It is of significant interest in many applications to sample from a high-dimensional target distribution $\pi$ with the density $\pi(\text{d} x) \propto e^{-U(x)} (\text{d} x) $, based on the temporal discretization of the Langevin stochastic differential equations (SDEs). In this paper, we propose an explicit projected Langevin Monte Carlo (PLMC) algorithm with non-convex potential $U$ and super-linear gradient of $U$ and investigate the non-asymptotic analysis of its sampling error in total variation distance. Equipped with time-independent regularity estimates for the corresponding Kolmogorov equation, we derive the non-asymptotic bounds on the total variation distance between the target distribution of the Langevin SDEs and the law induced by the PLMC scheme with order $\mathcal{O}(h |\ln h|)$. Moreover, for a given precision $\epsilon$, the smallest number of iterations of the classical Langevin Monte Carlo (LMC) scheme with the non-convex potential $U$ and the globally Lipshitz gradient of $U$ can be guaranteed by order ${\mathcal{O}}\big(\tfrac{d^{3/2}}{\epsilon} \cdot \ln (\tfrac{d}{\epsilon}) \cdot \ln (\tfrac{1}{\epsilon}) \big)$. Numerical experiments are provided to confirm the theoretical findings.

翻译：在许多应用中，基于朗之万随机微分方程（SDEs）的时间离散化方法，从具有密度 $\pi(\text{d} x) \propto e^{-U(x)} (\text{d} x)$ 的高维目标分布 $\pi$ 中采样具有重要意义。本文针对非凸势函数 $U$ 及其超线性梯度，提出一种显式投影朗之万蒙特卡洛（PLMC）算法，并在总变差距离下研究其采样误差的非渐近分析。借助相应的Kolmogorov方程的时间无关正则性估计，我们推导了朗之万SDEs目标分布与PLMC方案诱导分布之间总变差距离的非渐近界，其阶为 $\mathcal{O}(h |\ln h|)$。此外，对于给定精度 $\epsilon$，当经典朗之万蒙特卡洛（LMC）方案处理非凸势函数 $U$ 且其梯度满足全局Lipschitz条件时，其最小迭代次数可保证为 ${\mathcal{O}}\big(\tfrac{d^{3/2}}{\epsilon} \cdot \ln (\tfrac{d}{\epsilon}) \cdot \ln (\tfrac{1}{\epsilon}) \big)$ 阶。数值实验验证了理论结果。