This paper proposes stochastic spectral representation for Bayesian calibration of computer simulators with parametric and model structure uncertainty with unknown/poorly known prior hyper-parameters. The methodology is specifically developed for calibration of simulators with spatially/temporally varying parameters. Uncertainty in parameters and model structure is represented using independent stationary Gaussian processes with uncertain hyper-parameters. Gaussian processes are spectrally represented using Karhunnen-Loeve expansion. A methodology based on decomposition of parametric space and orthogonal polynomials defined on the decomposed space is developed for evaluating coefficients of Karhunnen-Loeve expansion of Gaussian process with uncertain hyper-parameters. Galerkin projection method is used to evaluate the resultant stochastic spectral decomposition of predicted system response. Bayesian inference is used to update the prior probability distribution of the polynomial chaos basis. The proposed method is demonstrated for calibration of a simulator of quasi-one dimensional flow through a convergent-divergent nozzle.

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