An improved sufficient dimension reduction-based Kriging modeling method for high-dimensional evaluation-expensive problems

Abstract

Kriging is a powerful surrogate modeling method for the analysis and optimization of computationally expensive problems. However, the efficient construction of high-precision models for high-dimensional expensive problems is an important challenge for Kriging method and has received much attention recently. To address this challenge, we propose an improved sufficient dimension reduction-based Kriging modeling method (KISDR) for high-dimensional evaluation-expensive problems. First, the martingale difference divergence is introduced into the sufficient dimension reduction to obtain a more accurate and stable estimate of the projection matrix, which can project the high-dimensional inputs into a low-dimensional latent space while maintaining sufficient information for response prediction. Second, we propose to utilize the ladle estimator to identify the dimension of latent space. The ladle estimator combines both eigenvalues and eigenvectors of the matrix and can identify the latent dimensionality more precisely. After that, a new Kriging correlation function is constructed by integrating the information of dimension reduction into the correlation structure, which significantly reduces the number of hyperparameters to be estimated. Finally, we devise a local optimization approach to fine-tune the Kriging hyperparameters to further improve the modeling accuracy. In this study, six mathematical examples and three engineering examples with dimensions varying from 30-D to 100-D are employed for performance analysis and comparison. The results indicate that the proposed KISDR can precisely identify and exploit the low-dimensional linear structure in the data to improve modeling accuracy and efficiency for high-dimensional evaluation-expensive problems.