  Kinetic transport equations are notoriously difficult to simulate because of
their complex multiscale behaviors and the need to numerically resolve a high
dimensional probability density function. Past literature has focused on
building reduced order models (ROM) by analytical methods. In recent years,
there is a surge of interest in developing ROM using data-driven or
computational tools that offer more applicability and flexibility. This paper
is a work towards that direction.
  Motivated by our previous work of designing ROM for the stationary radiative
transfer equation in [30] by leveraging the low-rank structure of the solution
manifold induced by the angular variable, we here further advance the
methodology to the time-dependent model. Particularly, we take the celebrated
reduced basis method (RBM) approach and propose a novel micro-macro decomposed
reduced basis method (MMD-RBM). The MMD-RBM is constructed by exploiting, in a
greedy fashion, the low-rank structures of both the micro- and macro-solution
manifolds with respect to the angular and temporal variables. Our reduced order
surrogate consists of: reduced bases for reduced order subspaces and a reduced
quadrature rule in the angular space. The proposed MMD-RBM features several
structure-preserving components: 1) an equilibrium-respecting strategy to
construct reduced order subspaces which better utilize the structure of the
decomposed system, and 2) a recipe for preserving positivity of the quadrature
weights thus to maintain the stability of the underlying reduced solver. The
resulting ROM can be used to achieve a fast online solve for the angular flux
in angular directions outside the training set and for arbitrary order moment
of the angular flux.
