Sequential design of computer experiments to minimize integrated response functions

In the last ten to fifteen years many phenomena that could be studied only 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 augment or replace physical experiments with computer experiments. In a computer experiment, a response z( x), usually deterministic, is computed for each set of input variables, x, according to an experimental design strategy. This strategy is determined by the goal of the experiment and depends, for example, on whether response prediction at unsampled input sites or response optimization is of primary interest. We are concerned with the commonly occuring situation in which there are two types of input variables: suppose x = ( xc, x e) where xc is a set of “control” (manufacturing) variables and xe is a set of “environmental” (noise) variables. Manufacturing variables can be controlled while noise variables are not controllable but have values governed by some probability distribution. For single response settings, we introduce a sequential experimental design for finding the optimum of e(x c) = E[z(x c, Xe)], where the expectation is taken over the distribution of the environmental variables. For bivariate response settings, we introduce a sequential experimental design for finding the constrained optimum of e1( xc)) = E[z( xc, X e)], subject to e2 (x c) = E[z2(x c, Xe)] ≤ U. The approach is Bayesian; the prior information is that the responses are a draw from a stationary Gaussian stochastic process with correlation function belonging to a parametric family with unknown parameters. The idea of the methods 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. Both procedures are illustrated by examples utilizing test functions from the numerical optimization literature.