# A multilevel Monte Carlo method for asymptotic-preserving particle schemes in the diffusive limit

## TL;DR

This paper introduces a multilevel Monte Carlo method for particle schemes in kinetic equations that effectively reduces bias and computational cost in the diffusive limit, improving long-time particle simulations.

## Contribution

The paper develops a multilevel Monte Carlo approach that combines different time step estimates to reduce bias and computational effort in asymptotic-preserving particle schemes.

## Key findings

- Significant bias reduction with larger time steps.
- Reduced computational cost compared to classical Monte Carlo.
- Effective correlation of trajectories across levels.

## Abstract

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.