  Linear kinetic transport equations play a critical role in optical
tomography, radiative transfer and neutron transport. The fundamental
difficulty hampering their efficient and accurate numerical resolution lies in
the high dimensionality of the physical and velocity/angular variables and the
fact that the problem is multiscale in nature. Leveraging the existence of a
hidden low-rank structure hinted by the diffusive limit, in this work, we
design and test the angular-space reduced order model for the linear radiative
transfer equation, the first such effort based on the celebrated reduced basis
method (RBM).
  Our method is built upon a high-fidelity solver employing the discrete
ordinates method in the angular space, an asymptotic preserving upwind
discontinuous Galerkin method for the physical space, and an efficient
synthetic accelerated source iteration for the resulting linear system.
Addressing the challenge of the parameter values (or angular directions) being
coupled through an integration operator, the first novel ingredient of our
method is an iterative procedure where the macroscopic density is constructed
from the RBM snapshots, treated explicitly and allowing a transport sweep, and
then updated afterwards. A greedy algorithm can then proceed to adaptively
select the representative samples in the angular space and form a surrogate
solution space. The second novelty is a least-squares density reconstruction
strategy, at each of the relevant physical locations, enabling the robust and
accurate integration over an arbitrarily unstructured set of angular samples
toward the macroscopic density. Numerical experiments indicate that our method
is effective for computational cost reduction in a variety of regimes.
