# Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations

## TL;DR

This paper evaluates a micro-macro acceleration method for stiff SDEs with scale separation, demonstrating it can use larger time steps and achieve higher accuracy than traditional microscopic or macroscopic approaches.

## Contribution

The paper introduces and numerically assesses a micro-macro acceleration algorithm that efficiently handles scale-separated stochastic differential equations, outperforming existing methods.

## Key findings

- Allows larger time steps than microscopic integrators
- Achieves higher accuracy than approximate macroscopic models
- Effectiveness depends on extrapolation step size and macroscopic variables

## Abstract

We discuss through multiple numerical examples the accuracy and efficiency of a micro-macro acceleration method for stiff stochastic differential equations (SDEs) with a time-scale separation between the fast microscopic dynamics and the evolution of some slow macroscopic state variables. The algorithm interleaves a short simulation of the stiff SDE with extrapolation of the macroscopic state variables over a longer time interval. After extrapolation, we obtain the reconstructed microscopic state via a matching procedure: we compute the probability distribution that is consistent with the extrapolated state variables, while minimally altering the microscopic distribution that was available just before the extrapolation. In this work, we numerically study the accuracy and efficiency of micro-macro acceleration as a function of the extrapolation time step and as a function of the chosen macroscopic state variables. Additionally, we compare the effect of different hierarchies of macroscopic state variables. We illustrate that the method can take significantly larger time steps than the inner microscopic integrator, while simultaneously being more accurate than approximate macroscopic models.