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 ones 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 a prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given scheme.
翻译:我们研究了低劣的兰埃文动态的离散方案在整体变差和以美元-美元-最低值为单位的趋同情况,这种算法在分子动态和计算统计中非常受欢迎,通常用来从利息目标分配中进行近似抽样。我们首先表明,对于一大批方案来说,分级办法有一个细化条件的统一,这一类包括欧勒-马鲁山方案和以分化战略为基础的不同方法等流行方法。第二,我们提供了温和的条件,确保我们认为符合几何Foster-Lyapunov漂移条件的一类方案,在分级办法中再次统一。这使我们能够得出几何趋同界限,与分级办法的递增率成直线缩。这种结果对于获得与某一方案相关的Poisson方程式解决办法准则的估计极感兴趣。