Moving Particles: a parallel optimal Multilevel Splitting method with application in quantiles estimation and meta-model based algorithms

摘要

Considering the issue of estimating small probabilities p, ie. measuring arare domain F = {x | g(x) > q} with respect to the distribution of a randomvector X, Multilevel Splitting strategies (also called Subset Simulation) aimat writing F as an intersection of less rare events (nested subsets) such thattheir measures are conditionally easily computable. However the definition ofan appropriate sequence of nested subsets remains an open issue. We introduce here a new approach to Multilevel Splitting methods in terms ofa move of particles in the input space. This allows us to derive two mainresults: (1) the number of samples required to get a realisation of X in F isdrastically reduced, following a Poisson law with parameter log 1/p (to becompared with 1/p for naive Monte-Carlo); and (2) we get a parallel optimalMultilevel Splitting algorithm where there is indeed no subset to define anymore. We also apply result (1) in quantile estimation producing a new parallelalgorithm and derive a new strategy for the construction of first Design OfExperiments in meta-model based algorithms.