A non-stationary covariance-based Kriging method for metamodelling in engineering design

Metamodels are widely used to facilitate the analysis and optimization of engineering systems that involve computationally expensive simulations. Kriging is a metamodelling technique that is well known for its ability to build surrogate models of responses with non-linear behaviour. However, the assumption of a stationary covariance structure underlying Kriging does not hold in situations where the level of smoothness of a response varies significantly. Although non-stationary Gaussian process models have been studied for years in statistics and geostatistics communities, this has largely been for physical experimental data in relatively low dimensions. In this paper, the non-stationary covariance structure is incorporated into Kriging modelling for computer simulations. To represent the non-stationary covariance structure, we adopt a non-linear mapping approach based on parameterized density functions. To avoid over-parameterizing for the high dimension problems typical of engineering design, we propose a modified version of the non-linear map approach, with a sparser, yet flexible, parameterization. The effectiveness of the proposed method is demonstrated through both mathematical and engineering examples. The robustness of the method is verified by testing multiple functions under various sampling settings. We also demonstrate that our method is effective in quantifying prediction uncertainty associated with the use of metamodels. Copyright (c) 2006 John Wiley & Sons, Ltd.