  Adding inequality constraints (e.g. boundedness, monotonicity, convexity)
into Gaussian processes (GPs) can lead to more realistic stochastic emulators.
Due to the truncated Gaussianity of the posterior, its distribution has to be
approximated. In this work, we consider Monte Carlo (MC) and Markov Chain Monte
Carlo (MCMC) methods. However, strictly interpolating the observations may
entail expensive computations due to highly restrictive sample spaces.
Furthermore, having (constrained) GP emulators when data are actually noisy is
also of interest for real-world implementations. Hence, we introduce a noise
term for the relaxation of the interpolation conditions, and we develop the
corresponding approximation of GP emulators under linear inequality
constraints. We show with various toy examples that the performance of MC and
MCMC samplers improves when considering noisy observations. Finally, on 2D and
5D coastal flooding applications, we show that more flexible and realistic GP
implementations can be obtained by considering noise effects and by enforcing
the (linear) inequality constraints.
