Keywords: surrogate modelling, space-filling, sequential design, active learning, Latin hypercube, experimental design, Kriging, Voronoi tessellation, Delaunay triangulation.

Simulated computer experiments have become a viable cost-effective alternative for controlled real-life experiments. However, the simulation of complex systems with multiple input and output parameters can be a very time-consuming process. Many of these high-fidelity simulators need minutes, hours or even days to perform one simulation. The goal of global surrogate modeling is to create a model that mimics the original simulator, but can be computed much faster.

Sequential design (which is also known as adaptive sampling [1] or active learning [2]) is often used to reduce the number of sample evaluations needed to achieve the desired accuracy. Sequential design is an iterative technique in which samples are selected and evaluated sequentially, allowing the algorithm to use information from previously evaluated samples to determine where to sample next. This results in a more efficient distribution of samples compared to traditional design of experiments, where all the samples are selected up front and models are only trained after all the samples are evaluated.

This paper shows that Voronoi and Delaunay-based space-filling sequential design strategies have several major advantages over traditional Latin hypercubes. Firstly, they are more flexible, avoiding undersampling and oversampling by selecting new samples one by one and stopping when the desired accuracy is met. When using a one-shot experimental design, the engineer has to guess the required number of samples in advance, and if the desired accuracy is not met, must resort to sequential design strategies to generate more samples.

Secondly, creating an optimally space-filling Latin hypercube is a very difficult task, and has a large computational cost because of the high dimensionality of the search space. Suboptimal Latin hypercubes often leave large gaps in the design space, which substantially reduces the accuracy of the model. Both Delaunay and Voronoi-based sequential design strategies produce much better space-filling designs and require less time to compute than an optimized Latin hypercube.

In order to determine which space-filling sampling strategy is the best choice, the two sequential design methods in this paper were compared with each other. Voronoi-based sampling uses an approximation of the Voronoi tessellation, making it a much faster alternative than Delaunay-based sampling, which requires the computation of the Delaunay triangulation. One additional disadvantage of Delaunay-based sampling is that it does not have good space-filling properties near the edges of the design space, due to the nature of the triangulation. This may have a negative impact on the accuracy of the model. Thus, Voronoi-based sampling should in most cases be prefered over Delaunay-based sampling and self-generated Latin hypercubes.

1

R. Lehmensiek, P. Meyer, M. Müller, "Adaptive sampling applied to multivariate, multiple output rational interpolation models with application to microwave circuits", International Journal of RF and Microwave Computer-Aided Engineering, 12(4):332-340, 2002. doi:10.1002/mmce.10032

2

M. Sugiyama, "Active Learning in Approximately Linear Regression Based on Conditional Expectation of Generalization Error", Journal of Machine Learning Research, 7:141-166, 2006.

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