Rare events occur in many different applications including finance, nuclear engineering and health care. Efficiently simulating these scenarios and accurately estimating the probability of such events occurring is challenging, yet crucial, for prediction, uncertainty quantification, safety and regulation. Since exact solutions are rarely available and standard Monte Carlo techniques are usually inefficient, it is necessary to develop clever, more advanced techniques to simulate rare events well. This event will consist of three talks covering different elements of theory and applications of rare event sampling.
Timetable:
1:30-2:30pm: Dr Mathias Rousset
2:30-3:30pm: Dr Francesco Romano Crucinio
3:30-4pm: Coffee break
4-5pm: Dr Tobias Grafke
5pm: Wrap up + close
Mathematical Sciences Building, University of Warwick (room 2.22)
Speaker: Mathias Rousset
Title: Fluctuations of Rare Event Simulation with Monte Carlo Splitting in the Small Noise Asymptotics.
Abstract: Rare events modeled by diffusion processes with small noise conditioned to reach a target set are considered. The AMS algorithm is a SMC-like Monte Carlo method that is used to
sample such rare events by iteratively simulating clones of the process and selecting trajectories that have reached the highest value of a so-called importance scalar function. In this paper, the relative variance of the AMS small probability estimator for large sample size is considered. The main result is a logarithmic equivalent of the latter in the small noise asymptotics, which will be rigorously derived using large deviations theory of Freidlin-Wentzel type. Interpretations and practical consequences will be discussed.
Speaker: Francesca Romano Crucinio
Title: Towards a turnkey approach to unbiased Monte Carlo estimation of smooth functions of expectations
Abstract: Given a smooth function f, we develop a general approach to turn Monte Carlo samples with expectation m into an unbiased estimate of f(m). Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion of f and estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters -- which depend on m -- automatically, with a view to make the whole approach simple to use. We develop our methods for the specific functions f(x)=logx and f(x)=1/x, as they arise in several statistical applications such as maximum likelihood estimation of latent variable models and Bayesian inference for un-normalised models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.
Speaker: Tobias Grafke
Title: Rare events via sharp large deviations estimates
Abstract: Rare events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. While sampling techniques can be modified via biasing, leading to importance sampling or multilevel splitting methods, there is a natural counterpoint for treating extremely rare events: Large deviation theory, which gets more accurate the more rare an event becomes. Unfortunately, large deviation estimates usually only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single sampling-free computation and without fitting parameters. The applicability of this method to high-dimensional real-world systems, for example coming from fluid dynamics or molecular dynamics, are discussed.
RSS Fellows - free to attend
Non-fellows - £10