Efficient and robust gradient enhanced Kriging emulators.

%E2%80%9CNaive%E2%80%9D or straight-forward Kriging implementations can often perform poorly in practice. The relevant features of the robustly accurate and efficient Kriging and Gradient Enhanced Kriging (GEK) implementations in the DAKOTA software package are detailed herein. The principal contribution is a novel, effective, and efficient approach to handle ill-conditioning of GEK's %E2%80%9Ccorrelation%E2%80%9D matrix, RN%CC%83, based on a pivoted Cholesky factorization of Kriging's (not GEK's) correlation matrix, R, which is a small sub-matrix within GEK's RN%CC%83 matrix. The approach discards sample points/equations that contribute the least %E2%80%9Cnew%E2%80%9D information to RN%CC%83. Since these points contain the least new information, they are the ones which when discarded are both the easiest to predict and provide maximum improvement of RN%CC%83's conditioning. Prior to this work, handling ill-conditioned correlation matrices was a major, perhaps the principal, unsolved challenge necessary for robust and efficient GEK emulators. Numerical results demonstrate that GEK predictions can be significantly more accurate when GEK is allowed to discard points by the presented method. Numerical results also indicate that GEK can be used to break the curse of dimensionality by exploiting inexpensive derivatives (such as those provided by automatic differentiation or adjoint techniques), smoothness in the response being modeled, and adaptive sampling. Developmentmore » of a suitable adaptive sampling algorithm was beyond the scope of this work; instead adaptive sampling was approximated by omitting the cost of samples discarded by the presented pivoted Cholesky approach.« less