MIXED COVARIANCE FUNCTION KRIGING MODEL FOR UNCERTAINTY QUANTIFICATION

摘要

In this paper, we develop a mixed covariance function Kriging (MCF-Kriging) model for uncertainty quantification. The mixed covariance function is a linear combination of a traditional stationary covariance function and a nonstationary covariance function constructed by the inner product of orthonormal polynomial basis functions. We use a weight matrix to control the contribution of each polynomial basis to the whole model representation, and a trade-off parameter is used to balance the contribution of the two different covariance functions. The optimal values of these model hyperparameters are obtained through an iterative algorithm derived by maximum likelihood estimation (MLE), and sparse representation is achieved automatically in the MLE step by removing the basis functions with small contribution. Additionally, the hyperparameters of stationary covariance function are tuned by minimizing the leave-one-out cross-validation error of the surrogate model. For validation, we investigate three benchmark test functions with different dimensionalities, and compare the accuracy and efficiency with the state-of-art sequential PC-Kriging and optimal PC-Kriging models. The results show that the MCF-Kriging model provides comparable performance compared to the two PC-Kriging models for nonlinear problems, that are moderate and even high-dimensional. Finally, we apply our model to a heat conduction problem to demonstrate its effectiveness in engineering application.