一种面向非线性偏微分方程的结构感知数据同化方法
A Structurally Informed Data Assimilation Approach for Nonlinear Partial Differential Equations
AI总结：
针对非线性偏微分方程中不连续状态变量导致ETKF数据同化因高斯假设产生偏差的问题，提出基于状态梯度二阶矩并结合聚类纳入不连续区域信息的结构感知非高斯先验方法，在浅水波方程上的表现优于含增强技术的ETKF。
摘要：
集合变换卡尔曼滤波（Ensemble transform Kalman filtering, ETKF）数据同化方法常被用于将已有观测数据与数值模拟相结合，以获取动力系统中统计上准确可靠的状态表征。但众所周知，对于非线性偏微分方程中普遍存在的具有不连续剖面的状态变量，常用的高斯分布假设会引入偏差。本研究设计了一种全新的结构感知非高斯先验，该先验利用模拟状态变量的统计信息。具体而言，我们基于状态变量梯度信息的二阶矩构建了新的权重矩阵，用以替代ETKF数据同化框架中用于模型与数据折衷的先验协方差矩阵。我们还通过聚类技术调整权重矩阵，使其纳入不连续区域的信息。数值实验表明，即使ETKF采用膨胀和局地化技术进行增强，该新方法在浅水波方程上得到的估计结果也比ETKF的结果更准确。
英文摘要：
Ensemble transform Kalman filtering (ETKF) data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed non-Gaussian prior that exploits statistical information from the simulated state variables. In particular, we construct a new weighting matrix based on the second moment of the gradient information of the state variable to replace the prior covariance matrix used for model/data compromise in the ETKF data assimilation framework. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using ETKF on shallow water equations, even when ETKF is enhanced with inflation and localization techniques.
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