The accuracy of the UQ analysis and the surrogate model approximation is directly related to the data sampling for model evaluations. A Latin hypercube sampling (LHS) is a design given by an n × m matrix in which each column is a random permutation of {1, 2,  …, n}. Here, n is the number of model evaluations, and m is the number of input parameters. A LHS has good projection properties on any single dimension, which means that, in any one-dimensional input parameter space, the sampled data always has a good space-filling property. For UQ studies using LHS for input parameter sampling, an optimal LHS is generated for n number of model evaluations in the [0, 1]m hypercube and then mapped to the input parameters’ probability distribution. An optimal LHS is space filling, which means that the sample points should spread over the entire input parameter space as evenly as possible. Meanwhile, it has no replications and sample points clustering in the projection onto any subspaces of the input parameter space. This is important because many problems could have only a subset of significant input parameters. The process of constructing an optimal LHS is formulated as a global optimization problem. The Uncertainty Quantification Module provides an optimal LHS algorithm that is efficient both in finding the global optimum and in computing the optimality criteria in every iteration. It uses the global optimization method and centered L2 discrepancy criterion described in Ref. 1. Note that depending on the starting design, the optimal LHS obtained from the optimization could be of lower quality. Therefore, the global optimization is repeated from different starting LHS designs.

In addition to constructing optimal LHS for a one-stage study, it is important to generate sequential (also known as follow-up) optimal LHS based on the current sampling for both ASPCE surrogate models and appending more model evaluation to improve a UQ analysis. The optimization of the sequential sampling uses the same criterion as for optimizing the initial optimal LHS. During the optimization, the sequential sampling does not modify the initial samples and tries to spread the sequential samples away from them. Because it is computationally expensive to run many model evaluations, it is possible to use any number of model evaluations for sequential sampling following the sequential optimization method described in Ref. 2.