On the Instability Issue of Gradient-Enhanced Gaussian Process Emulators for Computer Experiments
Xu He, Peter Chien · SIAM/ASA Journal on Uncertainty Quantification · 2018
How to incorporate the gradient information of a computer code is an important problem in computer experiments. The gradient-enhanced Gaussian process emulator is widely used to jointly model the output and its gradients from a computer model. The emulator can then be used for prediction, optimization, inverse analysis, sensitivity analysis, and uncertainty propagation. Often there is a trade-off between the statistical and numeric accuracy of an emulator. Indeed, the more data that are available, the better the statistical accuracy that the emulator will typically possess; on the other hand, the emulator can encounter greater numerical problems as the sample size grows, which adversely affects its accuracy. The gradient-enhanced Gaussian process emulator is known to possess more numeric problems than in many multivariate cases because of the dependence of the model output and each gradient output. We derive a statistical theory to understand why this problem occurs for cases with the squared-exponential correlation function and Cartesian product and sparse grid designs. This theory compares the smallest eigenvalue of the covariance matrix of the ordinary Gaussian process emulator and its counterpart of the gradient-enhanced Gaussian process emulator for this case. These results explicitly show that the latter will decay much faster as the distance of points approaches to zero. Examples are provided to illustrate the derived results.