A Local Variational Inference Framework for the Orthogonal Gaussian Process Calibration

Computer simulations are commonly used to mimic, understand, and predict true physical phenomena in many applications. Calibration aims to adjust the computer model parameters, called calibration parameters, so that the computer outputs match the physical observations as closely as possible. The orthogonal Gaussian Process calibration (OGP calibration) [J Am Stat Assoc. 112(519) (2017) 1274-1285] offers a fast convergence rate, high estimation accuracy, and convenient uncertainty quantification for the parameter estimates. However, the OGP calibration suffers from high computational complexity and is prone to local optima. It is extremely difficult for the OGP calibration to identify the optimal set of calibration parameters. In this work, we propose a gradient-free local variational inference framework for the OGP calibration to address these two issues simultaneously. Specifically, to reduce the computational complexity, an efficient algorithm is proposed to evaluate the variational density of the calibration parameters and the discrepancy between the true physical process and the calibrated computer simulation model. To prevent the estimator from falling into a local optimum, the least-squares estimate of the calibration parameters is introduced as the prior information in the variational inference framework. Results of a numerical simulation and a real-world case study demonstrate that, compared to the OGP calibration, the complexity of the proposed algorithm is greatly reduced without sacrificing too much accuracy.