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

Abstract

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.

KEYWORDS:

• Bayesian calibration
• Dimension reduction
• Functional input
• Sequential design

Supplementary Materials

Appendices.pdf: Appendix A plots the KL basis functions for various stochastic process priors. Appendix B provides a proof of Proposition 3.1. Appendix C proposes an MCMC algorithm for sampling from the emulator-based posterior distribution (16). Appendix D gives a proof of Theorem 5.1. Appendix E presents modified versions of two existing sequential design criteria for high-dimensional parameter calibration. Appendix F provides an example based on the steady state groundwater flow equation. Appendix G assesses the effects of using a truncated KL expansion to represent the functional input and replacing the simulator with an emulator on posterior inference.

code.zip file: This file gives Matlab codes for reproducing the results in Section 6, Appendix F, and Appendix G, where the codes were run in Matlab® R2020a (MathWorks, Inc., Natick, MA, USA) to obtain the results reported in the article.

Acknowledgments

We thank the editor, associate editor, two reviewers, and C. F. Jeff Wu for comments that helped improve the article significantly.

Additional information

Funding

The work described in this article was supported by two grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (General Research Fund Project Nos.: CityU 11201117 and CityU 11205118).