  We propose a data fusion method based on multi-fidelity Gaussian process
regression (GPR) framework. This method combines available data of the quantity
of interest (QoI) and its gradients with different fidelity levels, namely, it
is a Gradient-enhanced Cokriging method (GE-Cokriging). It provides the
approximations of both the QoI and its gradients simultaneously with
uncertainty estimates. We compare this method with the conventional
multi-fidelity Cokriging method that does not use gradients information, and
the result suggests that GE-Cokriging has a better performance in predicting
both QoI and its gradients. Moreover, GE-Cokriging even shows better
generalization result in some cases where Cokriging performs poorly due to the
singularity of the covariance matrix. We demonstrate the application of
GE-Cokriging in several practical cases including reconstructing the
trajectories and velocity of an underdamped oscillator with respect to time
simultaneously, and investigating the sensitivity of power factor of a load bus
with respect to varying power inputs of a generator bus in a large scale power
system. We also show that though GE-Cokriging method requires a little bit
higher computational cost than Cokriging method, the result of accuracy
comparison shows that this cost is usually worth it.
