This paper deals with the Gaussian process based approximation of a code
which can be run at different levels of accuracy. This method, which is a
particular case of co-kriging, allows us to improve a surrogate model of a
complex computer code using fast approximations of it. In particular, we focus
on the case of a large number of code levels on the one hand and on a Bayesian
approach when we have two levels on the other hand. The main results of this
paper are a new approach to estimate the model parameters which provides a
closed form expression for an important parameter of the model (the scale
factor), a reduction of the numerical complexity by simplifying the covariance
matrix inversion, and a new Bayesian modelling that gives an explicit
representation of the joint distribution of the parameters and that is not
computationally expensive. A thermodynamic example is used to illustrate the
comparison between 2-level and 3-level co-kriging
