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Long-Run Accuracy of Variational Integrators in the Stochastic Context

Bou-Rabee, Nawaf and Owhadi, Houman (2010) Long-Run Accuracy of Variational Integrators in the Stochastic Context. SIAM Journal on Numerical Analysis, 48 (1). pp. 278-297. ISSN 0036-1429. https://resolver.caltech.edu/CaltechAUTHORS:20100609-111014824

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Abstract

This paper presents a Lie–Trotter splitting for inertial Langevin equations (geometric Langevin algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein–Uhlenbeck flow. Assuming that the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin equations to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin equations without error. Numerical validation is provided using explicit variational integrators with first-, second-, and fourth-order accuracy.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/090758842 DOIArticle
http://arxiv.org/abs/0712.4123arXivDiscussion Paper
ORCID:
AuthorORCID
Owhadi, Houman0000-0002-5677-1600
Additional Information:© 2010 Society for Industrial and Applied Mathematics. Received by the editors May 12, 2009; accepted for publication (in revised form) January 25, 2010; published electronically April 2, 2010. This work was supported in part by DARPA DSO under AFOSR contract FA9550-07-C-0024. We wish to thank Christof Schutte and Eric Vanden-Eijnden for valuable advice. Denis Talay and Nicolas Champagnat helped sharpen the main result of the paper and put the paper in a better context.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-07-C-0024
Berlin Mathematical School (BMS)UNSPECIFIED
NSFDMS-0803095
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Subject Keywords:Lie-Trotter splitting; variational integrators; Ornstein-Uhlenbeck equations; Langevin equations; Boltzmann-Gibbs measure; geometric ergodicty
Issue or Number:1
Classification Code:AMS subject classifications: 65C30, 65C05, 60J05, 65P10
Record Number:CaltechAUTHORS:20100609-111014824
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20100609-111014824
Official Citation:Long-Run Accuracy of Variational Integrators in the Stochastic Context Nawaf Bou-Rabee and Houman Owhadi, SIAM J. Numer. Anal. 48, 278 (2010), DOI:10.1137/090758842
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18618
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:01 Jul 2010 18:54
Last Modified:03 Oct 2019 01:45

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