Riemannian manifold Hamiltonian Monte Carlo is traditionally carried out
using the generalized leapfrog integrator. However, this integrator is not the
only choice and other integrators yielding valid Markov chain transition
operators may be considered. In this work, we examine the implicit midpoint
integrator as an alternative to the generalized leapfrog integrator. We discuss
advantages and disadvantages of the implicit midpoint integrator for
Hamiltonian Monte Carlo, its theoretical properties, and an empirical
assessment of the critical attributes of such an integrator for Hamiltonian
Monte Carlo: energy conservation, volume preservation, and reversibility.
Empirically, we find that while leapfrog iterations are faster, the implicit
midpoint integrator has better energy conservation, leading to higher
acceptance rates, as well as better conservation of volume and better
reversibility, arguably yielding a more accurate sampling procedure.

本文比较了广义 leapfrog integrator 和 implicit midpoint integrator 在 Hamiltonian Monte Carlo 中的优缺点和理论性质，并通过实验证明 implicit midpoint integrator 不仅能更好地保持能量和体积守恒，而且采样效果更准确。

评估引理中点积分器在黎曼流形哈密尔顿蒙特卡罗中的应用

Evaluating the Implicit Midpoint Integrator for Riemannian Manifold  Hamiltonian Monte Carlo

We show how to derive a simple integrator for the Langevin equation and
illustrate how it is possible to check the accuracy of the obtained
distribution on the fly, using the concept of effective energy introduced in a
recent paper [J. Chem. Phys. 126, 014101 (2007)]. Our integrator leads to
correct sampling also in the difficult high-friction limit. We also show how
these ideas can be applied in practical simulations, using a Lennard-Jones
crystal as a paradigmatic case.

本文介绍了如何为随机 Langevin 方程导出简单的积分器，并展示了如何使用最近一篇论文中引入的有效能概念实时检查所获得分布的准确性。我们的积分器在高摩擦极限下也能正确采样。同时，我们还展示了如何将这些思想应用于实际模拟，以 Lennard-Jones 晶体为典型案例。