Gradient-Enhanced Kriging Method

Kriging [9] approximates f (|) by a weighted linear combination of samples, i. e.


f (I) = r(I) + T w (S®)f (S®)


where f (%l’i)) are N samples of the SRQ. у and wi are determined by minimizing the variance of the error e = f — f with the assumptions that the expectation of e is zero and that f (|) honors a spatial correlation model. We use direct gradient – enhanced Kriging (GEK) [8] that incorporates gradient information as secondary samples by an extended spatial correlation model that accommodates gradients. We implement GEK using the Surrogate-Modeling for Aero-Data Toolbox (SMART) [13] developed at DLR, opting for ordinary Kriging and a correlation model of spline type which is considered the most efficient in similar situations in [17]. The internal parameters of the correlation model are fine-tuned to fit the sampled data by a maximum likelihood estimation [24].

For this UQ job we first establish a GEK surrogate model of f (|) based on QMC samples of the CFD model, and integrate for the target statistics and pdf by a large number (105) of QMC samples on the surrogate model.