- We developed a unified performance model for the various PinT algorithms and presented it at the PinT Workshop in Marseilles. A paper describing it is currently under review. Improvements regarding the load balancing strategies have been identified based on this model and more systematic studies are currently underway.
- We made significant progress towards establishing MPI extensions supporting dynamic resource utilisation in an efficient way based on MPI Sessions. Time-X is currently the main driver behind this progress in the MPI Session working group, targeting the genericity of these interfaces also beyond Time-X. This already resulted in two publications.
- We started to implement time-adaptive spectral deferred correction methods within the pySDC framework. First steps have been taken to extend this to time-parallel methods on small- and larger-scales. Furthermore, we published a theoretical analysis of the impact of reduced coarse level resolution on convergence of Parareal.
- We developed two new PinT methods for solving optimal control problems or more generally time dependent PDE constraint optimization problems, called ParaOpt1 and ParaOpt2, which use the structure of the coupled forward and backward problems directly. We obtained a complete converge analysis of the new ParaOpt1 algorithm, which appeared in a joint paper. Time-X enabled an additional contribution, unforeseen at the time of writing the proposal, that complements the above work, involving a collaboration with application experts in turbulent flow simulations. We also have a complete convergence analysis of ParaOpt2.
- We formulated general iterative PinT algorithms together with other project partners using the technique of block iterations, which led to a general framework of analysis using generating functions for Parareal, MGRIT, PFASST and STMG. A manuscript is currently in revision.
- We investigated how the model error of the approximate model influences the convergence of the micro-macro Parareal algorithms, both from a theory side as using numerical experiments. We performed numerical experiments for various (nonlinear) stochastic differential equations. We are developing methods that aim at speeding up the simulation of stochastic differential equations by using moment models. Several publications are in preparation. The upscaling for differential algebraic equations similar to the micro/macro Parareal was investigated and the results are published. A publication describing the application of the developed approach is in preparation.
- We designed an adaptive variant of the Parareal algorithm, specifically tailored to molecular dynamics simulations. A significant gain in efficiency is obtained in comparison to the standard Parareal algorithm. We then implemented this algorithm in LAMMPS (a very well distributed software in the material science community), that will open the way to the simulation of realistic physical systems. A manuscript collecting the results is currently in preparation and should be submitted soon.
- We accomplished the design of a PinT multirate explicit stabilised method with applications to cardiac electrophysiology. Preliminary results are promising. A conference proceeding has appeared and a paper is in preparation.
- We worked on space/time multigrid preconditioners for fractional diffusion equations (FDEs). We started by considering a FDE with non-smooth solutions and built tailored multigrid preconditioners. Then we considered the case of tempered FDEs. We provided a spectral study and exploited it to build tailored multigrid solvers. Finally, we dealt with a new method for solving time-dependent n-dimensional PDEs through a particular approach. Each of the first two topics led to a publication, while the paper related to the third one is in preparation.
