Abstract
In the field of design of computer experiments (DoCE), Latin hypercube designs are frequently used for the approximation and optimization of black-boxes. In certain situations, we need a special type of designs consisting of two separate designs, one being a subset of the other. These nested designs can be used to deal with training and test sets, models with different levels of accuracy, linking parameters, and sequential evaluations. In this paper, we construct nested maximin Latin hypercube designs for up to ten dimensions. We show that different types of grids should be considered when constructing nested designs and discuss how to determine which grid to use for a specific application. To determine nested maximin designs for dimensions higher than two, four variants of the ESE algorithm of Jin et al. (J Stat Plan Inference 134(1):268–287, 2005) are introduced and compared. Our main focus is on GROUPRAND, the most successful of these four variants. In the numerical comparison, we consider the calculation times, space-fillingness of the obtained designs and the performance of different grids. Maximin distances for different numbers of points are provided; the corresponding nested maximin designs can be found on the website http://www.spacefillingdesigns.nl.
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Barthelemy JFM, Haftka RT (1993) Approximation concepts for optimum structural design—a review. Struct Multidisc Optim 5(3):129–144
Booker AJ, Dennis JE, Frank PD, Serafini DB, Torczon V, Trosset MW (1999) A rigorous framework for optimization of expensive functions by surrogates. Struct Multidisc Optim 17(1):1–13
Cherkassky V, Mulier F (1998) Learning from data: concepts, theory, and methods. Wiley, New York
Cressie NAC (1993) Statistics for spatial data (revised ed), vol 605. Wiley, New York
Den Hertog D, Stehouwer HP (2002) Optimizing color picture tubes by high-cost nonlinear programming. Eur J Oper Res 140(2):197–211
Forrester AIJ, Keane AJ, Bressloff NW (2006) Design and analysis of “noisy” computer experiments. AIAA J 44(10):2331–2339
Forrester AIJ, Sóbester A, Keane AJ (2007) Multi-fidelity optimization via surrogate modelling. In: Proceedings of the royal society a: mathematical, physical and engineering sciences, vol 463. The Royal Society, London, pp 3251–3269
Forrester AIJ, Sóbester A, Keane AJ (2008) Engineering design via surrogate modelling: a practical guide. Wiley, Chichester
Goel T, Haftka RT, Shyy W, Watson LT (2008) Pitfalls of using a single criterion for selecting experimental designs. Int J Numer Methods Eng 75(2):127–155
Grosso A, Jamali ARMJU, Locatelli M (2009) Finding maximin Latin hypercube designs by iterated local search heuristics. Eur J Oper Res 197(2):541–547
Husslage BGM, Van Dam ER, Den Hertog D, Stehouwer HP, Stinstra ED (2003) Collaborative metamodeling: coordinating simulation-based product design. Concurr Eng Res Appl 11(4):267–278
Husslage BGM, Van Dam ER, Den Hertog D (2005) Nested maximin Latin hypercube designs in two dimensions. CentER Discussion Paper 2005-79. Tilburg University, Tilburg, pp 1–11
Husslage BGM, Rennen G, Van Dam ER, Den Hertog D (2008) Space-filling Latin hypercube designs for computer experiments. CentER Discussion Paper 2008-104. Tilburg University, Tilburg, pp 1–14
Jin R, Chen W, Sudjianto A (2002) On sequential sampling for global metamodeling in engineering design. In: Proceedings of the ASME 2002 design engineering technical conferences and computers and information in engineering conference. Montreal, pp 1–10
Jin R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. J Stat Plan Inference 134(1):268–287
Johnson ME, Moore LM, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26:131–148
Jones DR (2001) A taxonomy of global optimization methods based on response surfaces. J Glob Optim 21(4):345–383
Kennedy MC, O’Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13
Kleijnen JPC (2008) Design and analysis of simulation experiments. In: International series in operations research & management science, vol 111. Springer, New York
Montgomery DC (1984) Design and analysis of experiments, 2nd ed. Wiley, New York
Morris MD, Mitchell TJ (1995) Exploratory designs for computer experiments. J Stat Plan Inference 43:381–402
Myers RH (1999) Response surface methodology—current status and future directions. J Qual Technol 31:30–74
Qian Z, Seepersad CC, Joseph VR, Allen JK, Wu CFJ (2006) Building surrogate models based on detailed and approximate simulations. J Mech Des 128(4):668–677
Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28
Sacks J, Schiller SB, Welch WJ (1989a) Designs for computer experiments. Technometrics 31:41–47
Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989b) Design and analysis of computer experiments. Stat Sci 4:409–435
Santner ThJ, Williams BJ, Notz WI (2003) The design and analysis of computer experiments. Springer Series in Statistics. Springer, New York
Simpson TW, Booker AJ, Ghosh D, Giunta AA, Koch PN, Yang R-J (2004) Approximation methods in multidisciplinary analysis and optimization: a panel discussion. Struct Multidisc Optim 27(5):302–313
Simpson TW, Toropov VV, Balabanov V, Viana FAC (2008) Design and analysis of computer experiments in multidisciplinary design optimization: a review. In: Proceedings of the 12th AIAA/ISSMO multidisciplinary analysis and optimization conference, pp 1–22
Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Struct Multidisc Optim 14(1):1–23
Van Dam ER, Husslage BGM, Den Hertog D, Melissen JBM (2007) Maximin Latin hypercube designs in two dimensions. Oper Res 55(1):158–169
Van Dam ER, Husslage BGM, Den Hertog D (2009a) One-dimensional nested maximin designs. J Glob Optim (in press)
Van Dam ER, Rennen G, Husslage BGM (2009b) Bounds for maximin Latin hypercube designs. Oper Res 57:595–608
Viana FAC, Balabanov V, Venter G, Garcelon J, Steffen V (2007) Generating optimal Latin hypercube designs in real time. In: 7th world congress on structural and multidisciplinary optimization, pp 2310–2315
Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129(4):370–380
Ye KQ, Li W, Sudjianto A (2000) Algorithmic construction of optimal symmetric Latin hypercube designs. J Stat Plan Inference 90(1):145–159
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The research of B.G.M. Husslage has been financially supported by the SamenwerkingsOrgaan Brabantse Universiteiten (SOBU).
The research of E.R. Van Dam has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
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Rennen, G., Husslage, B., Van Dam, E.R. et al. Nested maximin Latin hypercube designs. Struct Multidisc Optim 41, 371–395 (2010). https://doi.org/10.1007/s00158-009-0432-y
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DOI: https://doi.org/10.1007/s00158-009-0432-y