Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations

Particle simulations are required in many application domains. Often such simulations suffer from time-scale separation: the particles have fast characteristic dynamics, requiring small time steps, while the population-level (macroscopic) behaviour of large ensembles of such particles occurs on a much slower time scale, implying long time horizons for the simulation. This time-scale separation results in a high computational cost for classical simulation schemes. We consider hyperbolic transport equations, in which we introduce a scaling parameter $\varepsilon$ (related to the mean free particle path) to characterize the time-scale separation. In the limit where this scaling parameter tends to zero, the hyperbolic transport equation converges to a parabolic diffusion equation. Simulations of the transport equation in the small $\varepsilon$ region however, suffer from extreme time step reduction constraints to maintain stability. Asymptotic-preserving schemes, such as that proposed in~\cite{Dimarco2018}, avoid this issue, but do so while adding a model error which is linear in the time step size. In recent work~\cite{Loevbak2019}, we reduced this model error, while leveraging the computational advantage of the asymptotic-preserving scheme, using multilevel Monte Carlo~\cite{Giles2008}. The method first computes a coarse estimate with a large time step size, and correspondingly large bias. This estimate is then improved upon via a hierarchy of increasingly accurate Monte Carlo simulations using increasingly finer time step. In general, this approach reduces the computational cost, compared to directly simulating using a finer time step. However, the bias corrections and variances of differences of coupled simulations in this scheme have a different structure than typically observed in multilevel Monte Carlo applications, providing some novel insights into both the asymptotic-preserving schemes and efficient multilevel Monte Carlo simulations in this setting. In this talk, we will present some of these insights. Additionally, we will present an approach for selecting a suitable set of levels in the multilevel Monte Carlo scheme. \begin{thebibliography}{1} \bibitem{Dimarco2018} G.~Dimarco, L.~Pareschi and G.~Samaey. \newblock Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit. \newblock {\em SIAM Journal on Scientific Computing}, 40: A504--A528, 2018. \bibitem{Giles2008} M.B.~Giles. \newblock Multilevel Monte Carlo Path Simulation. \newblock {\em Operations Research}, 56(3): 607--617, 2008. \bibitem{Loevbak2019} E.~L{\o}vbak, G.~Samaey, S.~Vandewalle. \newblock A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit. \newblock {\em Submitted}, arXiv:1902.04347, 2019. \end{thebibliography}status: publishe