Bounds for Maximin Latin Hypercube Designs

Latin hypercube designs (LHDs) play an important role when approximating computer simula- tion models. To obtain good space-filling properties, the maximin criterion is frequently used. Unfortunately, constructing maximin LHDs can be quite time-consuming when the number of dimensions and design points increase. In these cases, we can use approximate maximin LHDs. In this paper, we construct bounds for the separation distance of certain classes of maximin LHDs. These bounds are useful for assessing the quality of approximate maximin LHDs. Until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs, i.e. for maximin designs without a Latin hypercube struc- ture. The separation distance of maximin LHDs also satisfies these “unrestricted” bounds. By using some of the special properties of LHDs, we are able to find new and tighter bounds for maximin LHDs. Within the different methods used to determine the upper bounds, a vari- ety of combinatorial optimization techniques are employed. Mixed Integer Programming, the Travelling Salesman Problem, and the Graph Covering Problem are among the formulations used to obtain the bounds. Besides these bounds, also a construction method is described for generating LHDs that meet Baer’s bound for the ℓ1 distance measure for certain values of n.Latin hypercube design;maximin;space-filling;mixed integer programming;trav- elling salesman problem;graph covering.

Maximin Designs for Computer Experiments.

Decision processes are nowadays often facilitated by simulation tools. In the field of engineering, for example, such tools are used to simulate the behavior of products and processes. Simulation runs, however, are often very time-consuming, and, hence, the number of simulation runs allowed is limited in practice. The problem then is to determine which simulation runs to perform such that the maximal amount of information about the product or process is obtained. This problem is addressed in the first part of the thesis. It is proposed to use so-called maximin Latin hypercube designs and many new results for this class of designs are obtained. In the second part, the case of multiple interrelated simulation tools is considered and a framework to deal with such tools is introduced. Important steps in this framework are the construction and the use of coordination methods and of nested designs in order to control the dependencies present between the various simulation tools

One-Dimensional Nested Maximin Designs

The design of computer experiments is an important step in black box evaluation and optimization processes.When dealing with multiple black box functions the need often arises to construct designs for all black boxes jointly, instead of individually.These so-called nested designs are used to deal with linking parameters and sequential evaluations.In this paper we discuss one-dimensional nested maximin designs.We show how to nest two designs optimally and develop a heuristic to nest three and four designs.Furthermore, it is proven that the loss in space-fillingness, with respect to traditional maximin designs, is at most 14:64 percent and 19:21 percent, when nesting two and three designs, respectively.simulation;computers;integer programming

Coordination of Coupled Black Box Simulations in the Construction of Metamodels

This paper introduces methods to coordinate black box simulations in the construction of metamodels for situations in which we have to deal with coupled black boxes.We de.ne three coordination methods: parallel simulation, sequential simulation and sequential modeling.To compare these three methods we focus on .ve aspects: throughput time, .exibility, simulated product designs, coordination complexityand the use of prior information.Special attention is given to the throughput time aspect.For this aspect we derive mathematical formulas and we give relations between the throughput times of the three coordination methods.At the end of this paper we summarize the results and give recommendations on the choice of a suitable coordination method.simulation;simulation models;coordination;black box;metamodels

Space-Filling Latin Hypercube Designs For Computer Experiments (Revision of CentER DP 2006-18)

In the area of computer simulation, Latin hypercube designs play an important role. In this paper the classes of maximin and Audze-Eglais Latin hypercube designs are considered. Up to now only several two-dimensional designs and a few higher dimensional designs for these classes have been published. Using periodic designs and the Enhanced Stochastic Evolutionary algorithm of Jin et al. (2005), we obtain new results which we compare to existing results. We thus construct a database of approximate maximin and Audze-Eglais Latin hypercube designs for up to ten dimensions and for up to 300 design points. All these designs can be downloaded from the website http://www.spacefillingdesigns.nl.Audze-Eglais;computer experiment;Enhanced Stochastic Evolutionary algorithm;Latin hypercube design;maximin;non-collapsing;packing problem;simulated annealing;space-filling