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IMA Journal of Numerical Analysis Advance Access originally published online on May 9, 2008
IMA Journal of Numerical Analysis 2009 29(2):421-443; doi:10.1093/imanum/drn018
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Stochastic variational integrators

Nawaf Bou-Rabee{dagger}

Applied and Computational Mathematics (ACM), Caltech, Pasadena, CA 91125, USA

Houman Owhadi{ddagger}

Applied and Computational Mathematics (ACM) and Control and Dynamical Systems (CDS), Caltech, Pasadena, CA 91125, USA

{dagger} Email: nawaf{at}acm.caltech.edu

{ddagger} Email: owhadi{at}acm.caltech.edu

Received on 21 October 2007. Revised on 13 February 2008.


  Abstract

This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds, akin to the Ornstein–Uhlenbeck theory of Brownian motion in a force field. The main result is to derive governing SDEs for such systems from a critical point of a stochastic action. Using this result, the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analogue of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discrete variational principle. The paper shows that the discrete flow of an SVI is almost surely symplectic and in the presence of symmetry almost surely momentum-map preserving. A first-order mean-squared convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid bodies interacting via a potential.

Key Words: variational integrators; Ornstein-Uhlenbeck process; stochastic Hamiltonian systems


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