A Gaussian Process Emulator Based Approach for Bayesian Calibration of a Functional Input

Bayesian calibration of a functional input/parameter to a time-consuming simulator based on a Gaussian process (GP) emulator involves two challenges that distinguish it from other parameter calibration problems. First, one needs to specify a flexible stochastic process prior for the input, and reduce it to a tractable number of random variables. Second, a sequential experiment design criterion that decreases the effect of emulator prediction uncertainty on calibration results is needed and the criterion should be scalable for high-dimensional input and output. In this article, we address these two issues. For the first issue, we employ a GP with a prior density for its correlation parameter as prior for the functional input, and the Karhunen-Loève (KL) expansion of this non-Gaussian stochastic process to reduce its dimension. We show that this prior gives far more robust inference results than a GP with a fixed correlation parameter. For the second issue, we propose the weighted prediction variance (WPV) criterion (with posterior density of the calibration parameter as weight) and prove the consistency of the sequence of emulator-based likelihoods given by the criterion. The proposed method is illustrated with examples on hydraulic transmissivity estimation for groundwater models.