A Generalized Polynomial Chaos-Based Method for Efficient Bayesian Calibration of Uncertain Computational Models

Abstract

This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent stationary Gaussian processes with uncertain hyper-parameters describe uncertainties of the model structure and parameters, while Karhunen-Loeve expansion is adopted to spectrally represent these Gaussian processes. The Karhunen-Loeve expansion of a prior Gaussian process is projected on a generalized Polynomial Chaos basis, whereas intrusive Galerkin projection is utilized to calculate the associated coefficients of the simulator output. Bayesian inference is used to update the prior probability distribution of the generalized Polynomial Chaos basis, which along with the chaos expansion coefficients represent the posterior probability distribution. The proposed method is demonstrated for calibration of a simulator of quasi-one-dimensional flow through a divergent nozzle with uncertain nozzle area profile. The posterior distribution of the nozzle area profile and the hyper-parameters obtained using the proposed method are found to match closely with the direct Markov Chain Monte Carlo-based implementation of the Bayesian framework. Efficacy of the proposed method is demonstrated for various choices of prior. Posterior hyper-parameters of the model structural uncertainty are shown to quantify acceptability of the simulator model.