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Preprints, Working Papers, ... Year : 2021

Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics

Abstract

We study the convergence in total variation and $V$-norm of discretization schemes of the underdamped Langevin dynamics. Such algorithms are very popular and commonly used in molecular dynamics and computational statistics to approximatively sample from a target distribution of interest. We show first that, for a very large class of schemes, a minorization condition uniform in the stepsize holds. This class encompasses popular methods such as the Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we provide mild conditions ensuring that the class of schemes that we consider satisfies a geometric Foster--Lyapunov drift condition, again uniform in the stepsize. This allows us to derive geometric convergence bounds, with a convergence rate scaling linearly with the stepsize. This kind of result is of prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given numerical method.

Dates and versions

hal-03333501 , version 1 (03-09-2021)

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Alain Durmus, Aurélien Enfroy, Éric Moulines, Gabriel Stoltz. Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics. 2021. ⟨hal-03333501⟩
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