Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds

Abstract

We derive a new methodology for the construction of high order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge-Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

arXiv comment40 pages

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Cited by in corpus (6)

• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• Multiple projection MCMC algorithms on submanifolds
• The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators
• A uniformly accurate scheme for the numerical integration of penalized Langevin dynamics
• Non-reversible sampling schemes on submanifolds
• Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations