Bayesian calibration of spatially varying model parameters with high-dimensional response

This paper proposes a methodology for Bayesian calibration of spatially varying model parameters, which can be represented as random fields to account for the variability over space and across specimens. Since the calibration of such parameters is affected by observation locations, a sensor placement optimization methodology is also developed to maximize the information gain from the collected calibration data. Two approaches are incorporated to increase the computational efficiency of sensor placement optimization and Bayesian calibration: (1) the expensive physics simulation model is first substituted with a set of surrogate models using the Kriging approach; (2) when the response is high-dimensional, offering numerous candidate observation locations, the high-dimensional response is transformed to a low-dimensional space based on singular value decomposition (SVD), thus facilitating a smaller set of surrogate models. Since the model discrepancy of the simulation model with spatially varying parameters is also spatially varying, the discrepancy term is modeled as a random field, following the Kennedy and O’Hagan (KOH) framework. Due to the randomness of the calibration parameters over specimens, the computational effort in Bayesian calibration is increased, which brings significant challenge to evaluation of the expected information gain. A Gaussian copula method integrated with a particle filter approach is developed to tackle this challenge. This integration significantly increases the efficiency in estimating the information gain measured by the Kullback-Leibler (KL) divergence. The optimal sensor locations are then identified using a simulated annealing algorithm by maximizing the expected information gain. A heat transfer problem of a panel, in which the spatially varying conduction coefficient and the convection coefficient need to be calibrated, demonstrates the effectiveness of the proposed framework.