高振荡力学系统的平衡资料同化
Balanced data assimilation for highly-oscillatory mechanical systems
摘要：
数据同化算法是利用部分观测和噪声观测来估计动力系统的状态。集合卡尔曼滤波由于其简单性和鲁棒性而成为一种流行的数据同化方案。然而，集合卡尔曼滤波器也有局限性，由于其固有的高斯和线性假设。这些限制可以在动态不一致的状态估计中表现出来。本文研究了具有满足一定平衡关系的动力学行为的高振荡哈密顿系统。我们首先证明，标准的集合卡尔曼滤波器可以导致不满足这些平衡关系的估计，最终导致滤波发散。我们还提出了两种补救措施，这种现象的混合时间步进计划和合奏为基础的惩罚方法。这些修改的标准集合卡尔曼滤波器的效果进行了讨论，并证明了两个模型的情况下数值。首先，我们考虑高度振荡的哈密顿系统的平衡运动，其次，我们研究热嵌入高度振荡的哈密顿系统。第一种情况是相关的应用气象学，而第二种情况是相关的数据同化分子动力学的应用。
英文摘要：
Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, the ensemble Kalman filter also has limitations due to its inherent Gaussian and linearity assumptions. These limitations can manifest themselves in dynamically inconsistent state estimates. We investigate this issue in this paper for highly oscillatory Hamiltonian systems with a dynamical behavior which satisfies certain balance relations. We first demonstrate that the standard ensemble Kalman filter can lead to estimates which do not satisfy those balance relations, ultimately leading to filter divergence. We also propose two remedies for this phenomenon in terms of blended time-stepping schemes and ensemble-based penalty methods. The effect of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for two model scenarios. First, we consider balanced motion for highly oscillatory Hamiltonian systems and, second, we investigate thermally embedded highly oscillatory Hamiltonian systems. The first scenario is relevant for applications from meteorology while the second scenario is relevant for applications of data assimilation to molecular dynamics.
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