CONSTRUCTION OF NESTED SPACE-FILLING DESIGNS

New types of designs called nested space-filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy. In this article, we develop several approaches to constructing such designs. The development of these methods also leads to the introduction of several new discrete mathematics concepts, including nested orthogonal arrays and nested difference matrices.

[1]  R. C. Bose,et al.  Orthogonal Arrays of Strength two and three , 1952 .

[2]  N. S. Mendelsohn,et al.  Orthomorphisms of Groups and Orthogonal Latin Squares. I , 1961, Canadian Journal of Mathematics.

[3]  O. Kempthorne,et al.  Some Main-Effect Plans and Orthogonal Arrays of Strength Two , 1961 .

[4]  S. S. Shrikhande,et al.  Generalized Hadamard Matrices and Orthogonal Arrays of Strength Two , 1964, Canadian Journal of Mathematics.

[5]  Jennifer Seberry,et al.  Some remarks on generalised Hadamard matrices and theorems of Rajkundlia on SBIBDs , 1979 .

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

[7]  Changbao Wu,et al.  An Approach to the Construction of Asymmetrical Orthogonal Arrays , 1991 .

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

[9]  Henry P. Wynn,et al.  Screening, predicting, and computer experiments , 1992 .

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

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

[12]  A. Owen Lattice Sampling Revisited: Monte Carlo Variance of Means Over Randomized Orthogonal Arrays , 1994 .

[13]  Boxin Tang A theorem for selecting OA-based Latin hypercubes using a distance criterion , 1994 .

[14]  Jung-Chao Wang,et al.  Mixed difference matrices and the construction of orthogonal arrays , 1996 .

[15]  M. Floater,et al.  Multistep scattered data interpolation using compactly supported radial basis functions , 1996 .

[16]  Wei-Liem Loh,et al.  A combinatorial central limit theorem for randomized orthogonal array sampling designs , 1996 .

[17]  A. S. Hedayat,et al.  On difference schemes and orthogonal arrays of strength t , 1996 .

[18]  Wei-Liem Loh On Latin hypercube sampling , 1996 .

[19]  Kenny Q. Ye Orthogonal Column Latin Hypercubes and Their Application in Computer Experiments , 1998 .

[20]  Colin L. Mallows,et al.  Factor-covering designs for testing software , 1998 .

[21]  Boxin Tang,et al.  SELECTING LATIN HYPERCUBES USING CORRELATION CRITERIA , 1998 .

[22]  Koen Dewettinck,et al.  Modeling the steady-state thermodynamic operation point of top-spray fluidized bed processing , 1999 .

[23]  Lih-Yuan Deng,et al.  Orthogonal Arrays: Theory and Applications , 1999, Technometrics.

[24]  C. F. Jeff Wu,et al.  Experiments: Planning, Analysis, and Parameter Design Optimization , 2000 .

[25]  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.

[26]  Yong Zhang,et al.  Uniform Design: Theory and Application , 2000, Technometrics.

[27]  A. O'Hagan,et al.  Predicting the output from a complex computer code when fast approximations are available , 2000 .

[28]  Neil A. Butler,et al.  Optimal and orthogonal Latin hypercube designs for computer experiments , 2001 .

[29]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[30]  William J. Welch,et al.  Uniform Coverage Designs for Molecule Selection , 2002, Technometrics.

[31]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[32]  Dick den Hertog,et al.  Collaborative Metamodeling: Coordinating Simulation-based Product Design , 2003, Concurr. Eng. Res. Appl..

[33]  Stephen J. Leary,et al.  Optimal orthogonal-array-based latin hypercubes , 2003 .

[34]  Alyson G. Wilson,et al.  Integrated Analysis of Computer and Physical Experiments , 2004, Technometrics.

[35]  Dave Higdon,et al.  Combining Field Data and Computer Simulations for Calibration and Prediction , 2005, SIAM J. Sci. Comput..

[36]  Michael Goldstein,et al.  Probabilistic Formulations for Transferring Inferences from Mathematical Models to Physical Systems , 2005, SIAM J. Sci. Comput..

[37]  Runze Li,et al.  Design and Modeling for Computer Experiments , 2005 .

[38]  Madhusudan K. Iyengar,et al.  Challenges of data center thermal management , 2005, IBM J. Res. Dev..

[39]  Carolyn Conner Seepersad,et al.  Building Surrogate Models Based on Detailed and Approximate , 2004, DAC 2004.

[40]  Dennis K. J. Lin,et al.  A construction method for orthogonal Latin hypercube designs , 2006 .

[41]  M. J. Bayarri,et al.  Computer model validation with functional output , 2007, 0711.3271.

[42]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[43]  A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments , 2007, 0708.0656.

[44]  Peter Z. G. Qian,et al.  On the existence of nested orthogonal arrays , 2008, Discret. Math..

[45]  Peter Z. G. Qian,et al.  Gaussian Process Models for Computer Experiments With Qualitative and Quantitative Factors , 2008, Technometrics.

[46]  Timothy G. Trucano,et al.  Verification and validation benchmarks , 2008 .

[47]  Mathias Wintzer,et al.  Multifidelity design optimization of low-boom supersonic jets , 2008 .

[48]  Peter Z. G. Qian,et al.  Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments , 2008, Technometrics.

[49]  Boxin Tang,et al.  NESTED SPACE-FILLING DESIGNS FOR COMPUTER EXPERIMENTS WITH TWO LEVELS OF ACCURACY , 2009 .

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

[51]  Peter Z. G. Qian,et al.  Sliced space-filling designs , 2009 .

[52]  Thomas J. Santner,et al.  Prediction for Computer Experiments Having Quantitative and Qualitative Input Variables , 2009, Technometrics.

[53]  Stanley H. Cohen,et al.  Design and Analysis , 2010 .

[54]  Peter Z. G. Qian,et al.  Nested Lattice Sampling: A New Sampling Scheme Derived by Randomizing Nested Orthogonal Arrays , 2010 .

[55]  Patrick R. Palmer,et al.  Multi-fidelity simulation modelling in optimization of a submarine propulsion system , 2010, 2010 IEEE Vehicle Power and Propulsion Conference.

[56]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[57]  Peter Z. G. Qian,et al.  An Approach to Constructing Nested Space-Filling Designs for Multi-Fidelity Computer Experiments. , 2010, Statistica Sinica.

[58]  Shiyu Zhou,et al.  A Simple Approach to Emulation for Computer Models With Qualitative and Quantitative Factors , 2011, Technometrics.

[59]  Peter Z. G. Qian,et al.  Accurate emulators for large-scale computer experiments , 2011, 1203.2433.

[60]  Peter Z. G. Qian,et al.  Sudoku-based space-filling designs , 2011 .