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Probabilistic learning for time-parallel solvers of complex models

Supervisors: Massimiliano Tamborrino and Tim Sullivan
Supervisors: Massimiliano Tamborrino and Tim Sullivan


Complex models in science often involve solving large systems of differential equations (DEs), whose study may be strongly limited by the wallclock time to numerically integrate them in time. For example, turbulent fusion plasma simulation can take 100-200 days to integrate over a time interval of 1s.
To tackle this, time-parallel integration methods have been proposed, but they: 1) do not account for the underlying uncertainty (e.g. model misspecification, numerical or observation errors); 2) do not scale well for high-dimensional DEs; 3) have not yet been embedded within parameter estimation algorithms. This PhD project aims to fill in one or more of these gaps.


Parareal (P) is one of the most popular time-parallel numerical schemes, combining a coarse but fast numerical solver, running over the entire time interval, with an accurate but slow solver, running in parallel over time sub-intervals. P then iteratively locates a numerical solution at each sub-interval, until a convergence criterion is met. Two variants of P inspired by stochastic approaches (SParareal, SP, in the figure) and probabilistic numerics (GParareal, GP, using Gaussian processes emulators to learn the difference between coarse and fine solvers) have been recently proposed by the Warwick supervisors.

This PhD project offers three possible avenues of investigations:

1) Development of a time-parallel solver carrying over the uncertainty of the numerical solution, returning a well calibrated probability distribution over (rather than point estimates of) the solution.

2) Development of a solver faster than P/SP/GP to solve high-dimensional systems of DEs arising from applications.
3) Embedment of P/SP/GP into simulation-based schemes (inference problems where the likelihood is unknown/intractable, but simulations from the model are possible) to bring a computational gain and offer a natural way of updating the information and inference throughout the scheme iterations.

This project is suitable for anyone with a background in mathematics, statistics, or computer science. It will involve both programming and theoretical analysis, although the exact balance can be tuned to the interests and skills of the successful candidate.

Links to HetSys Training

This project combines the classical numerical analysis with a statistical perspective, a novel combination known as probabilistic numerics (PN). This will be applications to high-dimensional systems of differential equations arising from physical sciences. Moreover, since the coarse/fine solvers used in time-parallel integration methods are often derived from simplified physical modelling assumptions, the project will encourage a multiscale/fidelity modelling approach.

The statistical/PN approach has the explicit aim of incorporating uncertainty into physical modelling, since PN methods aim to derive Bayesian posterior distributions over numerical solutions and associated quantities of interest, reflecting all sources of uncertainty, especially numerical. In practice, PN methods depart from this ideal and one also seeks to quantify this approximation error. All developed source codes will be available in open- source repositories. The project will aim to integrate them into existing ‘probnum’ open- source libraries.

Are you interested in applying for this project? Head over to our Study with Us page for information on the application process, and the HetSys training programme.

For the 2023/24 academic year, UK Research and Innovation (UKRI) funding is open to both UK and International research students. Awards pay a stipend to cover maintenance as well as paying the university fees and providing a research training support grant. For further details, please visit the HetSys Funding Page

At the University of Warwick, we strongly value equity, diversity and inclusion, and HetSys will provide a healthy working environment dedicated to outstanding scientific guidance, mentorship and personal development. Read more about life in the HetSys CDT here.

HetSys is proud to be a part of the Physics Department which holds an Athena SWAN Silver award, a national initiative to promote gender equality for all staff and students. The Physics Department is also a Juno Champion, which is an award from the Institute of Physics to recognise our efforts to address the under-representation of women in university physics and to encourage better practice for both women and men.