We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nystr\"{o}m methods, tailored for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers, such as unadjusted Hamiltonian Monte Carlo and unadjusted kinetic Langevin Monte Carlo. We establish quantitative $5/2$-order $L^2$-accuracy upper bounds under gradient and Hessian Lipschitz assumptions on the potential energy function. The numerical experiments demonstrate the superior efficiency of the proposed unadjusted samplers on a variety of well-behaved, high-dimensional target distributions.

翻译：本文针对非可逆马尔可夫链蒙特卡洛采样器（如非调整哈密顿蒙特卡洛和非调整动能朗之万蒙特卡洛）中哈密顿流的近似问题，提出了具有$5/2$阶与$7/2$阶$L^2$精度的随机化龙格-库塔-尼斯特伦方法。在势能函数满足梯度与海森矩阵利普希茨连续性的假设下，我们建立了严格的$5/2$阶$L^2$精度上界定量分析。数值实验表明，所提出的非调整采样器在多种性质良好的高维目标分布上均表现出卓越的计算效率。