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

TL;DR

This paper develops a new high-order integrator construction method for sampling invariant measures of ergodic SDEs on manifolds, extending aromatic Butcher-series formalism and validating with numerical experiments.

Contribution

It introduces a novel methodology for deriving high-order sampling integrators for constrained ergodic SDEs, applicable to various manifolds.

Findings

Order conditions for Runge-Kutta methods on manifolds derived

A second-order method demonstrated with numerical experiments

Validation on sphere, torus, and special linear group

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.