Stochastic Polynomial Interpolation for Uncertainty Quantification With Computer Experiments

Abstract

Multivariate polynomials are increasingly being used to construct emulators of computer models for uncertainty quantification. For deterministic computer codes, interpolating polynomial metamodels should be used instead of noninterpolating ones for logical consistency and prediction accuracy. However, available methods for constructing interpolating polynomials only provide point predictions. There is no known method that can provide probabilistic statements about the interpolation error. Furthermore, there are few alternatives to grid designs and sparse grids for constructing multivariate interpolating polynomials. A significant disadvantage of these designs is the large gaps between allowable design sizes. This article proposes a stochastic interpolating polynomial (SIP) that seeks to overcome the problems discussed above. A Bayesian approach in which interpolation uncertainty is quantified probabilistically through the posterior distribution of the output is employed. This allows assessment of the effect of interpolation uncertainty on estimation of quantities of interest based on the metamodel. A class of transformed space-filling design and a sequential design approach are proposed to efficiently construct the SIP with any desired number of runs. Simulations demonstrate that the SIP can outperform Gaussian process (GP) emulators. This article has supplementary material online.

• Bayesian approach
• Design and analysis of experiments
• Deterministic simulations
• Interpolation uncertainty
• Polynomial metamodels.

ACKNOWLEDGMENTS

We thank the associate editor and two referees for comments that helped improve the article significantly. This research is supported by City University of Hong Kong Start-Up Grant 7200364 and Early Career Scheme (ECS) project No. 21201414 sponsored by the Research Grants Council of Hong Kong.