterizations for time-dependent inverse scattering problems.