  Gaussian Process (GP) models are popular statistical surrogates used for
emulating computationally expensive computer simulators. The quality of a GP
model fit can be assessed by a goodness of fit measure based on optimized
likelihood. Finding the global maximum of the likelihood function for a GP
model is typically very challenging as the likelihood surface often has
multiple local optima, and an explicit expression for the gradient of the
likelihood function is typically unavailable. Previous methods for optimizing
the likelihood function (e.g. MacDonald et al. (2013)) have proven to be robust
and accurate, though relatively inefficient. We propose several likelihood
optimization techniques, including two modified multi-start local search
techniques, based on the method implemented by MacDonald et al. (2013), that
are equally as reliable, and significantly more efficient. A hybridization of
the global search algorithm Dividing Rectangles (DIRECT) with the local
optimization algorithm BFGS provides a comparable GP model quality for a
fraction of the computational cost, and is the preferred optimization technique
when computational resources are limited. We use several test functions and a
real application from an oil reservoir simulation to test and compare the
performance of the proposed methods with the one implemented by MacDonald et
al. (2013) in the R library GPfit. The proposed method is implemented in a
Matlab package, GPMfit.
