  In many real-world applications we are interested in approximating costly
functions that are analytically unknown, e.g. complex computer codes. An
emulator provides a fast approximation of such functions relying on a limited
number of evaluations. Gaussian processes (GPs) are commonplace emulators due
to their statistical properties such as the ability to estimate their own
uncertainty. GPs are essentially developed to fit smooth, continuous functions.
However, the assumptions of continuity and smoothness is unwarranted in many
situations. For example, in computer models where bifurcations or tipping
points occur, the outputs can be discontinuous. This work examines the capacity
of GPs for emulating step-discontinuous functions. Several approaches are
proposed for this purpose. Two special covariance functions/kernels are adapted
with the ability to model discontinuities. They are the neural network and
Gibbs kernels whose properties are demonstrated using several examples. Another
approach, which is called warping, is to transform the input space into a new
space where a GP with a standard kernel, such as the Matern family, is able to
predict the function well. The transformation is perform by a parametric map
whose parameters are estimated by maximum likelihood. The results show that the
proposed approaches have superior performance to GPs with standard kernels in
capturing sharp jumps in the true function.
