A uniformly accurate scheme for the numerical integration of penalized Langevin dynamics

Abstract

In molecular dynamics, penalized overdamped Langevin dynamics are used to model the motion of a set of particles that follow constraints up to a parameter $\epsilon$. The most used schemes for simulating these dynamics are the Euler integrator in ${\mathbb{R}}^d$ and the constrained Euler integrator. Both have weak order one of accuracy, but work properly only in specific regimes depending on the size of the parameter $\epsilon$. We propose in this paper a new consistent method with an accuracy independent of $\epsilon$ for solving penalized dynamics on a manifold of any dimension. Moreover, this method converges to the constrained Euler scheme when $\epsilon$ goes to zero. The numerical experiments confirm the theoretical findings, in the context of weak convergence and for the invariant measure, on a torus and on the orthogonal group in high dimension and high codimension.

arXiv comment27 pages

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