We propose a novel discrete Poisson equation approach to estimate the statistical error of a broad class of numerical integrators for the underdamped Langevin dynamics. The statistical error refers to the mean square error of the estimator to the exact ensemble average with a finite number of iterations. With the proposed error analysis framework, we show that when the potential function $U(x)$ is strongly convex in $\mathbb R^d$ and the numerical integrator has strong order $p$, the statistical error is $O(h^{2p}+\frac1{Nh})$, where $h$ is the time step and $N$ is the number of iterations. Besides, this approach can be adopted to analyze integrators with stochastic gradients, and quantitative estimates can be derived as well. Our approach only requires the geometric ergodicity of the continuous-time underdamped Langevin dynamics, and relaxes the constraint on the time step.

翻译：我们提出了一种新颖的离散泊松方程方法，用于估计一类广泛使用的欠阻尼朗之万动力学数值积分器的统计误差。统计误差是指有限迭代次数下估计量相对于精确系综平均的均方误差。借助所提出的误差分析框架，我们证明：当势函数 $U(x)$ 在 $\mathbb R^d$ 中强凸且数值积分器具有强阶 $p$ 时，统计误差为 $O(h^{2p}+\frac1{Nh})$，其中 $h$ 为时间步长，$N$ 为迭代次数。此外，该方法可推广至分析具有随机梯度的积分器，并推导出定量估计。我们的方法仅要求连续时间欠阻尼朗之万动力学满足几何遍历性，并放宽了对时间步长的约束。