# A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit

## TL;DR

This paper introduces a multilevel Monte Carlo particle method that efficiently simulates kinetic equations in the diffusion limit, overcoming stability and accuracy issues of classical techniques.

## Contribution

It combines multilevel Monte Carlo with asymptotic-preserving schemes to reduce bias and computational cost in simulating kinetic equations in the diffusive regime.

## Key findings

- Effective bias reduction in diffusive limit simulations
- Reduced computational cost compared to traditional methods
- Successful numerical validation of the approach

## Abstract

We propose a multilevel Monte Carlo method for a particle-based asymptotic-preserving scheme for kinetic equations. Kinetic equations model transport and collision of particles in a position-velocity phase-space. With a diffusive scaling, the kinetic equation converges to an advection-diffusion equation in the limit of zero mean free path. Classical particle-based techniques suffer from a strict time-step restriction to maintain stability in this limit. Asymptotic-preserving schemes provide a solution to this time step restriction, but introduce a first-order error in the time step size. We demonstrate how the multilevel Monte Carlo method can be used as a bias reduction technique to perform accurate simulations in the diffusive regime, while leveraging the reduced simulation cost given by the asymptotic-preserving scheme. We describe how to achieve the necessary correlation between simulation paths at different levels and demonstrate the potential of the approach via numerical experiments.