A non‐Gaussian Bayesian filter for sequential data assimilation with non‐intrusive polynomial chaos expansion

Abstract

Non-Gaussian data assimilation is vital for several applications with nonlinear dynamical systems, including geosciences, socio-economics, infectious disease modeling, and autonomous navigation. Widespread adoption of non-Gaussian data assimilation requires easy-to-implement schemes. We develop, implement, and apply an efficient nonlinear non-Gaussian data assimilation scheme using non-intrusive stochastic collocation-based polynomial chaos expansion (PCE) and Gaussian mixture model (GMM) priors fit to the state's uncertainty. 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 filter.

Sampling‐Based Methods for Uncertainty Propagation in Flood Modeling Under Multiple Uncertain Inputs: Finding Out the Most Efficient Choice

“…Therefore, only the QMC alternative with the Hammersley‐based sequences is considered for probabilistic flood modeling. In the non‐intrusive stochastic collocation approach (Avasarala & Subramani, 2021; Eldred, 2009; Zio & Rochinha, 2012), global orthogonal polynomials can be used as continuous basis functions to span the uncertainty space (Shaw & Kesserwani, 2020; Xiu & Karniadakis, 2002). However, this choice is not suited for probabilistic flood modeling that needs an accurate representation of non‐smooth responses in the quantities of interest that manifest in discrete histograms with multimodalities (Abgrall & Mishra, 2017; Shaw et al., 2020).…”