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Langevin Algorithms : Questions at the Numerical Analysis / Applied Probability Interface

Langevin 算法：数值分析/应用概率接口的问题

基本信息

批准号：
EP/D505607/1
负责人：
Gareth Roberts
金额：
$ 15.93万
依托单位：
Lancaster University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2006
资助国家：
英国
起止时间：
2006 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FD505607%2F1
关键词：
Langevin Algorithms Questions Numerical Analysis

项目摘要

Randomly evolving phenomena are often mathematically described by so-called Stochastic (Partial) Differential Equations (SPDE), which essentially give update rules for how the system is to evolve in any given infinitesimally small time instance. Mathematicians, statisticians and more generally scientists in many areas are interested in this area as they form plausible models for diverse problems ranging from the price of a stock to the movement of a molecule. They are also of interest in more abstract settings for exploring complex parameter spaces for statistical inference.Commonly, interest is in how such a system will behave "on average'over a long period of time. This is termed the * steady state' behaviour of the system. For example, a piece of string tied at both ends is subject to constant perturbations from surrounding molecules, and thus continually flutuates, but we might be interested in its average position. If we try to simulate such a random system, we need to follow the update rules in discrete time. This is an approximation of the true continuous-time dynamics, but for sufficiently fine time-discretisation, this approximation ought to have some claim to being a good one. However, the steady state of the system might not be adequately approximated by this method. Fortunately an easy way to correct for the error in estimating averages exists in the guise of the so-called Metropolis-Hastings algorithm which occasionally rejects an update move in favour of staying at the same location.If we use large time steps in this discretisation, the Metropolis-Hastings rejection moves will be required often, and the overall continuous-time dynamics will not be well approximated. However since interest is in steady state behaviour, this is not necessarily a problem. On the other hand if time steps are too large, then a large proportion of proposed moves will be rejected which will adversely affect the estimation of steady state.This project is all about devising random algorithms which can efficiently simulate from approximations to SPDEs in order to estimate properties of their steady state behaviour. There are two types of question we will answer. Firstly we will consider the problem of choosing an efficient SPDE which resembles its "average1 behaviour in a relatively short time period. Secondly, we shall consider how to optimally choose the time-dicretisation rules to maximise the efficiency of the algorithm for estimating steady state properties.

随机演化现象通常由所谓的随机(偏)微分方程(SPDE)来数学描述，它本质上给出了系统在任何给定的无穷小时间实例中如何演化的更新规则。数学家、统计学家和更广泛地说，许多领域的科学家都对这一领域感兴趣，因为他们为从股票价格到分子运动的各种问题建立了可信的模型。他们还对探索复杂参数空间以进行统计推理的更抽象的设置感兴趣。通常，感兴趣的是这样一个系统在很长一段时间内的平均表现。这被称为系统的*稳态行为。例如，绑在两端的一根线受到周围分子的持续扰动，因此不断地颤动，但我们可能对它的平均位置感兴趣。如果我们试图模拟这样一个随机系统，我们需要遵循离散时间的更新规则。这是真正的连续时间动力学的近似值，但对于足够精细的时间离散化，这个近似值应该是一个很好的近似值。然而，这种方法可能不能很好地逼近系统的稳态。幸运的是，在所谓的Metropolis-Hastings算法的伪装下，有一种简单的方法可以纠正估计平均值中的错误，该算法偶尔会拒绝更新移动，而支持停留在同一位置。如果我们在这种离散化中使用大的时间步长，将经常需要Metropolis-Hastings拒绝移动，并且整体连续时间动态将不能很好地逼近。然而，由于人们对稳定状态的行为感兴趣，这不一定是一个问题。另一方面，如果时间步长太大，那么很大一部分建议的移动将被拒绝，这将对稳态估计产生不利影响。该项目旨在设计能够有效地模拟从近似到SPDEs的随机算法，以便估计其稳态行为的性质。我们将回答两类问题。首先，我们将考虑选择一个有效的SPDE的问题，该SPDE类似于它在相对较短的时间段内的“平均行为”。其次，我们将考虑如何最优地选择时间分解规则，以最大化估计稳态性质的算法的效率。