Accelerated uncertainty propagation in two-level probabilistic studies under monotony

Double-level probabilistic uncertainty models that separate aleatory and epistemic components enjoy significant interest in risk assessment. But the expensive computational costs associated with calculations of rare failure probabilities are still a large obstacle in practice. Computing accurately a risk lower than 10−3 with 95% epistemic confidence usually requires 107–108 runs in a brute-force double Monte Carlo. For single-level probabilistic studies, FORM (First Order Reliability Analysis) is a classical recipe allowing fast approximation of failure probabilities while MRM (Monotonous Reliability Method) recently proved an attractive robust alternative under monotony. This paper extends these methods to double-level probabilistic models through two novel algorithms designed to compute a set of failure probabilities or an aleatory risk level with an epistemic confidence quantile. The first, L2-FORM (level-2 FORM), allows a rapid approximation of the failure probabilities through a combination of FORM with new ideas to use similarity between computations. L2-MRM (level-2 MRM), a quadrature approach, provides 100%-guaranteed error bounds on the results. Experiments on three flood prediction problems showed that both algorithms approximate a set of 500 failure probabilities of 10−3–10−2 or derived 95% epistemic quantiles with a total of only 500–1000 function evaluations, outperforming importance sampling, iterative FORM and regression splines metamodels.