Quantifying uncertainties on excursion sets under a Gaussian random field prior

TL;DR

This paper introduces a Bayesian method for efficiently estimating and quantifying uncertainties of excursion sets of functions modeled as Gaussian random fields, using optimal Monte Carlo sampling and affine predictors to reduce computational costs.

Contribution

It proposes an optimal sampling strategy and a reconstruction approach for Gaussian random fields to accurately quantify excursion set uncertainties with lower computational effort.

Findings

Reduces Monte Carlo simulation costs significantly.

Enables detailed uncertainty quantification on fine grids.

Successfully applied to a safety engineering case study.

Abstract

We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective function gives rise to a posterior distribution on excursion sets. Several approaches exist to summarize the distribution of such sets based on random closed set theory. While the recently proposed Vorob'ev approach exploits analytical formulae, further notions of variability require Monte Carlo estimators relying on Gaussian random field conditional simulations. In the present work we propose a method to choose Monte Carlo simulation points and obtain quasi-realizations of the conditional field at fine designs through affine predictors. The points are chosen optimally in the sense that they minimize the posterior expected distance in measure between the excursion set and its reconstruction. The proposed method reduces the computational costs due to Monte Carlo simulations and enables the computation of quasi-realizations on fine designs in large dimensions. We apply this reconstruction approach to obtain realizations of an excursion set on a fine grid which allow us to give a new measure of uncertainty based on the distance transform of the excursion set. Finally we present a safety engineering test case where the simulation method is employed to compute a Monte Carlo estimate of a contour line.