Abstract :
[en] In many applications, including in evaluations of failure regions and in geophysical hazard assessment, we
are interested in evaluating excursion sets, that is, regions in the spatial domain where the response function
exceeds some critical value. Determining such excursion sets in the presence of uncertainties in a model is
an interesting problem connected with the theory of random sets.
A first issue connected with random excursion sets is in defining confidence regions that can properly
represent the uncertainty in the excursion sets . Here, we adopt a definition based on a generalization of the
concept of confidence regions based on previous works by [Bolin, 2015] and [French, 2015]. An outer or
inner confidence region is defined as a region that contains or is contained in the excursion set with a given
level of probability, respectively.
Such confidence regions are approximated numerically as optimal subsets within a parametric family of
subsets with the appropriate coverage probability, which provides nested approximations for the confidence
regions. A second issue, related to this numerical approximation of confidence regions, stems from the
numerical approximation of the coverage probability, which may prove challenging for computationally
intensive models and small probability levels. Here, we explore methods based on a hybrid surrogate-based
approach [Li, 2010] and subset simulation [Au, 2001] to evaluate the coverage probability.
We apply this methodology to the evaluation of confidence regions for the retreat of grounded ice in largescale simulation of the Antarctic ice sheet subject to parametric uncertainties.
[1] [Au, 2001] Au, S.-K. and Beck, J. L. (2001). Estimation of small failure probabilities in high dimensions
by subset simulation. Probabilistic Engineering Mechanics, 16(4):263–277.
[2] [Bolin, 2015] Bolin, D. and Lindgren, F. (2015). Excursion and contour uncertainty regions for latent
gaussian models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(1):85–106.
[3] [French, 2015] French, J. P. and Hoeting, J. A. (2015). Credible regions for exceedance sets of
geostatistical data. Environmetrics, 27(1):4–14.
[4] [Li, 2010] Li, J. and Xiu, D. (2010). Evaluation of failure probability via surrogate models. Journal of
Computational Physics, 229(23):8966–8980.
