Data assimilation for models with parametric uncertainty

Modeling the behavior of complex systems is notoriously difficult given approximate simulation models, parametric uncertainty, and limited and noisy data. To address such difficulty and harness information from both model forecast and observation, we propose a novel particle filter framework with the generalized polynomial chaos (gPC) method. By constructing a gPC expansion for the system state of interest, our framework delivers a system assimilation procedure that updates gPC coefficients when observations of the system are available, and whose forward model is defined by the stochastic Galerkin method. In this way, one can not only estimate the system state for specific realizations but also its statistical moments, and even the probability density function. The effectiveness of the proposed scheme is demonstrated through four numerical examples.