Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a Quantity of Interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in high-collisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research.

翻译：动力学方程描述了粒子在输运和碰撞效应下的位置-速度分布。在扩散标度下，当碰撞率趋于无穷大时，这些综合效应收敛为位置密度的扩散方程。尽管存在明确的极限，但当碰撞率虽高但有限时，数值模拟代价高昂，因为需要很小的时间步长。本文提出了一种渐近保持的多级蒙特卡罗粒子方案，利用扩散极限加速计算。在该方案中，我们首先对扩散极限模型进行采样，使用大时间步长计算目标量的有偏初始估计；随后通过少量更精细的输运-碰撞动力学模拟来修正偏差。多级方法的效率取决于在离散化层级体系上实现粒子相关性模拟的能力。我们提出了一种关联粒子轨迹的方法，并给出了分析和数值实验。结果表明，与先前工作相比，本方法显著降低了高碰撞率状态下粒子模拟的计算成本，展示了此类方案在多个活跃研究领域的巨大应用潜力。