## Abstract

The accurate estimation of rare event probabilities is a crucial problem in engineering to characterize the reliability of complex systems. Several methods such as Importance Sampling or Importance Splitting have been proposed to perform the estimation of such events more accurately (i.e., with a lower variance) than crude Monte Carlo method. However, these methods assume that the probability distributions of the input variables are exactly defined (e.g., mean and covariance matrix perfectly known if the input variables are defined through Gaussian laws) and are not able to determine the impact of a change in the input distribution parameters on the probability of interest. The problem considered in this paper is the propagation of the input distribution parameter uncertainty defined by intervals to the rare event probability. This problem induces intricate optimization and numerous probability estimations in order to determine the upper and lower bounds of the probability estimate. The calculation of these bounds is often numerically intractable for rare event probability (say 10^{−5}), due to the high computational cost required. A new methodology is proposed to solve this problem with a reduced simulation budget, using the adaptive Importance Sampling. To this end, a method for estimating the Importance Sampling optimal auxiliary distribution is proposed, based on preceding Importance Sampling estimations. Furthermore, a Kriging-based adaptive Importance Sampling is used in order to minimize the number of evaluations of the computationally expensive simulation code. To determine the bounds of the probability estimate, an evolutionary algorithm is employed. This algorithm has been selected to deal with noisy problems since the Importance Sampling probability estimate is a random variable. The efficiency of the proposed approach, in terms of accuracy of the found results and computational cost, is assessed on academic and engineering test cases.

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Balesdent, M., Morio, J. & Brevault, L. Rare Event Probability Estimation in the Presence of Epistemic Uncertainty on Input Probability Distribution Parameters.
*Methodol Comput Appl Probab* **18, **197–216 (2016). https://doi.org/10.1007/s11009-014-9411-x

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DOI: https://doi.org/10.1007/s11009-014-9411-x

### Keywords

- Epistemic uncertainty
- Surrogate model
- Importance sampling
- Rare event estimation
- Input–Output function
- Kriging

### AMS 2000 Subject Classification

- Monte Carlo methods 65C05
- Simulation 68U20