Gaussian Process-Based Dimension Reduction for Goal-Oriented Sequential Design

Several methods are available for goal-oriented sequential design of expensive black-box functions. Yet, it is a difficult task when the dimension increases. A classical approach is two-stage. First, sensitivity analysis is performed to reduce the dimension of the input variables. Second, the goal-oriented sampling is achieved by considering only the selected influential variables. This approach can be computationally expensive and may lack flexibility since dimension reduction is done once and for all. In this paper, we propose a so-called Split-and-Doubt algorithm that performs sequentially both dimension reduction and the goal-oriented sampling. The Split step identifies influential variables. This selection relies on new theoretical results on Gaussian process regression. We prove that large correlation lengths of covariance functions correspond to inactive variables. Then, in the Doubt step, a doubt function is used to update the subset of influential variables. Numerical tests show the efficiency of the Split-and-Doubt algorithm.