  In this paper we introduce a new hierarchical model reduction framework for
the Fokker-Planck equation. We reduce the dimension of the equation by a
truncated basis expansion in the velocity variable, obtaining a hyperbolic
system of equations in space and time. Unlike former methods like the Legendre
moment models, the new framework generates a suitable problem-dependent basis
of the reduced velocity space that mimics the shape of the solution in the
velocity variable. To that end, we adapt the framework of [M. Ohlberger and K.
Smetana. A dimensional reduction approach based on the application of reduced
basis methods in the framework of hierarchical model reduction. SIAM J. Sci.
Comput., 36(2):A714-A736, 2014] and derive initially a parametrized elliptic
partial differential equation (PDE) in the velocity variable. Then, we apply
ideas of the Reduced Basis method to develop a greedy-algorithm that selects
the basis from solutions of the parametrized PDE. Numerical experiments
demonstrate the potential of this new method.
