Computing Rare Event Probabilities Using Large Deviations Theory, Path Integrals, and Sampling Based Methods

Background

Extreme climate 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 just two days. Improving our ability to anticipate such extremes is critical for early warning systems, infrastructure planning, and climate adaptation. Reliable predictions support timely alerts and informed decisions. However, forecasting extreme events using historical records poses major challenges: observational records are too short to estimate the likelihood of events with return periods beyond a few decades. Researchers turn to large ensembles of climate model simulations to improve estimates of return levels. Nonetheless, capturing events with 100-year return periods still requires very large ensembles, which is computationally demanding. This project uses simulated data and advanced mathematical modelling to improve rare-event probability estimates for robust climate risk assessment.

PhD Opportunity

The main objective is to compute probabilities of extreme events in stochastic systems governed by SDEs. These equations introduce randomness into deterministic climate models, enabling realistic variability and explicit uncertainty modelling. 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.
This project focuses on extreme heat events, such as multi-day heatwaves, which pose serious risks to public health, and infrastructure. The student will develop mathematical tools to estimate the probability of such rare events using simulated data from climate models. These synthetic datasets allow controlled validation of rare-event estimation, without the limitations of short observational records. The project aims to address the challenge of estimating probabilities of events with long return periods, which require very large ensembles and are computationally demanding. The student will gain expertise in two classes of methods for rare-event probabilities. The first is sampling-based, focusing on Monte Carlo techniques and importance sampling, a variance reduction method that provide substantial computational gains. The problem reduces to solving a nonlinear PDE that is challenging in high dimensions. The student will explore dimensionality reduction and machine learning techniques to address this. The second class of methods is based on Large Deviations theory, which involves solving a deterministic control problem and yields accurate approximations of rare-event probabilities. The student will study the instanton, the most probable path leading to the rare event, and use it to design efficient importance sampling strategies.
By improving our ability to compute the likelihood of heat extremes, this project contributes directly to climate risk assessment, supporting early warning systems, resilience planning, and climate adaptation.

Applicant Profile

We are looking for a student with a strong background in mathematics and statistics who is eager to apply their skills to climate science. Essential requirements include Python programming, a solid understanding of probability and stochastic calculus, and familiarity with stochastic numerics. Knowledge of mathematical physics would be advantageous. The ideal candidate will be motivated to develop new mathematical and computational techniques and to use them in tackling the challenges of extreme events computation in climate systems.

Other Information

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.

S P Fitzgerald and T J W Honour 2024 J. Phys. A: Math. Theor. 57 175002