  Kriging-based surrogate models have become very popular during the last
decades to approximate a computer code output from few simulations. In
practical applications, it is very common to sequentially add new simulations
to obtain more accurate approximations. We propose in this paper a method of
kriging-based sequential design which combines both the error evaluation
providing by the kriging model and the observed errors of a Leave-One-Out
cross-validation procedure. This method is proposed in two versions, the first
one selects points one at-a-time. The second one allows us to parallelize the
simulations and to add several design points at-a-time. Then, we extend these
strategies to multi-fidelity co-kriging models which allow us to surrogate a
complex code using fast approximations of it. The main advantage of these
extensions is that it not only provides the new locations where to perform
simulations but also which versions of code have to be simulated (between the
complex one or one of its fast approximations). A real multi-fidelity
application is used to illustrate the efficiency of the proposed approaches. In
this example, the accurate code is a two-dimensional finite element model and
the less accurate one is a one-dimensional approximation of the system.
