Kinetic equations model distributions of particles in position-velocity phase space. Often, one is interested in studying the long-time behavior of particles in high-collisional regimes in which an approximate (advection)-diffusion model holds. In this paper we consider the diffusive scaling. Classical particle-based techniques suffer from a strict time-step restriction in this limit, to maintain stability. Asymptotic-preserving schemes avoid this problem, but introduce an additional time discretization error, possibly resulting in an unacceptably large bias for larger time steps. Here, we present and analyze a multilevel Monte Carlo scheme that reduces this bias by combining estimates using a hierarchy of different time step sizes. We demonstrate how to correlate trajectories from this scheme, using different time steps. We also present a strategy for selecting the levels in the multilevel scheme. Our approach significantly reduces the computation required to perform accurate simulations of the considered kinetic equations, compared to classical Monte Carlo approaches.

翻译：位置速度相位空间中粒子的动因方程式分布模型。 人们通常有兴趣研究在高银河系中颗粒的长期行为, 在这种高银河系中, 存在一种近似( 向导) 扩散模型。 在本文中, 我们考虑 diffusive 缩放。 经典粒子基技术在这一限制中受到严格的时间限制, 以维持稳定性 。 Asympt- 保留计划避免了这个问题, 但是引入了额外的时间分解错误, 可能导致对更大的时间步骤产生令人无法接受的巨大偏差 。 在此, 我们提出并分析一个多等级的蒙特卡洛 方案, 通过使用不同时间级大小的等级合并估计来减少这种偏差。 我们演示如何使用不同时间步骤从这个方案中得出相关的轨迹 。 我们还提出了一个选择多层次方案水平的战略 。 我们的方法大大降低了对被认为具有动动方程式进行精确模拟所需的计算, 与经典的蒙特卡洛 方法相比 。