AI 中文总结
提出一种在线辛滤波算法，利用移动传感器测量重构参数化哈密顿偏微分方程的解，保证辛结构和能量守恒，并证明动态传感器布置优于静态布置。
AI 中文摘要
虽然哈密顿系统前向模拟的保结构方法已得到充分发展，但其数据同化和滤波对应部分的数学基础仍相对欠发达，尤其是在严格精度保证方面。为弥补这一差距，我们提出一种在线滤波算法，从有限测量中重构未知函数 $u^\dagger$，假设 $u^\dagger$ 求解一个具有未知输入的参数化哈密顿偏微分方程。该方法使用低维辛近似空间，这些空间根据偏微分方程模型随时间演化，并由测量数据提供信息。该方法还可与一种动态传感器布置策略耦合，该策略旨在优化重构稳定性。我们推导出关于稳定性和最佳逼近的误差界，并表明重构保持在辛空间中，且能量守恒误差不超过逼近误差。数值实验证明了该方法的有效性，以及动态传感器布置相对于静态布置的优势。
英文摘要
While structure-preserving methods for the forward simulation of Hamiltonian systems are well established, the mathematical foundations of their data assimilation and filtering counterparts remain comparatively less developed, particularly regarding rigorous accuracy guarantees. To address this gap, we propose an online filtering algorithm to reconstruct an unknown function $u^\dagger$ from finitely many measurements, assuming that $u^\dagger$ solves a parametric Hamiltonian PDE with unknown inputs. The method uses low-dimensional symplectic approximation spaces that evolve in time according to the PDE model and informed by the measurement data. The method can also be coupled with a dynamical sensor-placement strategy designed to optimize reconstruction stability. We derive error bounds in terms of stability and best approximation, and show that the reconstruction remains in a symplectic space and preserves energy up to the approximation error. Numerical experiments demonstrate the effectiveness of the method and the benefits of dynamic over static sensor placement.
