Micro-macro Parareal algoritmes voor de simulatie van multischaal gewone differentiaalvergelijkingen en stochastische differentiaalvergelijkingen

Parallel-in-time (PinT) methods aim to parallelise the computer simulation of initial-value problems. Research on PinT methods has a long history. Recently, renewed attention has been given to this topic in the light of (i) the development and the availability of new hardware, more specifically of massively parallel machines, and (ii) the saturation of other, perhaps more straightforward, sources of parallelism (such as parallelisation over the spatial dimension). PinT methods have been designed for simulation, for optimisation with differential equations as constraints, and for data assimilation. Our research builds on a line of research where Parareal, a prototypical PinT method, has been extended to the simulation of multiscale systems, under the name micro-macro Parareal. In this thesis we analyse, through convergence theory and numerical simulations, the use of micro-macro Parareal for the simulation of (i) multiscale ordinary differential equations (ODEs) and (ii) (McKean-Vlasov) stochastic differential equations (SDEs). Multiscale ODEs are characterised by the occurrence of different scales, here, time scales. McKean-Vlasov SDEs model the time-evolution of an ensemble of particles that is subject to noise. For each particle, the evaluation of the drift and diffusion coefficients not only depends on the individual particle itself but also on the distribution of all particles in the ensemble. Our main contribution is the Monte Carlo-moments (MC-moments) Parareal algorithm. It combines a very cheap coarse solver that approximates the statistical moments of an (McKean-Vlasov) SDE, with a time-parallel application of an expensive Monte Carlo method.