Mathematical Colloquium

Friday, January 11, 2019, 2:30pm

The parareal in time method or how to use different processors to simulate in parallel time dependant PDE’s

Yvon Maday (Sorbonne U./IUF/Brown U.)

Solving complex models with high accuracy and within a reasonable computing time has motivated the search for numerical schemes that exploit efficiently parallel computing architectures. For a given Partial Differential Equation (PDE), one of the main ideas to parallelize a simulation is to break the problem into subproblems defined over subdomains of a partition of the original domain. The domain can potentially have high dimensionality and be composed of different variables like space, time, velocity or even more specific variables for some problems. While there exist algorithms with very good scalability properties for the decomposition of the spatial variable in elliptic and saddle-point problems, the same cannot be said for the decomposition of time of even simple systems of ODEs. This is despite the fact that research on time domain decomposition is currently very active and has by now a history of at least 50 years (back to at least to work of J. Nievergelt in 1964) during which several algorithms have been explored. As a consequence, time domain decomposition is to date only a secondary option when it comes to deciding what algorithm/method distributes the tasks in a parallel cluster… and this is certainly a pity.

In 2001, with J.L. Lions and G. Turinici, we have introduced the parareal algorithm that has put back some attention on this issue. Our algorithm has achieved accelerations in wall clock restitution thanks to the use of decomposition of the time interval into sub-intervals each of them being treated on independent processors. Since then, the number of contributions has increased and a community is meeting regularly to treat the remaining difficulties, with a success that leads groups in industries to consider parallelism in time as a realistic avenue.

We shall present in details the parareal in time method, with the current achievements, in particular we shall present a recent version that allow to expect full efficiency of the algorithms in terms of parallel speed-up. We shall also present some open problems that remain to be solved.

Der Vortrag beginnt um 14.30 Uhr in Raum 008 (SeMath, Pontdriesch 14-16).

Im Anschluss wird zu Kaffee/Tee eingeladen.

Location: Raum 008/SeMath, Pontdriesch 14-16, 52062 Aachen