Geometric integrators and the Hamiltonian Monte Carlo method

IF 16.3 1区数学 Q1 MATHEMATICS Acta Numerica Pub Date : 2017-11-14 DOI:10.1017/S0962492917000101

Nawaf Bou-Rabee, J. Sanz-Serna

引用次数: 86

Abstract

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications for the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behaviour of HMC as the dimensionality of the target distribution increases.

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几何积分器和哈密顿蒙特卡罗方法

本文详细研究了数值积分与哈密顿（或混合）蒙特卡罗方法（HMC）之间的关系。由于HMC的计算成本主要在于数值积分，因此应尽可能有效地进行这些计算。然而，HMC需要具有保体积和可逆的几何特性的方法，这限制了可以使用的积分器的数量。另一方面，这些几何特性对积分误差具有重要的定量影响，而积分误差又对提案的接受率产生影响。虽然目前选择velocity-Verlet算法是有充分理由的，但我们认为可以对其进行改进。我们还详细讨论了HMC随着目标分布维度的增加而表现出的行为。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.

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