论文标题
子曼群上的非可逆抽样方案
Non-reversible sampling schemes on submanifolds
论文作者
论文摘要
在分子动力学，计算统计力学和贝叶斯统计的各个应用领域，通常需要计算亚策略上的概率度量的平均值。近年来，文献中已经提出了各种数值方案，以根据适当可逆的约束随机动力学研究此问题。在本文中，我们介绍并分析了由一位作者[Esaim：M2an，54（2020），第391-430页]开发的基于投射方案的非可逆概括。该方案由两个步骤组成 - 从Submanifold上的状态开始，我们首先使用非可逆随机微分方程更新状态，该方程将状态带离了子曼群，在第二步中，我们使用普通微分方程的长期限制将状态投射到歧管上。我们证明了此数值方案的一致性，并根据有限的运行平均值为估计器提供了定量误差估计。此外，我们提供了理论分析，该分析表明，该方案在渐近方差方面的表现优于其可逆的对应物。我们在一个说明性的测试示例中演示了我们的发现。
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.
