Sequentially Refined Latin Hypercube Designs: Reusing Every Point

The use of iteratively enlarged Latin hypercube designs for running computer experiments has recently gained popularity in practice. This approach conducts an initial experiment with a computer code using a Latin hypercube design and then runs a follow-up experiment with additional runs elaborately chosen so that the combined design set for the two experiments forms a larger Latin hypercube design. This augmenting process can be repeated multiple stages, where in each stage the augmented design set is guaranteed to be a Latin hypercube design. We provide a theoretical framework to put this approach on a firm footing. Numerical examples are given to corroborate the derived theoretical results. Supplementary materials for this article are available online.

[1]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[2]  Charles Tong,et al.  Refinement strategies for stratified sampling methods , 2006, Reliab. Eng. Syst. Saf..

[3]  T. J. Mitchell,et al.  Exploratory designs for computational experiments , 1995 .

[4]  V. R. Joseph,et al.  ORTHOGONAL-MAXIMIN LATIN HYPERCUBE DESIGNS , 2008 .

[5]  Jason L. Loeppky,et al.  Projection array based designs for computer experiments , 2012 .

[6]  T. J. Mitchell,et al.  Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction , 1993 .

[7]  G. G. Wang,et al.  Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points , 2003 .

[8]  M. Stein.  Large sample properties of simulations using latin hypercube sampling , 1987 .

[9]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[10]  Tito Homem-de-Mello,et al.  Some Large Deviations Results for Latin Hypercube Sampling , 2005, Proceedings of the Winter Simulation Conference, 2005..

[11]  Jason L. Loeppky,et al.  Batch sequential designs for computer experiments , 2010 .

[12]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[13]  G. Rennen,et al.  Nested maximin Latin hypercube designs , 2009 .

[14]  Robert W. Mee,et al.  Semifolding 2 k–P Designs , 2000, Technometrics.

[15]  A. Owen.  Controlling correlations in latin hypercube samples , 1994 .

[16]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[17]  J. S. Hunter,et al.  The 2 k—p Fractional Factorial Designs Part I , 2000, Technometrics.

[18]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[19]  Boxin Tang.  Orthogonal Array-Based Latin Hypercubes , 1993 .

[20]  Ronald L. Iman.  Latin Hypercube Sampling , 2008 .

[21]  J. S. Hunter,et al.  The 2 k — p Fractional Factorial Designs , 1961 .

[22]  H. Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing , 1995 .

[23]  Jon C. Helton,et al.  Extension of Latin hypercube samples with correlated variables , 2008, Reliab. Eng. Syst. Saf..

[24]  David Mease,et al.  Latin Hyperrectangle Sampling for Computer Experiments , 2006, Technometrics.

[25]  Peter Z. G. Qian,et al.  Nested orthogonal array-based Latin hypercube designs , 2011 .

[26]  Peter Z. G. Qian.  Nested Latin hypercube designs , 2009 .