PhD Studentship: Probabilistic Learning for Time-Parallel Solvers of Complex Models

University of Warwick


Massimiliano Tamborrino (Statistics), Tim Sullivan (MathematicsEngineering)


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 notscale 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.

For further details about the project and how it links to the training included in the HetSys PhD programme, please visit:

Probabilistic learning for time-parallel solvers of complex models (


Awards for both UK residents and international applicants pay a stipend to cover maintenance as well as paying the university fees and a research training support. The stipend is at the standard UKRI rate. For more details visit:

If you’re from outside the UK, the final application deadline for all courses starting in September/October is 23:59 (GMT) on 25 January 2023.

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