Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the conditional GP YN (maximum a posteriori ) converges uniformly to the interpolation thin plate splines with in equality constraints. This extends to the case of inequality constraints the correspondence established by Kimeldorf and Wahba (1970) between Bayesian estimation on stochastic process and smoothing by splines.