We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $\pi$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusions. This method inspired by \cite{mainPPlangevin} and \cite{giles_szpruch_invariant} relies on a multilevel occupation measure, $i.e.$ on an appropriate combination of $R$ occupation measures of (constant-step) Euler schemes with respective steps $\gamma_r = \gamma_0 2^{-r}$, $r=0,\ldots,R$. We first state a quantitative result under general assumptions which guarantees an \textit{$\varepsilon$-approximation} (in a $L^2$-sense) with a cost of the order $\varepsilon^{-2}$ or $\varepsilon^{-2}|\log \varepsilon|^3$ under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential $U:\mathbb{R}^d\rightarrow\mathbb{R}$ and obtain an \textit{$\varepsilon$-complexity} of the order ${\cal O}(d\varepsilon^{-2}\log^3(d\varepsilon^{-2}))$ or ${\cal O}(d\varepsilon^{-2})$ under additional assumptions on $U$. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by ${(\bar{\lambda}_U\vee 1)^2}{\underline{\lambda}_U^{-3}} d\varepsilon^{-2}$ (where $\bar{\lambda}_U$ and $\underline{\lambda}_U$ respectively denote the supremum and the infimum of the largest and lowest eigenvalue of $D^2U$). We finally complete these theoretical results with some numerical illustrations including comparisons to other algorithms in Bayesian learning and opening to non strongly convex setting.

翻译：摘要： 我们提出并研究了一种基于（过阻尼）朗之万扩散的数值逼近吉布斯分布π（定义在ℝ^d上）的新多层方法。该方法受文献\cite{mainPPlangevin}和\cite{giles_szpruch_invariant}启发，依赖于多层占据测度，即利用R个具有步长γ_r = γ_0 2^{-r}（r=0,…,R）的（常步长）欧拉格式的占据测度的适当组合。首先，我们在一般假设下建立定量结果，保证在L^2意义下达到ε-逼近，其计算复杂度在较弱压缩性假设下为ε^{-2}或ε^{-2}|\log ε|^3阶。随后，我们将该方法应用于具有强凸势函数U:ℝ^d→ℝ的过阻尼朗之万扩散，并得到在U的附加假设下为O(dε^{-2}log^3(dε^{-2}))或O(dε^{-2})阶的ε-复杂度。更精确地，在忽略普适常数的前提下，适当参数选择可将计算成本控制为{(∨1)^2}{\underline{\lambda}_U^{-3}} dε^{-2}（其中和分别表示D^2U最大特征值的上确界与最小特征值的下确界）。最后，我们给出包括贝叶斯学习算法对比及向非强凸情形拓展在内的数值实验，以验证上述理论结果。