Importance Sampling for Computing Extremes

Background: Extreme climate events such as prolonged heatwaves, heavy rainfall, and severe windstorms with return periods of hundreds of years or more have severe impacts when they occur. In August 2003, a 10-day heatwave resulted in 2,000 deaths in the UK and more than 70,000 deaths in western Europe (Robine et al. 2008). In winter 2013/2014, rainfall over the UK led to significant damage with 18,700 insurance claims, and clean-up costs of over £1 billion in the Thames River valley (Thompson et al. 2017). More recently, the record-setting UK heatwave of July 2022 led to an estimated 850 excess deaths over a 2-day period (Kendon 2022, Mitchell & Lo 2022).

Therefore, predicting the occurrence of such critical events is of paramount practical interest to several sectors. Particularly, it would help to make alerts by detecting possible hazards. This is crucial at both national (measures adaptation) and international (policy and strategy design) levels.

The question is why do not we rely on historical data to predict these extremes? Obviously, historical data are too short to give reliable predictions for events with a return period longer than a few decades. To surmount this challenge, researchers in climate and weather science traditionally turn to extreme value modelling as their conventional approach. However, due to the limited number of observations on the tails, large errors can result from extrapolating the extreme value statistical distribution.

An alternative approach is to use climate models to provide a much larger sample of events, potentially providing a more accurate estimate of return periods. Running a large ensemble from climate models is shown to be efficient, as Thomson et al. 2017 have shown that the 2013/2014 extreme rainfall and flooding could have been anticipated. However, the approach presented in Thompson et al. 2017 is computationally expensive for extreme events with a return period of hundred years, which requires running the climate model long enough to get a sample in the rare/extreme region (Ragone et al. 2018). More specifically, a substantial number of samples needs to be generated by the model to get a “relevant/important” sample, i.e., a sample that belongs to the rare set of interest.

Project description and objectives:

Importance sampling is a very popular technique that, when appropriately used, can dramatically reduce the computational effort when computing extreme event return periods (Kroese et al. 2011). The core concept involves performing a change in the probability measure such that we sample more in the region of interest. The crucial challenge lies in determining this change of measure to yield substantial computational savings.  Despite the continuous advances in the development of importance sampling schemes, its adoption among climate researchers remains relatively limited (Ragone et al. 2018, Ragone et al. 2021).

This proposal aims to develop computationally efficient importance sampling schemes for the computation of extreme climate events. The project comprises the following objectives:

  • Initially, the student will design an importance sampling scheme tailored to a relatively simple yet crucial probabilistic rainfall model. This serves as a foundational step before delving into more complex climate models.
  • The second objective entails the development of an efficient importance sampling scheme for estimating return periods in scenarios where dynamics evolves according to the simple yet chaotic Lorenz climate model. The Lorenz model comprises a system of coupled stochastic differential equations, necessitating a change of measure in trajectory/path space.
  • Once the student has acquired proficiency in employing importance sampling for two relatively straightforward climate models, the overarching goal is to implement importance sampling techniques within a complex real climate model from the Met Office. The objective is to draw similar conclusions as those presented in Thompson et al. (2017) in a computationally efficient manner.


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  • F Ragone et al. Computation of extreme heat waves in climate models using a large deviation algorithm. Proceedings of the National Academy of Sciences of the USA, 115 (2018), 24–29,
  • V Thompson et al. High risk of unprecedented UK rainfall in the current climate. Nature Communications, 8 (2017), 107,
  • DP Kroese, T Taimre and ZI Botev. Handbook of Monte Carlo Methods. Wiley Series in Probability and Statistics. Wiley, 2011,
  • M Kendon, Unprecedented extreme heatwave, July 2022. Met Office report.
  • DM Mitchell and Y T E Lo. Downplaying the catastrophic health impact of heatwaves costs lives. The BMJ, 378 (2022),