Simulation plays a major in the conception, the optimization and the certification of complex systems. Of particular interest here is the calibration of the parameters of computer models from high-dimensional physical observations. When the run times of these computer codes is high, this work focuses on the numerical challenges associated with the statistical inference. In particular, several adaptations of the Gaussian Process Regression (GPR) to the high-dimensional or functional output case are presented for the em-ulation of computer codes from limited data. Then, an adaptive procedure is detailed to minimize the calibration parameters uncertainty at the minimal computational cost. The proposed method is eventually applied to two applications that are based on dynamic simulators.