Approximating Gaussian Process Emulators with Linear Inequality Constraints and Noisy Observations via MC and MCMC
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Adding inequality constraints (e.g. positivity, monotonicity, convexity) in Gaussian processes (GPs) leads 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 MC (MCMC) methods. However, strictly interpolating the observations may entail expensive computations due to highly restrictive sample spaces. Furthermore, having emulators when data are actually noisy is also of interest for real-world applications. 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 demonstrate on various synthetic 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 emulators are obtained by considering noise effects and by enforcing the inequality constraints.
KeywordsLinear Inequality Constraints Gaussian Processes Emulators Monte Carlo and Markov Chain Monte Carlo Methods Uncertainty Quantification with Noisy Observations
This research was conducted within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research (BRGM, CEA, IFPEN, IRSN, Safran, Storengy) and academia (CNRS, Ecole Centrale de Lyon, Mines Saint-Etienne, Univ. Grenoble, Univ. Nice, Univ. Toulouse) around advanced methods for computer experiments.
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