Statistics and Computing

, Volume 22, Issue 3, pp 703–712 | Cite as

Maximin design on non hypercube domains and kernel interpolation

  • Yves Auffray
  • Pierre Barbillon
  • Jean-Michel MarinEmail author


In the paradigm of computer experiments, the choice of an experimental design is an important issue. When no information is available about the black-box function to be approximated, an exploratory design has to be used. In this context, two dispersion criteria are usually considered: the minimax and the maximin ones. In the case of a hypercube domain, a standard strategy consists of taking the maximin design within the class of Latin hypercube designs. However, in a non hypercube context, it does not make sense to use the Latin hypercube strategy. Moreover, whatever the design is, the black-box function is typically approximated thanks to kernel interpolation. Here, we first provide a theoretical justification to the maximin criterion with respect to kernel interpolations. Then, we propose simulated annealing algorithms to determine maximin designs in any bounded connected domain. We prove the convergence of the different schemes. Finally, the methodology is applied on a challenging real example where the black-blox function describes the behaviour of an aircraft engine.


Computer experiments Kernel interpolation Kriging maximin designs Simulated annealing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Auffray, Y., Barbillon, P., Marin, J.-M.: Modèles réduits à partir d’expériences numériques. J. Soc. Fr. Stat. 152(1), 89–102 (2011) MathSciNetGoogle Scholar
  2. Bartoli, N., Del Moral, P.: Simulation & algorithmes stochastiques. Cépaduès (2001) Google Scholar
  3. Bratley, P., Fox, B.L.: Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 14(1), 88–100 (1988) zbMATHCrossRefGoogle Scholar
  4. Bursztyn, D., Steinberg, D.M.: Comparison of designs for computer experiments. J. Stat. Plan. Inference 136(3), 1103–1119 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  5. Chib, S., Greenberg, E.: Understanding the Metropolis-Hastings algorithm. Am. Stat. 49(4), 327–335 (1995) CrossRefGoogle Scholar
  6. Cressie, N.A.C.: Statistics for Spatial Data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York (1993) Google Scholar
  7. den Hertog, D., Kleijnen, J., Siem, A.: The correct Kriging variance estimated by bootstrapping. J. Oper. Res. Soc. 57(4), 400–409 (2006) zbMATHCrossRefGoogle Scholar
  8. Fang, K.-T., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments (Computer Science & Data Analysis). Chapman & Hall/CRC, London/Boca Raton (2005) Google Scholar
  9. Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970) zbMATHCrossRefGoogle Scholar
  10. Johnson, M.E., Moore, L.M., Ylvisaker, D.: Minimax and maximin distance designs. J. Stat. Plan. Inference 26(2), 131–148 (1990) MathSciNetCrossRefGoogle Scholar
  11. Joseph, V.R.: Limit kriging. Technometrics 48(4), 458–466 (2006) MathSciNetCrossRefGoogle Scholar
  12. Koehler, J.R., Owen, A.B.: Computer experiments. In: Design and Analysis of Experiments, Handbook of Statistics, vol. 13, pp. 261–308. North-Holland, Amsterdam (1996) CrossRefGoogle Scholar
  13. Laslett, G.M.: Kriging and splines: an empirical comparison of their predictive performance in some applications. J. Am. Stat. Assoc. 89(426), 391–409 (1994) MathSciNetCrossRefGoogle Scholar
  14. Li, R., Sudjianto, A.: Analysis of computer experiments using penalized likelihood in Gaussian Kriging models. Technometrics 47(2), 111–120 (2005) MathSciNetCrossRefGoogle Scholar
  15. Locatelli, M.: Convergence properties of simulated annealing for continuous global optimization. J. Appl. Probab. 33(4), 1127–1140 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  16. Lophaven, N.S., Nielsen, H.B., Sondergaard, J.: DACE, a Matlab Kriging toolbox. Technical Report IMM-TR-2002-12, DTU (2002) Google Scholar
  17. Madych, W.R., Nelson, S.A.: Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70(1), 94–114 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  18. Matheron, G.: Principles of Geostatistics. Econ. Geol. 58(8), 1246–1266 (1963) CrossRefGoogle Scholar
  19. McKay, M., Beckman, R., Conover, W.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Mease, D., Bingham, D.: Latin hyperrectangle sampling for computer experiments. Technometrics 48(4), 467–477 (2006) MathSciNetCrossRefGoogle Scholar
  21. Morris, M.D., Mitchell, T.J.: Exploratory designs for computational experiments. J. Stat. Plan. Inference 43, 381–402 (1995) zbMATHCrossRefGoogle Scholar
  22. Roberts, G.O., Rosenthal, J.S.: Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains. Ann. Appl. Probab. 16(4), 2123–2139 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  23. Sacks, J., Schiller, S., Mitchell, T., Wynn, H.: Design and analysis of computer experiments (with discussion). Stat. Sci. 4(4), 409–435 (1989a) zbMATHCrossRefGoogle Scholar
  24. Sacks, J., Schiller, S.B., Welch, W.J.: Designs for computer experiments. Technometrics 31(1), 41–47 (1989b) MathSciNetCrossRefGoogle Scholar
  25. Santner, T.J., Williams, B.J., Notz, W.I.: The Design and Analysis of Computer Experiments. Springer Series in Statistics. Springer, New York (2003) zbMATHGoogle Scholar
  26. Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3(3), 251–264 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  27. Schaback, R.: Kernel-based meshless methods. Technical report, Institute for Numerical and Applied Mathematics, Georg-August-University Goettingen (2007) Google Scholar
  28. Stein, M.L.: Interpolation of Spatial Data. Some Theory for Kriging. Springer Series in Statistics. Springer, New York (1999) zbMATHCrossRefGoogle Scholar
  29. Stein, M.L.: The screening effect in Kriging. Ann. Stat. 30(1), 298–323 (2002) zbMATHCrossRefGoogle Scholar
  30. Stinstra, E., den Hertog, D., Stehouwer, P., Vestjens, A.: Constrained maximin designs for computer experiments. Technometrics 45(4), 340–346 (2003) MathSciNetCrossRefGoogle Scholar
  31. Tierney, L.: A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab. 8(1), 1–9 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  32. van Dam, E.R., Husslage, B., den Hertog, D., Melissen, H.: Maximin Latin hypercube designs in two dimensions. Oper. Res. 55(1), 158–169 (2007) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Yves Auffray
    • 1
  • Pierre Barbillon
    • 2
  • Jean-Michel Marin
    • 3
    Email author
  1. 1.Dassault Aviation & Département de MathématiquesUniversité Paris-SudOrsayFrance
  2. 2.INRIA Saclay, Projet Select, Département de MathématiquesUniversité Paris-SudOrsayFrance
  3. 3.Institut de Mathématiques et Modélisation de MontpellierUniversité Montpellier 2Montpellier cedex 5France

Personalised recommendations