Branin Function Benchmark

This benchmark performs convergence tests on multiple realizations of the 2D Branin function. Details of the 2D Branin function can be found at https://www.sfu.ca/~ssurjano/branin.html. This particular version uses 8 realizations of the Branin function, each with a different set of parameters. The code samples these 8 realizations simultaneously using a spacefilling Latin Hypercube experimental design with a varying number of target points, and then tests the convergence of the resulting emulators. As the number of targe points increases, the prediction error and prediction variance should decrease.

(Note however that eventually, the predictions worsen once the number of target points becomes large enough that the points become too densely sampled. In this case, the points become co-linear and the resulting covariance matrix is singular and cannot be inverted. To avoid this problem, the code iteratively adds additional noise to the covariance function to stabilize the inversion. However, this noise reduces the accuracy of the predictions. The values chosen for this benchmark attempt to avoid this, but in some cases this still becomes a problem due to the inherent smoothness of the squared exponential covariance function.)