Optimal potential functions for the interacting particle system method

Abstract

The assessment of the probability of a rare event with a naive Monte Carlo method is computationally intensive, so faster estimation or variance reduction methods are needed. We focus on one of these methods which is the interacting particle system (IPS) method. The method is not intrusive in the sense that the random Markov system under consideration is simulated with its original distribution, but selection steps are introduced that favor trajectories (particles) with high potential values. An unbiased estimator with reduced variance can then be proposed. The method requires to specify a set of potential functions. The choice of these functions is crucial because it determines the magnitude of the variance reduction. So far, little information was available on how to choose the potential functions. This paper provides the expressions of the optimal potential functions minimizing the asymptotic variance of the estimator of the IPS method and it proposes recommendations for the practical design of the potential functions.

Extreme heat wave sampling and prediction with analog Markov chain and comparisons with deep learning

“…Data-driven approaches could be used to improve such rare event simulations. This is because knowing the probability of an event gives a prior information to the rare event algorithm (Chraibi et al, 2021). A recent work (Jacques-Dumas et al, 2023) compares SWG and CNN in computing importance functions for two simple Atlantic meridional overturning circulation models.…”

Application of the interacting particle system method to piecewise deterministic Markov processes used in reliability

“…where z − j = Φ z j−1 (t j−1 ), and t * n = t − n−1 i=0 t i . In the rest of the paper we will denote E t the set of trajectories of size t that satisfy (1), and by M(E t ) the set of bounded (Θ −1 t (S t ), E t )-measurable functions.…”

“…We display here an important hint on how to select potential functions that yield a small variance. Indeed, the theoretical expressions of the potential functions that minimize the asymptotic variance of the IPS estimator are known [1].…”

An ergodic theorem for the weighted ensemble method

“…Other unbiased methods, differing from weighted ensemble in that they usually sample finite-time quantities rather than ergodic averages, include Adaptive Multilevel Splitting [7,8,10], Forward Flux Sampling [1], and some sequential Monte Carlo methods [13,20,49]. This unbiased property allows for a relatively straightforward study of variance using martingale techniques [2,7,18,20].…”