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

随机进化现象通常用所谓的随机（偏）微分方程（SPDE）在数学上描述，它本质上给出了系统在任何给定的无限小时间实例中如何进化的更新规则。数学家、统计学家和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合理的模型。他们也对更抽象的设置感兴趣，以探索复杂的参数空间进行统计推断。通常，人们关心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，一根两端系在一起的绳子受到周围分子的不断扰动，因此不断地波动，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。这是真正连续时间动力学的近似，但对于足够精细的时间离散，这个近似应该是一个很好的近似。然而，这种方法可能不能充分地近似系统的稳态。幸运的是，有一种简单的方法可以纠正估计平均值时的错误，即所谓的Metropolis-Hastings算法，该算法偶尔会拒绝更新移动，而倾向于保持在同一位置。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抑制移动，并且整体连续时间动力学将不能很好地近似。然而，由于兴趣是在稳态行为，这并不一定是一个问题。另一方面，如果时间步长过大，则很大一部分提议的移动将被拒绝，这将对稳态的估计产生不利影响。这个项目是关于设计随机算法，可以有效地模拟从近似值到spde，以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，它与其在相对较短时间内的“平均行为”相似。其次，我们将考虑如何最优地选择时间离散规则，以最大限度地提高算法估计稳态特性的效率。

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.