Exploiting active subspaces of hyperparameters for efficient high-dimensional Kriging modeling

Kriging surrogate model is a powerful tool to facilitate engineering system analysis and mechanical design by emulating the time-consuming simulations. However, estimating the hyper parameters of Kriging for high-dimensional problems can itself be computationally-expensive, because the large correlation matrix needs to be inverted a lot of times. This paper investigates a method for accelerating the training process of Kriging with a relatively trivial loss of model accuracy. The main idea is to reduce the number of hyperparameters by projecting the original high-dimensional hyperparameter space onto a low-dimensional subspace spanned by a small set of orthogonal directions. Specifically, the identification of the subspace whose bases are linear combinations of the original hyperparameters is accomplished by employing the Active Subspace Method (ASM). Provided that the primary variability of the likelihood function is well described in the active subspace of hyperparameters, the maximum likelihood estimation process is expected to be facilitated. The main steps of the whole modeling procedure are demonstrated, and practical implementation details are given. An experimental study including a set of analytical functions from 20-D to 80-D and a rear subframe modal analysis problem parameterized with 35 design variables is conducted for performance analysis and comparison. Results indicate that for high-dimensional problems a reasonable trade-off between the modeling efficiency and model accuracy of Kriging can be achieved by the proposed method.