Statistical Error of Numerical Integrators for Underdamped Langevin Dynamics with Deterministic And Stochastic Gradients

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.