The Vlasov equation describes the motion of charged particles in phase space. It is non linearly coupled to a field solver. We shall consider the Poisson equation in this talk.
Many simulations can be performed using a cartesian grid or a mapped cartesian grid. The particle distribution function, which is the solution of the Vlasov equations, can then be represented on a 6D logical grid. This yields a huge amount of data for which computations and storage is necessary. We are going to present two ideas, which enable to reduce considerably the amount of data that is actually needed at the price of a small loss of accuracy compared to a full grid simulation on the same grid.
Our simulations were performed using the semi-Lagrangian method, which consists in solving particle trajectories and interpolations, these will need to be done with our reduced data format. The first approach we are going to describe is a sparse grid method, specially adapted to our problem by using a multiplicative delta f idea, and the second one is based on a low rank decomposition of the 6D tensor representing the distribution function, for which we tried a hierarchical Tucker decomposition and the Tensor Train decomposition. These ideas are quite general and can be used either only for data compression for storage or message passing or directly for computations. (Joint work with Katharina Kormann)