A self-similarity principle for the computation of rare event probability

Abstract

The probability of rare and extreme events is an important quantity for design purposes. However, computing the probability of rare events can be expensive because only a few events, if any, can be observed. To this end, it is necessary to accelerate the observation of rare events using methods such as the importance splitting technique, which is the main focus here. In this work, it is shown how a genealogical importance splitting technique can be made more efficient if one knows how the rare event occurs in terms of the mean path followed by the observables. Using Monte Carlo simulations, it is shown that one can estimate this path using less rare paths. A self-similarity model is formulated and tested using an a priori and a posteriori analysis. The self-similarity principle is also tested on more complex systems including a turbulent combustion problem with 10 7 degrees of freedom. While the self-similarity model is shown to not be strictly valid in general, it can still provide a good approximation of the rare mean paths and is a promising route for obtaining the statistics of rare events in chaotic high-dimensional systems.

GANISP: a GAN-assisted Importance SPlitting Probability Estimator

“…For both L96 and KSE, the GAMS algorithm is run with 100 concurrent simulations. The weights assigned to each simulation (that are used to decide how many simulations are cloned or pruned) are obtained using the method of Hassanaly and Raman (2019) where the most likely average path is computed with 100 simulations. In both simulations, the target level of Q is the one that corresponds to a probability of the order of 10 −1 (Q = 2.0 for KSE and Q = 1300 for L96).…”

“…As is common practice, the QoI is also the reaction coordinate (Wouters and Bouchet 2016;Lestang, Bouchet, and Lévêque 2020). The instantaneous value of the reaction coordinate along with heuristics on the most likely rare path (Hassanaly and Raman 2019) are used to dictate which realizations to clone or prune. In the original formulation of the GAMS method, a small perturbation of the form εη is added to every variable that defines the state of the cloned trajectories, where ε is sufficiently small to not affect the probability to estimate, and η is drawn from a standard normal distribution.…”

“…The "no-perturbation" strategy was successfully used to model zonal jet instabilities (Bouchet, Rolland, and Simonnet 2019;Simonnet, Rolland, and Bouchet 2021), drifting equation with Brownian noise (Grafke and Vanden-Eijnden 2019;Wouters and Bouchet 2016), and molecular dynamics (Teo et al 2016). When applied to deterministic systems, random perturbations have been also been successful, such as for the Lorenz 96 equation (Wouters and Bouchet 2016;Hassanaly and Raman 2019). However, when applied to fluid flow behind a bluff body, the random perturbation strategy was observed to fail at generating diverse rare event trajectories (Lestang, Bouchet, and Lévêque 2020).…”

“…The data used to train the model can be collected from unbiased trajectories simulated. In the GAMS algorithm, it is common to first perform a rough MC estimate to determine how to appropriately choose the number of clones to generate (Wouters and Bouchet 2016;Hassanaly and Raman 2019). These realizations are also leveraged here to collect the data used by the cGAN.…”

“…Importance splitting is one such approach that creates a bias towards trajectories that trend towards the desired rare event (Kahn and Harris 1951). This work focuses on a variant of importance splitting called genealogical adaptive multilevel splitting (GAMS) (Del Moral and Garnier 2005;Cérou and Guyader 2007) that can be used for deterministic systems (Wouters and Bouchet 2016;Hassanaly and Raman 2019). A graphical illustration of the method is shown in Fig.…”