Nested maximin Latin hypercube designs

In the field of design of computer experiments (DoCE), Latin hypercube designs are frequently used for the approximation and optimization of black-boxes. In certain situations, we need a special type of designs consisting of two separate designs, one being a subset of the other. These nested designs can be used to deal with training and test sets, models with different levels of accuracy, linking parameters, and sequential evaluations. In this paper, we construct nested maximin Latin hypercube designs for up to ten dimensions. We show that different types of grids should be considered when constructing nested designs and discuss how to determine which grid to use for a specific application. To determine nested maximin designs for dimensions higher than two, four variants of the ESE algorithm of Jin et al. (J Stat Plan Inference 134(1):268-287, 2005) are introduced and compared. Our main focus is on GROUPRAND, the most successful of these four variants. In the numerical comparison, we consider the calculation times, space-fillingness of the obtained designs and the performance of different grids. Maximin distances for different numbers of points are provided; the corresponding nested maximin designs can be found on the website http://www.spacefillingdesigns.nl.