About the Project
Background
Extreme events such as prolonged heatwaves can have devastating societal and economic impacts. The July 2022 UK heatwave led to record-breaking temperatures and about 850 excess deaths in two days. Given their severe consequences, improving our ability to anticipate such extremes is critical for early warning systems, infrastructure planning, and long-term climate adaptation. However, forecasting extreme events using historical records poses major challenges: observational records are too short…
to reliably estimate the likelihood of events with return periods beyond a few decades, leading to substantial uncertainty in our understanding of climate extremes. Researchers therefore turn to large ensembles of climate model, which generate a richer sample of possible outcomes. Nonetheless, capturing events with 100-year return periods still requires very large ensembles and is computationally demanding, limiting our ability to reliably characterise their probabilities and uncertainties.
PhD project
The main objective is to compute probabilities of extreme events in stochastic systems governed by stochastic differential equations (SDEs). SDEs introduce randomness into deterministic climate and weather models, enabling realistic variability and explicit representation of uncertainty. In climate science, SDEs are used to model wind speed variability, quantify uncertainty in global temperature and moisture, and represent the stochastic evolution of global mean temperature and atmospheric CO₂. The project will focus on extreme heat events using simulated climate data.
The student will investigate two complementary approaches for characterising rare events and the uncertainty associated with their probabilities. The first is sampling-based, focusing on Monte Carlo methods. Standard Monte Carlo becomes prohibitively expensive for very rare events, resulting in large statistical uncertainty unless enormous numbers of simulations are used. The student will explore importance sampling, which generates rare events more frequently while retaining unbiased probability estimates. Designing efficient importance sampling methods requires solving challenging mathematical problems, particularly in high dimensions. The student will explore dimensionality reduction and machine learning techniques to address this challenge.
The second approach is based on Large Deviations for SDEs. It provides approximations of rare event probabilities through a deterministic control problem. The student will study the instanton, representing the most probable pathway leading to an extreme event. This provides information not only about its probability but also about how uncertain stochastic dynamics can generate extreme behaviour.
Finally, the student will connect the two approaches by using the instanton to design efficient importance sampling strategies. This will provide new tools to better characterise uncertainty in rare climate events while substantially reducing the computational effort.
Applicant Profile
We are looking for a student with a strong background in mathematics or statistics who is eager to apply their skills to climate science. The ideal candidate will have experience in Python programming and a solid understanding of probability. Familiarity with stochastic calculus and stochastic numerical methods would be advantageous. Knowledge of mathematical physics would also be beneficial but is not essential. The student should be motivated to develop new mathematical and computational techniques and apply them to understanding uncertainty and rare extreme events in climate systems.
Funding Notes
This project is part of the UNRISK CDT, which offers 15-18 fully-funded NERC studentships, covering full university tuition fees; a personal stipend at standard UKRI rates; £6000 individual research and training costs; £5000 (per student) of cohort-level training; and a ‘Flexible Fund’ for special projects.
International applicants will need to cover costs related to applying for a student visa and the international health surcharge (IHS).
Applications are open to UK and international applicants. The number of awards for international applicants is limited by UKRI rules.
More information is available on the UNRISK website: View Website.
References
Ben Rached, N., Haji-Ali, A. L., Subbiah Pillai, S. M., & Tempone, R. (2025). Multilevel importance sampling for rare events associated with the McKean–Vlasov equation. Statistics and Computing, 35(1), 1.
Ben Rached, N., Haji-Ali, A. L., Subbiah Pillai, S. M., & Tempone, R. (2024). Double-loop importance sampling for McKean–Vlasov stochastic differential equation. Statistics and Computing, 34(6), 197.
Schorlepp, T., Tong, S., Grafke, T., & Stadler, G. (2023). Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems. Statistics and Computing, 33(6), 137.