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

Stanford University, Stanford CA

Investigators

Abstract

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.