  Running a reliability analysis on engineering problems involving complex
numerical models can be computationally very expensive, requiring advanced
simulation methods to reduce the overall numerical cost. Gaussian process based
active learning methods for reliability analysis have emerged as a promising
way for reducing this computational cost. The learning phase of these methods
consists in building a Gaussian process surrogate model of the performance
function and using the uncertainty structure of the Gaussian process to enrich
iteratively this surrogate model. For that purpose a learning criterion has to
be defined. Then, the estimation of the probability of failure is typically
obtained by a classification of a population evaluated on the final surrogate
model. Hence, the estimator of the probability of failure holds two different
uncertainty sources related to the surrogate model approximation and to the
sampling based integration technique. In this paper, we propose a methodology
to quantify the sensitivity of the probability of failure estimator to both
uncertainty sources. This analysis also enables to control the whole error
associated to the failure probability estimate and thus provides an accuracy
criterion on the estimation. Thus, an active learning approach integrating this
analysis to reduce the main source of error and stopping when the global
variability is sufficiently low is introduced. The approach is proposed for
both a Monte Carlo based method as well as an importance sampling based method,
seeking to improve the estimation of rare event probabilities. Performance of
the proposed strategy is then assessed on several examples.
