We consider the inverse problem of reconstructing an unknown function $u$ from a finite set of measurements, under the assumption that $u$ is the trajectory of a transport-dominated problem with unknown input parameters. We propose an algorithm based on the Parameterized Background Data-Weak method (PBDW) where dynamical sensor placement is combined with approximation spaces that evolve in time. We prove that the method ensures an accurate reconstruction at all times and allows to incorporate relevant physical properties in the reconstructed solutions by suitably evolving the dynamical approximation space. As an application of this strategy we consider Hamiltonian systems modeling wave-type phenomena, where preservation of the geometric structure of the flow plays a crucial role in the accuracy and stability of the reconstructed trajectory.

翻译：我们考虑在未知函数$u$为传输主导问题（含未知输入参数）轨迹的假设下，基于有限测量集重构该未知函数的逆问题。提出一种基于参数化背景数据弱方法（PBDW）的算法，将动态传感器布设与随时间演化的逼近空间相结合。理论证明该方法能保证所有时刻的重构精度，并通过动态演化逼近空间将相关物理性质融入重构解中。作为该策略的应用，我们考虑描述波型现象的哈密顿系统，其中保持流的几何结构对重构轨迹的精度与稳定性具有关键作用。