Theory and Algorithms for sequential non-Gaussian Bayesian filtering and estimation
Seamless integration of dynamical system models with sparse measurements, called as Data Assimilation, is important in many applications like weather forecasting, socio-economics, navigation, and beyond. In order to produce accurate and efficient state forecasts using data assimilation, one needs to account for non-linearities in the dynamics of state space, and the interaction between probabilistic information of both sensor measurements and state space. From a computational viewpoint, it is desirable to have schemes that require low computing cost and are easy to implement on a code. In this thesis, we develop an efficient non-intrusive sequential Data assimilation scheme that utilizes Stochastic collocation-based Polynomial Chaos expansion (PCE) to propagate the uncertainty in a non-linear dynamic system and Gaussian Mixture Model (GMM) priors to represent the statistics of PC expansion forecasts. First, we represent the uncertainty in a dynamical system using PCE and propagate it using the stochastic collocation method until an assimilation time. Then, we convert the polynomial basis prior to its equivalent Karhunen-Loeve (KL) form, fit a GMM in the subspace and perform a Bayesian filtering step. Thereafter, the posterior polynomial basis is recovered from the posterior GMM in the KL form, and uncertainty propagation is continued using the stochastic collocation method. The derivation and new equations required for the above conversions are presented. We apply the new scheme to an illustrative population growth dynamics application and a complex fluid flow problem for demonstrating its capabilities. In both cases, our filter accurately captures the non-Gaussian statistics compared to the Polynomial Chaos - Ensemble Kalman Filter and the Polynomial Chaos - Error Subspace Statistical Estimation.