Metropolis integration schemes for self-adjoint diffusions

Research output: Contribution to journalArticle

Abstract

We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits the user to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated in several examples, including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes.

Original languageEnglish (US)
Pages (from-to)781-831
Number of pages51
JournalMultiscale Modeling and Simulation
Volume12
Issue number2
DOIs
StatePublished - 2014

Fingerprint

high gain
numerical integration
Probability distributions
Tensors
divergence
DNA
generators
deoxyribonucleic acid
tensors
Computer simulation
Metropolis-Hastings
predictions
High Order Accuracy
Brownian Dynamics
Predictor-corrector
Equilibrium Distribution
simulation
Explicit Methods
Runge-Kutta
Dynamic Simulation

Keywords

  • Brownian dynamics with hydrodynamic interactions
  • DNA simulations
  • Ergodicity
  • Explicit integrators
  • Fluctuation-dissipation theorem
  • Metropolis-hastings algorithm
  • Predictor-corrector schemes
  • Small noise limit

ASJC Scopus subject areas

  • Modeling and Simulation
  • Chemistry(all)
  • Computer Science Applications
  • Ecological Modeling
  • Physics and Astronomy(all)

Cite this

Metropolis integration schemes for self-adjoint diffusions. / Bou-Rabee, Nawaf; Donev, Aleksandar; Vanden Eijnden, Eric.

In: Multiscale Modeling and Simulation, Vol. 12, No. 2, 2014, p. 781-831.

Research output: Contribution to journalArticle

@article{695fe3ebf5dc4395be382096967ce953,
title = "Metropolis integration schemes for self-adjoint diffusions",
abstract = "We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits the user to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated in several examples, including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes.",
keywords = "Brownian dynamics with hydrodynamic interactions, DNA simulations, Ergodicity, Explicit integrators, Fluctuation-dissipation theorem, Metropolis-hastings algorithm, Predictor-corrector schemes, Small noise limit",
author = "Nawaf Bou-Rabee and Aleksandar Donev and {Vanden Eijnden}, Eric",
year = "2014",
doi = "10.1137/130937470",
language = "English (US)",
volume = "12",
pages = "781--831",
journal = "Multiscale Modeling and Simulation",
issn = "1540-3459",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - Metropolis integration schemes for self-adjoint diffusions

AU - Bou-Rabee, Nawaf

AU - Donev, Aleksandar

AU - Vanden Eijnden, Eric

PY - 2014

Y1 - 2014

N2 - We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits the user to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated in several examples, including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes.

AB - We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits the user to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated in several examples, including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes.

KW - Brownian dynamics with hydrodynamic interactions

KW - DNA simulations

KW - Ergodicity

KW - Explicit integrators

KW - Fluctuation-dissipation theorem

KW - Metropolis-hastings algorithm

KW - Predictor-corrector schemes

KW - Small noise limit

UR - http://www.scopus.com/inward/record.url?scp=84903947565&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903947565&partnerID=8YFLogxK

U2 - 10.1137/130937470

DO - 10.1137/130937470

M3 - Article

AN - SCOPUS:84903947565

VL - 12

SP - 781

EP - 831

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 2

ER -