Fluctuations in the heterogeneous multiscale methods for fast–slow systems

David Kelly, Eric Vanden Eijnden

Research output: Contribution to journalArticle

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 theorem (CLT) and large deviation principle (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. All results in this article assume the HMM speedup factor λ to be constant and in particular independent of the scale parameter ε. Similar type of arguments can also be used to justify that the τ-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.

Original languageEnglish (US)
Article number23
JournalResearch in Mathematical Sciences
Volume4
Issue number1
DOIs
StatePublished - Dec 1 2017

Fingerprint

Multiscale Methods
Fluctuations
Large Deviation Principle
Central limit theorem
Markov Jump Processes
Stochastic Simulation
Scale Parameter
Justify
Speedup

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Theoretical Computer Science
  • Mathematics (miscellaneous)

Cite this

Fluctuations in the heterogeneous multiscale methods for fast–slow systems. / Kelly, David; Vanden Eijnden, Eric.

In: Research in Mathematical Sciences, Vol. 4, No. 1, 23, 01.12.2017.

Research output: Contribution to journalArticle

@article{46c98e67f0014af9a11015f4799d0a13,
title = "Fluctuations in the heterogeneous multiscale methods for fast–slow systems",
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 theorem (CLT) and large deviation principle (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. All results in this article assume the HMM speedup factor λ to be constant and in particular independent of the scale parameter ε. Similar type of arguments can also be used to justify that the τ-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.",
author = "David Kelly and {Vanden Eijnden}, Eric",
year = "2017",
month = "12",
day = "1",
doi = "10.1186/s40687-017-0112-2",
language = "English (US)",
volume = "4",
journal = "Research in Mathematical Sciences",
issn = "2522-0144",
publisher = "SpringerOpen",
number = "1",

}

TY - JOUR

T1 - Fluctuations in the heterogeneous multiscale methods for fast–slow systems

AU - Kelly, David

AU - Vanden Eijnden, Eric

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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 theorem (CLT) and large deviation principle (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. All results in this article assume the HMM speedup factor λ to be constant and in particular independent of the scale parameter ε. Similar type of arguments can also be used to justify that the τ-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.

AB - 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 theorem (CLT) and large deviation principle (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. All results in this article assume the HMM speedup factor λ to be constant and in particular independent of the scale parameter ε. Similar type of arguments can also be used to justify that the τ-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.

UR - http://www.scopus.com/inward/record.url?scp=85050372427&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050372427&partnerID=8YFLogxK

U2 - 10.1186/s40687-017-0112-2

DO - 10.1186/s40687-017-0112-2

M3 - Article

AN - SCOPUS:85050372427

VL - 4

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

SN - 2522-0144

IS - 1

M1 - 23

ER -