On the optimal designs for the prediction of Ornstein–Uhlenbeck sheets

Computer simulations are often used to replace physical experiments for exploring the complex relationships between input and output variables. We study the optimal design problem for the prediction of a stationary Ornstein–Uhlenbeck sheet on a monotonic set with respect to the integrated mean square prediction error criterion and the entropy criterion. We show that there is a substantial difference between the shapes of optimal designs for Ornstein–Uhlenbeck processes and sheets. In particular, we show that the optimal prediction based on the integrated mean square prediction error does not necessarily lead to space-filling designs.