卡尔曼滤波的时空分解
Space-Time Decomposition of Kalman Filter
AI总结：
本文提出DD-KF方法，将Schwarz域分解与时间并行思想结合，对卡尔曼滤波在时空方向上进行划分，通过修正项保证全局解一致性，并用浅水方程验证其在天气与洋流监测预报中的可靠性。
摘要：
我们提出了一种对卡尔曼滤波（简称KF）的创新性解释，结合了Schwarz域分解（DD）与时间并行（PinT）方法的思想。此后我们将其称为DD-KF。与已经纳入KF及其他状态估计模型的标准DD方法（在时间循环内实现直接的数据并行）不同，DD-KF从一开始就在空间和时间两个方向/步骤上对整个模型进行划分，包括滤波方程和动态模型。因此，我们得到局部KF，在局部域上以更小的维度重现原始滤波器。同时，子问题可以并行求解。为了强制局部解在重叠区域上匹配，进而获得与KF相同的全局解，局部KF被略微修改，增加了一个修正项，用于跟踪相邻子域对重叠区域的贡献。该修正项平衡了重叠区域上的局部化误差，作为对局部解的正则化约束。此外，这种局部化将远程观测从每个被分析位置中排除，改善了误差协方差矩阵的条件数。作为动态模型，我们考虑浅水方程，它可以被视为一个可靠的工具，用于对DD-KF在天气系统和洋流的监测与预报中的可靠性评估进行概念验证。
英文摘要：
We present an innovative interpretation of Kalman Filter (KF, for short) combining the ideas of Schwarz Domain Decomposition (DD) and Parallel in Time (PinT) approaches. Thereafter we call it DD-KF. In contrast to standard DD approaches which are already incorporated in KF and other state estimation models, implementing a straightforward data parallelism inside the loop over time, DD-KF ab-initio partitions the whole model, including filter equations and dynamic model along both space and time directions/steps. As a consequence, we get local KFs reproducing the original filter at smaller dimensions on local domains. Also, sub problems could be solved in parallel. In order to enforce the matching of local solutions on overlapping regions, and then to achieve the same global solution of KF, local KFs are slightly modified by adding a correction term keeping track of contributions of adjacent subdomains to overlapping regions. Such a correction term balances localization errors along overlapping regions, acting as a regularization constraint on local solutions. Furthermore, such a localization excludes remote observations from each analyzed location improving the conditioning of the error covariance matrices. As dynamic model we consider Shallow Water equations which can be regarded a consistent tool to get a proof of concept of the reliability assessment of DD-KF in monitoring and forecasting of weather systems and ocean currents
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