Keywords: simulation, Latin hypercube sampling, correlation, progressive sampling, design of experiments, adaptive sample size, neural network learning, response surface, simulated annealing.

In many computer experiments the adequacy of a given sample to give acceptable estimates of the desired statistical quantities cannot be determined a priori, and thus the ability to extend or refine an experimental design may be important. This can be done using crude Monte Carlo sampling, though, running each realization (physical or virtual experiment) is often very expensive. Therefore the variance reduction techniques such as Latin hypercube sampling [1,2] is a suitable option, because it yields lower variance of estimates of statistical moments compared to crude Monte Carlo sampling at the same sample size. In conventional Latin hypercube sampling (LHS), however, it is the necessary to specify the number of simulations (or physical realizations in the design of experiments) in advance. However, in real life problems the sample size yielding stable and statistically significant estimations of output statistics is not known beforehand. For example, when an analyst is planning to run complex nonlinear finite element computations using LHS, it is unclear how many simulations are needed. If too small a sample set is used (i.e. a set that does not give acceptable statistical results), the analyst normally has to abandon the results and run new analyses with a larger sample set. It is thus desirable to start with a small sample and then extend (or refine) the design if deemed necessary. The extension would permit the use of a larger sample set without the loss of any of the already performed expensive calculations (experiments).

This problem has been overcome by the suggested methods. The paper present procedures for extending the size of a Latin hypercube sample (LHS) with rank correlated variables. The methods are based on a hierarchy of Latin hypercube-like samples which are proven to yield to smaller variances of results compared to the crude Monte Carlo method. It is called hierarchical because each refinement explores the design space with higher resolution than the previous (pre-refined) design. The subsets sampled by the proposed method can be merged together exploiting the property of variance reduction, yet retaining the sampling flexibility. The whole procedure of a cascade of LHS-like runs can be fully automated and the stopping criterion might be for example the significance of the output statistics or the desired computational time. Simply the simulation can be stopped in a run-time depending on the current accuracy of the results and analyst budget. In this way, e.g. some crude pilot studies can later be efficiently reused and refined.

1

W. Conover, "On a Better Method for Selecting Input Variables", unpublished Los Alamos National Laboratories manuscript, reproduced as Appendix A of "Latin Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems" by J.C. Helton and F.J. Davis, Sandia National Laboratories report SAND2001-0417, Nov 2002., 1975.

2

M.D. McKay, W.J. Conover, R.J. Beckman, "A comparison of three methods for selecting values of input variables in the analysis of output from a computer code", Technometrics, 21, 239-245, 1979. doi:10.2307/1268522

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