Cross-Validation Based Adaptive Sampling for Multilevel Gaussian Process Models

Abstract. Complex computer codes or models can often be run in a hierarchy of different levels of complexity ranging from the very basic to the sophisticated. The top levels in this hierarchy are typically expensive to run, which limits the number of possible runs. To make use of simulations over all levels, and crucially improve predictions at the top level, we use multilevel Gaussian process emulators (GPs). The accuracy of the GP greatly depends on the design of the training points. In this paper, we present a multilevel adaptive sampling algorithm to sequentially increase the set of design points to optimally improve the fit of the GP. The normalized squared expected leave-one-out cross-validation error (ES-LOO) is calculated at all unobserved locations, and a new design point is chosen using expected improvement combined with a repulsion function to find the maximum ES-LOO. We use ES-LOO to obtain a model-free measure of prediction error at each level. This criterion is calculated for each model level weighted by an associated cost for the code at that level. Hence, at each iteration, our algorithm optimises for both the new point location and the model level. The algorithm is extended to batch selection as well as single point selection, where batches can be designed for single levels or optimally across all levels. We apply the new multilevel ES-LOO algorithm to a range of examples, and make comparisons to existing design methods. This includes comparisons with single-shot methods as well as more recent sequential design algorithms. A toolbox containing relevant code can be found at: github.com/EXA-UQ/EXAUQ-Toolbox .