Statistica Sinica 10(2000), 1133-1152

SEQUENTIAL DESIGN OF COMPUTER EXPERIMENTS

TO MINIMIZE INTEGRATED RESPONSE FUNCTIONS

Brian J. Williams, Thomas J. Santner and William I. Notz

The Ohio State University

Abstract: In the last ten to fifteen years many phenomena that could only be studied using physical experiments can now be studied by computer experiments. Advances in the mathematical modeling of many physical processes, in algorithms for solving mathematical systems, and in computer speeds, have combined to make it possible to replace some physical experiments with computer experiments. In a computer experiment, a deterministic output, $y(\mbox{\boldmath$x$})$, is computed for each set of input variables, $\mbox{\boldmath$x$}$. This paper is concerned with the commonly occuring situation in which there are two types of input variables: suppose $\mbox{\boldmath$x$} = (\mbox{\boldmath$x$}_c,\mbox{\boldmath$x$}_e)$ where $\mbox{\boldmath$x$}_c$ is a set of ``manufacturing'' (control) variables and $\mbox{\boldmath$x$}_e$ is a set of ``environmental'' (noise) variables. Manufacturing variables can be controlled while environmental variables are not controllable but have values governed by some distribution. We introduce a sequential experimental design for finding the optimum of $\ell(\mbox{\boldmath$x$}_c) = E\{ y(\mbox{\boldmath$x$}_c,\mbox{\boldmath$X$}_e)\}$, where the expectation is taken over the distribution of the environmental variables. The approach is Bayesian; the prior information is that is a draw from a stationary Gaussian stochastic process with correlation function from the Matérn class having unknown parameters. The idea of the method is to compute the posterior expected ``improvement'' over the current optimum for each untested site; the design selects the next site to maximize the expected improvement. The procedure is illustrated with examples from the literature.

Key words and phrases: Computer experiments, control variables, expected improvement, noise variables, optimization, sequential design.