Stochastic Polynomial Interpolation for Uncertainty Quantification with Computer Experiments

Description

The research objective of this project is to construct a statistical methodology fordesigning and analyzing computer experiments using interpolating polynomials.Multivariate polynomial metamodels are increasingly being used in computerexperiments due to the development of polynomial chaos expansions and sparse gridinterpolation for uncertainty quantification, and the popularity of response surfacemethodology and regression for model building. For deterministic computer codes (thosethat yield fixed outputs/ responses for fixed inputs), interpolating polynomialmetamodels should be used instead of noninterpolating ones for logical consistency andprediction accuracy. However, methods proposed in the literature for constructinginterpolating polynomial metamodels only provide point predictions. There is no knownmethod that can provide probabilistic statements about interpolation errors, which iscrucial for assessing the impact of potential interpolation errors on decision-making.Furthermore, in the uncertainty quantification of engineering models, grid designs orsparse grids are used to construct interpolating polynomials, resulting in highlyinflexible designs with large number of runs for moderate to high dimension problems.These designs are also nonadaptive, i.e., they do not use information in the data at handto determine the best location for the next design points. Consequently, they can beinefficient.This research will develop a stochastic interpolating polynomial (SIP) that seeks tolems discussed above. A Bayesian approach will be taken in whichinterpolation uncertainty is quantified probabilistically through the posterior distributionof the output. The proposed model will allow assessment of the effect of interpolationuncertainty on estimation of quantities of interest, optimization, and decision-makingbased on metamodel predictions. Statistical methods for designing experiments toconstruct the SIP will be developed. Algorithms for generating designs that optimizeseveral posterior measures of uncertainty about the true output will be proposed. It isexpected that these designs can be of almost any size and can adapt to the peculiaritiesof the function being modeled.Finally, this research will apply the SIP to four important classes of engineeringproblems. The basic SIP model will be extended to model experiments with multiplelevels of fidelity and functional outputs. Applications of the SIP to optimization androbust parameter design (RPD) will also be considered.This project will provide novel, flexible, and powerful statistical methods for polynomialinterpolation. The proposed methods will be important contributions to the engineeringcommunity due to the widespread acceptance and use of polynomial metamodels foruncertainty quantification in the community.