子流形上的不可逆采样格式
Non-reversible sampling schemes on submanifolds
摘要：
在分子动力学、计算统计力学和贝叶斯统计等应用领域中，常常需要计算子流形上概率测度的平均值。近年来，在文献中提出了各种数值方案来研究这个问题的基础上适当的可逆约束随机动力学。在本文中，我们提出并分析了一位作者开发的基于投影的方案的不可逆推广[ESAIM：M2 AN，54（2020），pp. 391-430]。该方案由两个步骤组成-从子流形上的状态开始，我们首先使用不可逆的随机微分方程将状态从子流形上带走，并在第二步中，我们使用常微分方程的长时间限制将状态投影回流形上。我们证明了这种数值方案的一致性，并提供定量的误差估计的基础上有限时间运行平均值的估计。此外，我们提出的理论分析表明，该方案优于其可逆的对应的渐近方差。我们证明了我们的研究结果的一个说明性的测试例子。
英文摘要：
Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes have been proposed in the literature to study this problem based on appropriate reversible constrained stochastic dynamics. In this paper we present and analyse a non-reversible generalisation of the projection-based scheme developed by one of the authors [ESAIM: M2AN, 54 (2020), pp. 391-430]. This scheme consists of two steps - starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of an ordinary differential equation. We prove the consistency of this numerical scheme and provide quantitative error estimates for estimators based on finite-time running averages. Furthermore, we present theoretical analysis which shows that this scheme outperforms its reversible counterpart in terms of asymptotic variance. We demonstrate our findings on an illustrative test example.
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