Fluctuations in the heterogeneous multiscale methods for fast-slow systems

TL;DR

This paper analyzes how heterogeneous multiscale methods handle fluctuations in fast-slow systems, revealing limitations of standard HMM and proposing a modified parallel HMM that accurately captures fluctuations as per CLT and LDP.

Contribution

It introduces a simple modification to HMM, called parallel HMM, which correctly captures fluctuations in fast-slow systems, improving upon the standard HMM approach.

Findings

Standard HMM artificially amplifies fluctuations.

Parallel HMM accurately captures fluctuations at CLT and LDP levels.

The approach extends to justify tau-leaping method in stochastic simulations.

Abstract

How heterogeneous multiscale methods (HMM) handle fluctuations acting on the slow variables in fast-slow systems is investigated. In particular, it is shown via analysis of central limit theorems (CLT) and large deviation principles (LDP) that the standard version of HMM artificially amplifies these fluctuations. A simple modification of HMM, termed parallel HMM, is introduced and is shown to remedy this problem, capturing fluctuations correctly both at the level of the CLT and the LDP. Similar type of arguments can also be used to justify that the tau-leaping method used in the context of Gillespie's stochastic simulation algorithm for Markov jump processes also captures the right CLT and LDP for these processes.