An asymptotic preserving multilevel Monte Carlo method for particle based simulation of kinetic equations

Applications in particle simulation often suffer from the effects of a time-scale separation and stiffness. One has to perform simulations with a small time step in order to track the system's fast dynamics, but one also has to simulate over a long time-horizon in order to capture the system's slow dynamics. This time-scale separation is commonly characterized by means of a scaling parameter related to the mean free particle path, for example in hyperbolic particle-based transport equations. In the limit of a vanishing scaling parameter, the hyperbolic transport equation converges to a parabolic diffusion equation. However, simulations of the transport equation in this region suffer from extreme time step restriction constraints, and this because of stability reasons. In [1], a new Monte Carlo scheme was developed that converges in the diffusive limit, and avoids these stability constraints. This so-called asymptotic preserving Monte Carlo methods achieves its favorable convergence properties by introducing a certain linear and vanishing modelling error in the time step size. In the present work, we apply the multilevel Monte Carlo method [2] to this asymptotic preserving scheme. As such, the precise, but expensive, Monte Carlo simulation is replaced with a cheap, but biased, coarse simulation followed by a sequence of corrections that remove the bias. We will demonstrate that this multilevel method is able to accelerate the original scheme by one or two orders of magnitude. We also show that the bias corrections and variances in this scheme have a very different structure from what is typically observed in other multilevel Monte Carlo applications. In this talk some analytical results will be showcased for a two-speed equation, as well as an experimental verification thereof. An extension of the multilevel asymptotic preserving Monte Carlo method to more general transport equations will also be presented. References: [1] G. Dimarco, L. Pareschi and G. Samaey. Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit. SIAM Journal on Scientific Computing, 40: A504–A528, 2018. [2] M.B. Giles Multilevel Monte Carlo methods. Acta Numerica, 24: 259–328, 2015.status: publishe