Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-05-22 DOI:10.1137/23m1556289

Benedict J. Leimkuhler, Daniel Paulin, Peter A. Whalley

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1226-1258, June 2024.
Abstract. We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for [math]-[math]Lipschitz, [math]-convex potentials. Our approach gives convergence rates of [math], with explicit step size restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property “[math]-limit convergent” to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have step size restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure.

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离散动能朗万动力学的收缩与收敛速率

SIAM 数值分析期刊》第 62 卷第 3 期第 1226-1258 页，2024 年 6 月。摘要。我们提供了一个框架，用于分析[math]-[math]Lipschitz、[math]-凸势能的离散动力学 Langevin 动力学的收敛性。我们的方法给出了[math]的收敛率，并有明确的步长限制，与高斯目标的稳定阈值同阶，且对摩擦参数的大区间有效。我们将这一方法应用于分子动力学和机器学习领域流行的各种积分方案。此外，我们还引入了"[math]-limit convergent "属性，以描述在高摩擦极限下收敛于过阻尼动力学的欠阻尼 Langevin 方案，这些方案的步长限制与摩擦参数无关。最后，我们提供了 BAOAB 方案的渐近偏差估计值，通过与保留不变度量的修正随机动力学进行比较，BAOAB 方案在高摩擦极限下仍然保持精确。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.