Dynamical low-rank approaches for time-dependent PDEs

In this talk we present the so-called dynamical low-rank approximation for time-dependent PDEs. This approach consists in constraining the evolution of the system to a low-rank manifold, such that the time integration is performed on a lower dimensional space. By doing so, we end up with solving differential equations for the low-rank factors of the solution only and we never reconstruct the full approximation out of them, with clear computational savings.

We employ this strategy to solve matrix differential equations arising from the spatial discretization of PDEs. We also present a low-rank integrator for the solution of Vlasov-Maxwell equations, which are kinetic equations posed in an up to six-dimensional space.

This talk is based on joint work with A. Ostermann, H. Walach, and L. Einkemmer.