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.

动力学方程模拟了位置-速度相空间中颗粒的分布。通常，人们对研究在近似碰撞（对流）扩散模型成立的高碰撞状态下粒子的长期行为感兴趣。在本文中，我们考虑了扩散标度。基于经典粒子的技术在此限制范围内受到严格的时间步限制，以保持稳定性。渐近保存方案避免了此问题，但引入了额外的时间离散化误差，可能会导致较大的时间步长产生不可接受的较大偏差。在这里，我们介绍并分析了一种多级蒙特卡洛方案，该方案通过使用不同时间步长的层次结构组合估计来减少此偏差。我们演示了如何使用不同的时间步长来关联此方案中的轨迹。我们还提出了一种在多级方案中选择级别的策略。与经典的蒙特卡洛方法相比，我们的方法大大减少了对所考虑的动力学方程式进行精确仿真所需的计算量。