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

该研究解决了开发有效的计算工具来处理急剧增加的观测数据的迫切需要。许多复杂系统的管理（例如，交通）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，导致预测的准确性和有用性降低。因此，重要的是要制定实用的方法来“调整”系统的概率状态，并利用观测数据减少不确定性。这种方法一般称为数据同化。我们将开发新的技术，将观测数据，以减少在国家利益的两个特定领域的预测不确定性：流体动力学（例如，洪水预报和交通管理。本文提出了一个新的数据同化框架，用于时间动力学由双曲守恒律描述的物理过程。这个框架利用了双曲系统的动力学表示，因此，可用性明确的确定性方程的时间演化的概率密度函数的因变量。这些方程往往可以推导和精确求解，产生显式的边缘和联合概率密度函数的解析解。对于双曲守恒律方程组，需要一个适当的封闭性假设。因此，所提出的框架依赖于动力学表示，其形式为联合概率密度函数的线性方程。贝叶斯更新被用来将观察到的预测。

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.