# Approximating Gaussian Process Emulators with Linear Inequality Constraints and Noisy Observations via MC and MCMC

## TL;DR

This paper develops Monte Carlo and MCMC methods for Gaussian process emulators that incorporate linear inequality constraints and noisy observations, improving realism and computational efficiency in high-dimensional applications.

## Contribution

It introduces a noise relaxation approach for constrained Gaussian processes and demonstrates enhanced performance with noisy data in complex, real-world scenarios.

## Key findings

- MC and MCMC samplers perform better with noisy observations.
- Constrained GPs provide more realistic emulators in flood modeling.
- The methods are effective in high-dimensional applications.

## Abstract

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.