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Mathematics > Numerical Analysis

arXiv:0712.4123 (math)
[Submitted on 26 Dec 2007 (v1), last revised 13 Jan 2010 (this version, v5)]

Title:Long-Run Accuracy of Variational Integrators in the Stochastic Context

Authors:Nawaf Bou-Rabee, Houman Owhadi
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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 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 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 without error. Numerical validation is provided using explicit variational integrators with first, second, and fourth order accuracy.
Comments: 30 pages
Subjects: Numerical Analysis (math.NA)
MSC classes: 65C30
Cite as: arXiv:0712.4123 [math.NA]
  (or arXiv:0712.4123v5 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.0712.4123
Related DOI: https://doi.org/10.1137/090758842

Submission history

From: Nawaf Bou-Rabee [view email]
[v1] Wed, 26 Dec 2007 04:13:08 UTC (152 KB)
[v2] Mon, 28 Jan 2008 14:16:47 UTC (94 KB)
[v3] Mon, 2 Jun 2008 08:57:44 UTC (13 KB)
[v4] Mon, 11 May 2009 21:16:53 UTC (18 KB)
[v5] Wed, 13 Jan 2010 17:49:23 UTC (21 KB)
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