We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $\pi$ on R d , based on (over-damped) Langevin diffusions. This method both inspired by [PP18] and [GMS + 20] relies on a multilevel occupation measure, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with respective steps $\gamma$r = $\gamma$02 --r , r = 0,. .. , R. For a given diffusion, we first state a result under general assumptions which guarantees an $\epsilon$-approximation (in a L 2-sense) with a cost proportional to $\epsilon$ --2 (i.e. proportional to a Monte-Carlo method without bias) or $\epsilon$ --2 | log $\epsilon$| 3 under less contractive assumptions. This general result is then applied to over-damped Langevin diffusions in a strongly convex setting, with a study of the dependence in the dimension d and in the spectrum of the Hessian matrix D 2 U of the potential U : R d $\rightarrow$ R involved in the Gibbs distribution. This leads to strategies with cost in O(d$\epsilon$ --2 log 3 (d$\epsilon$ --2)) and in O(d$\epsilon$ --2) under an additional condition on the third derivatives of U. In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by ($\lambda$ U $\lor$1) 2 $\lambda$ 3 U d$\epsilon$ --2 (where$\lambda$U and $\lambda$ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U). In our numerical illustrations, we show that our theoretical bounds are confirmed in practice and finally propose an opening to some theoretical or numerical strategies in order to increase the robustness of the procedure when the largest and smallest eigenvalues of D 2 U are respectively too large or too small.

中文翻译：

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吉布斯近似的多级朗之万路径平均

我们提出并研究了一种新的多级方法，用于基于（过阻尼）朗之万扩散对 R d 上的吉布斯分布 $\pi$ 进行数值逼近。这种受[PP18]和[GMS + 20]启发的方法依赖于多级占用度量，即Langevin扩散的（恒定步长）离散化方案的R占用度量的适当组合，具有各自的步骤$\gamma$r = $\gamma$02 --r , r = 0,. .. , R. 对于给定的扩散，我们首先在一般假设下陈述一个结果，该结果保证了 $\epsilon$-近似值（在 L 2 意义上），其成本与 $\epsilon$ --2 成正比（即成比例到没有偏差的蒙特卡罗方法）或 $\epsilon$ --2 | 记录 $\epsilon$| 3 在较少收缩的假设下。然后将此一般结果应用于强凸设置中的过阻尼朗之万扩散，研究了维度 d 和潜在 U 的 Hessian 矩阵 D 2 U 的谱中的相关性：R d $\rightarrow$ R 参与吉布斯分布。这导致在第三个附加条件下，成本为 O(d$\epsilon$ --2 log 3 (d$\epsilon$ --2)) 和 O(d$\epsilon$ --2) 的策略derivatives of U. In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by ($\lambda$ U $\ lor$1) 2 $\lambda$ 3 U d$\epsilon$ --2 （其中$\lambda$U 和$\lambda$ U 分别表示D 2 U 的最大和最小特征值的上界和下界）。在我们的数字插图中，