A regularization method for constructing trend function in Kriging model

Kriging is a popular surrogate for approximating computationally expensive computer experiments. When sample points are limited, it is difficult to identify the overall trend of the problem at hand properly. Thanks to the interpolating characteristic of the Kriging model, a constant is widely used as the trend function, which neglects the overall trend presented by data. However, previous researches prove that an appropriate trend function considering high-order terms is able to enhance the approximation ability of the Kriging model. In this paper, a regularization approach is proposed to construct the trend function in the Kriging model to improve the prediction accuracy. First, a new weighting scheme, which is formulated as an optimization problem with regularization terms, is used to solve the regression coefficients. Then, the other model parameters are estimated by maximizing the likelihood function, which leads to a nested optimization problem. It is solved iteratively to obtain the final estimation of the model parameters. From a Bayesian point of view, the proposed regularization method can adaptively tune the parameter of the prior distribution on the regression coefficients in the iterative algorithm. To select good regularization parameters, a cross-validation method is used. The implementation is tested on several analytical examples and physical examples, and the experimental results confirm the effectiveness of the proposed approach.