A Hierarchical Tucker Solver for the Vlasov-Maxwell System

Abstract: The Vlasov-Maxwell equations is a common model to describe the behavior of a plasma, but come at the cost of their dimensionality since both spatial and veloocity data is stored in three dimensions. This causes a significant obstacle to be able solve them numerically since the size of the data is N^6, which yields a data size of 8.59 GB for as a small problem size as N=32 stored with double precision. The rememedy that we propose in this thesis is to use the fact thatseparable functions only have the storage requirement of the individual variable domains, and not their Carteesian product as is otherwise the case. However, this decomposition of a functionrequires a transfer tensor which contains the coefficients that dictate the connection between every possible combination of the values. Notably, for higher-dimensional problems, this tensor still grows proportionally to the problem size. The solution to that problem is to split it up into a tensor tree known as a hierarchical Tucker tensor. The property that we study in this thesis is how much the ranks of the tensors in this hierarchical tensor format grows as we store the distribution function for the plasma in that format when solving the Vlasov-Maxwell problem. Ourconclusion was that the compression rate remained high throughout the simulation, but that the issue of intermediate rank growth caused the simulation to run out of random access memory, which prompts further research into mitigating that issue to fully utilize the power of this method.