Rare Event Simulation and Splitting for Discontinuous Random Variables

Multilevel Splitting methods, also called Sequential Monte-Carlo or \emph{Subset Simulation}, are widely used methods for estimating extreme probabilities of the form $P[S(\mathbf{U}) > q]$ where $S$ is a deterministic real-valued function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Very often, $X := S(\mathbf{U})$ is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in $S$ and/or $\mathbf{U}$ is discrete or mixed discrete/continuous this assumption does not hold any more and the estimator is not consistent. In this paper, we study the impact of discontinuities in the \emph{cdf} of $X$ and present three unbiased \emph{corrected} estimators to handle them. These estimators do not require to know in advance if $X$ is actually discontinuous or not and become all equal if $X$ is continuous. Especially, one of them has the same statistical properties in any case. Efficiency is shown on a 2-D diffusive process as well as on the \emph{Boolean SATisfiability problem} (SAT).Comment: 16 pages (12 + Appendix 4 pages), 6 figure

Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.Comment: Submitted to Stochastic Processes and their Application

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative precision, where {\beta} is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page

Unbiased simulation of rare events in continuous time

For rare events described in terms of Markov processes, truly unbiased estimation of the rare event probability generally requires the avoidance of numerical approximations of the Markov process. Recent work in the exact and ε-strong simulation of diffusions, which can be used to almost surely constrain sample paths to a given tolerance, suggests one way to do this. We specify how such algorithms can be combined with the classical multilevel splitting method for rare event simulation. This provides unbiased estimations of the probability in question. We discuss the practical feasibility of the algorithm with reference to existing ε-strong methods and provide proof-of-concept numerical examples

Rare-event sampling applied to the simulation of extreme mechanical eorts exerted by a turbulent ow on a blu body

This study evaluates the relevance of rare-event sampling techniques to accelerate the simulation of extreme mechanical eorts exerted by a turbulent ow impinging onto a blu body. The main idea is to replace a long simulation by a set of much shorter ones, running in parallel, with dynamics that are replicated or pruned in order to sample large-amplitude events more frequently. Such techniques have been shown to be ecient for a wide range of problems in statistical physics, computer science, biochemistry, enabling the simulation of rare events otherwise out of reach by direct sampling. This work is the rst application to uid-structure interaction problems. The drag experienced by a squared obstacle placed in a turbulent ow (in two dimensions) is taken as a representative case study to investigate the performance of two major rare-event sampling algorithms, namely the Adaptive Multilevel Splitting (AMS) and the Giardina-Kurchan-Tailleur-Lecomte (GKTL) algorithms. Practical evidence is given that the fast sweeping-time of uid structures past the obstacle has a drastic inuence on the eciency of these two algorithms. While it is shown that the AMS algorithm does not yield signicant run-time savings, the GKTL algorithm appears to be ecient to sample extreme uctuations of the time-averaged drag and estimate related statistics such as return times. Beyond the study of applicability of rare-event sampling techniques to a uid-mechanical problem, this work also includes a detailed phenomenological description of extreme-drag events of a turbulent ow on a blu body