Collaborative Proposal: Density-enhanced data assimilation for hyperbolic balance laws

该研究解决了迫切需要开发有效的计算工具来处理急剧增加的观测数据量。许多复杂系统（如交通）的管理必须面对其当前和未来状态的不确定性。这种不确定性通常会随着时间的推移而增加，从而导致预测的准确性和有用性降低。因此，开发实用的方法来“调整”系统的概率状态并利用观测数据减少不确定性是很重要的。这种方法被广泛地称为数据同化。我们将开发结合观测数据的新技术，以减少流体动力学（例如，洪水预报）和交通管理这两个国家感兴趣的特定领域预测的不确定性。两者都对我们社会的可持续发展至关重要。我们建议为时间动力学由双曲守恒定律描述的物理过程开发一种新的数据同化框架。该框架利用了双曲系统的动力学表示，因此，因变量的概率密度函数的时间演化的显式确定性方程的可用性。这些方程通常可以精确地推导和求解，得到边际和联合概率密度函数的显式解析解。对于双曲守恒律系统，需要一个适当的闭包假设。因此，所提出的框架依赖于动力学表示，它采用联合概率密度函数的线性方程形式。利用贝叶斯更新将观测结果合并到预测中。

The research addresses the urgent need to develop efficient computational tools to process the dramatically increasing amounts of observational data. Management of many complex systems (e.g., traffic) has to confront the uncertainty in both their current and future state. This uncertainty typically increases with time, leading to less accurate and useful predictions. Thus, it is important to develop practical methods for "adjusting" the probabilistic state of the system and reducing uncertainty using observational data. This approach is broadly referred to as data assimilation. We will develop novel techniques for incorporating observational data to reduce uncertainty in predictions in two particular areas of national interest: fluid dynamics (e.g., flood forecasting) and traffic management. Both are of vital importance to sustainable development of our society.We propose to develop a novel data assimilation framework for physical processes whose time-dynamics is described by hyperbolic conservation laws. This framework takes advantage of a kinetic representation of hyperbolic systems and, thus, availability of explicit deterministic equations for the time evolution of probability density function for dependent variables. These equations can often be derived and solved exactly, yielding explicit analytical solutions for the marginal and joint probability density functions. For systems of hyperbolic conservation laws an appropriate closure assumption is needed. Thus, the proposed framework relies on the kinetic representation, which takes the form of linear equations for joint probability density functions. Bayesian updating is utilized to incorporate observations into the prediction.