Improvements to gradient-enhanced kriging using a Bayesian interpretation

Abstract

Cokriging is a flexible tool for constructing surrogate models on the outputs of computer models. It can readily incorporate gradient information, in which form it is named gradient-enhanced Kriging (GEK), and promises accurate surrogate models in >10 dimensions with a moderate number of sample locations for sufficiently smooth responses. However, GEK suffers from several problems: poor robustness and ill-conditionedness of the surface. Furthermore it is unclear how to account for errors in gradients, which are typically larger than errors in values. In this work we derive GEK using Bayes’ Theorem, which gives an useful interpretation of the method, allowing construction of a gradienterror contribution. The Bayesian interpretation suggests the “observation error” as a proxy for errors in the output of the computer model. From this point we derive analytic estimates of robustness of the method, which can easily be used to compute upper bounds on the correlation range and lower bounds on the observation error. We thus see that by including the observation error, treatment of errors and robustness go hand in hand. The resulting GEK method is applied to uncertainty quantification for two test problems.