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

TL;DR

This paper introduces a self-similarity principle to improve the efficiency of importance splitting methods for estimating rare event probabilities, especially in complex, high-dimensional chaotic systems.

Contribution

It proposes a novel self-similarity model to approximate rare event paths, enhancing importance splitting efficiency in high-dimensional systems.

Findings

Self-similarity model approximates rare event paths effectively.

Method reduces the number of rare paths needed for estimation.

Applicable to complex systems like turbulent combustion.

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.