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：我们考虑从有限测量集重构未知函数$u$的反问题，假设$u$是具有未知输入参数的输运主导问题的轨迹。我们提出一种基于参数化背景数据弱方法（PBDW）的算法，该方法将动态传感器布局与随时间演化的逼近空间相结合。我们证明该方法能确保在所有时刻实现精确重构，并通过适当演化动态逼近空间，使重构解能够融入相关物理特性。作为该策略的应用，我们考虑模拟波动型现象的哈密顿系统，其中流动几何结构的保持对重构轨迹的精度和稳定性起着至关重要的作用。