Theoretical investigations of the new Cokriging method for variable-fidelity surrogate modeling: Well-posedness and maximum likelihood training

Cokriging is a variable-fidelity surrogate modeling technique which emulates a target process based on the spatial correlation of sampled data of different levels of fidelity. In this work, we address two theoretical questions associated with the so-called new Cokriging method for variable-fidelity modeling: A mandatory requirement for the well-posedness of the Cokriging emulator is the positive definiteness of the associated Cokriging correlation matrix. Spatial correlations are usually modeled by positive definite correlation kernels, which are guaranteed to yield positive definite correlation matrices for mutually distinct sample points. However, in applications, low-fidelity information is often available at high-fidelity sample points and the Cokriging predictor may benefit from the additional information provided by such an inclusive sampling. We investigate the positive definiteness of the Cokriging covariance matrix in both of the aforementioned cases and derive sufficient conditions for the well-posedness of the Cokriging predictor.The approximation quality of the Cokriging predictor is highly dependent on a number of model- and hyper-parameters. These parameters are determined by the method of maximum likelihood estimation. For standard Kriging, closed-form optima of the model parameters along hyper-parameter profile lines are known. Yet, these do not readily transfer to the setting of Cokriging, since additional parameters arise, which exhibit a mutual dependence. In previous work, this obstacle was tackled via a numerical optimization. Here, we derive closed-form optima for all Cokriging model parameters along hyper-parameter profile lines. The findings are illustrated by numerical experiments.