Designing computer experiments to estimate integrated response functions

Complex physical systems can be modeled mathematically and then solved by appropriate numerical methods implemented by a complex computer code. Such computer codes allow us to construct analogs of physical experiments that would not be possible due to physical, financial and/or time constraints. In a computer experiment, a response, y(x), usually deterministic, is computed by the code for each set of input variables, x, according to an experimental design strategy. Then, as in physical experiments, the relationship between the inputs x and y(x) is studied.

We are concerned with the design of computer experiments when there are two types of inputs x=(x_c, x_e): control variables that can be set by researcher or product designer, x_c, and environmental variables that are not controlled in the field but have some probability distribution characterizing a population of interest, x_e. Our interest is in accurately predicting the mean of the deterministic response function μ(x_c)=E[y(x_c,X_e)] over the distribution of the environmental variables. We introduce a new method for constructing an “inexpensive” predictor of the mean response that is of greatest use when the complexity of the computer code or the high-dimensionality of inputs limit the number of runs possible and Var[y(x_c,X_e)] varies considerably as x_c varies. We also propose a sequential design strategy for constructing the training data on which to base the predictor in such problems where the variance of the response surface varies greatly over the control space. In such cases, all further computing effort is best spent taking more observations in the regions of the control space where the variance appears higher. The procedures introduced are illustrated by examples utilizing test functions from the numerical optimization literature.