Search for level sets of functions using computer experiments

In engineering and other fields, it is common to use a computer simulation to model a real world process. The inputs to a function f represent factors that influence the outcome. The output represents a quantity of interest. Often there will be a specified level L, and the objective is to find the inputs for which output is above L. L may be a tolerance level, and the inputs for which response is larger than L form a tolerance region. We might estimate the tolerance region by evaluating f on a grid, but even a coarse grid may have thousands of points in four or five dimensions. If the function f is costly to evaluate, we need to be able to estimate the tolerance region with as few evaluations as possible. We approach this problem with a sequential search. Use data at any stage to fit a spatial process that approximates the function. Fit a Gaussian spatial process, as described in Currin, Mitchell, Morris, and Ylvisaker [1991]. The spatial process gives an estimate of the L-contour. We can also use the process to estimate how much information would be gained if f is evaluated at point p. Choose points where it is estimated that f takes value L, but where uncertainty is high. Evaluate f at the chosen points. This will augment the set of data points and the vector of data values. Repeat the procedure with this augmented data. Calculate convergence criteria after each iteration, and stop when criteria reach predetermined goals.;The search process is applied to several functions defined in low dimensional space. Finally, it is applied to an actual simulation function