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Published 2010 | Published
Journal Article Open

Long-Run Accuracy of Variational Integrators in the Stochastic Context


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.

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.

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Published - BouRabee2010p10187Siam_J_Numer_Anal.pdf


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