  We develop a novel multilevel asymptotic-preserving Monte Carlo method,
called Multilevel Kinetic-Diffusion Monte Carlo (ML-KDMC), for simulating the
kinetic Boltzmann transport equation with a Bhatnagar-Gross-Krook (BGK)
collision operator. This equation occurs, for instance, in mathematical models
of the neutral particles in the plasma edge of nuclear fusion reactors. In this
context, the Kinetic-Diffusion Monte Carlo method is known to maintain accuracy
both in the low-collisional and the high-collisional limit, without an
exploding simulation cost in the latter. We show that, by situating this method
within a Multilevel Monte Carlo (MLMC) framework, using a hierarchy of larger
time step sizes, the simulation cost is reduced even further. The different
levels in our ML-KDMC method are connected via a new and improved recipe for
correlating particle trajectories with different time step sizes. Furthermore,
a new and more general level selection strategy is presented. We illustrate the
efficiency of our ML-KDMC method by applying it to a one-dimensional test case
with nonhomogeneous and anisotropic plasma background. Our method yields
significant speedups compared to the single-level KDMC scheme, both in the low
and high collisional regime. In the high-collisional case, our ML-KDMC
outperforms the single-level KDMC method by several orders of magnitude.
