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.
Highlights
We consider systems of stochastic differential equations (SDEs) in Rd subject to a smooth scalar constraint and a Stratonovich noise of the form dX (t) = M(X (t)) f (X (t))dt + M(X (t)) (X (t)) ◦ dW (t), X (0) = X0 ∈ M, (1.1)Communicated by Christian Lubich
Foundations of Computational Mathematics where M : Rd → Rd×d is the orthogonal projection on the tangent bundle of the manifold M = {x ∈ Rd, ζ (x) = 0} of codimension q, ζ : Rd → Rq is a given constraint, f : Rd → Rd is a smooth drift, : Rd → Rd×d is a smooth diffusion coefficient, and W is a standard d-dimensional Brownian motion in Rd on a probability space equipped with a filtration and fulfilling the usual assumptions
We mention the recent work [39], which introduced the exotic aromatic B-series for the computation of order conditions for sampling the invariant measure of ergodic SDEs in Rd, and that we extend in this paper to the context of SDEs on manifolds
Summary
We consider systems of stochastic differential equations (SDEs) in Rd subject to a smooth scalar constraint and a Stratonovich noise of the form dX (t) = M(X (t)) f (X (t))dt + M(X (t)) (X (t)) ◦ dW (t), X (0) = X0 ∈ M, (1.1)
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