Coupling and Control in Continuous Time

连续时间耦合与控制

基本信息

批准号：
EP/P003818/1
负责人：
Aleksandar Mijatovic
金额：
$ 42.02万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP003818%2F1
关键词：
Coupling Control Continuous Time

项目摘要

Randomness is ubiquitous in the natural world, and advances in understanding and modelling random events are key to making progress with many problems in the natural and social sciences, engineering, statistics, to name but a few. Coupling is a fundamental paradigm in probability through which probability distributions of random quantities (random variables, random processes) can be compared with each other via "pointwise" comparisons. It yields powerful techniques for analysing random systems. A Markov process is a random process whereby, conditional on the present, its future and past are independent. That is, if we know the present state of the process, we can gain no additional information on its future evolution by knowing more about its past. This paradigm describes many random processes used as models in the natural and social sciences. In coupling we are looking at two Markov processes that start from different locations and evolve jointly. We are interested in them meeting a number of criteria, e.g. the two processes meeting as soon as possible, staying close to each other for as long as possible, or other criteria (e.g. the large deviation behaviour of the coupling time, i.e. what the exponential rate of decay of the coupling time is). As well as being an interesting mathematical question in and of itself, this problem has significant potential applications. For example, the rate of convergence to stochastic equilibrium (a crucial question in many applications) is controlled by the rate at which coupling occurs.There is a natural lower bound in the speed of coupling. The "fastest" couplings, i.e. the couplings where the probability that the two processes have not met by any given time is smallest, are known as "maximal" couplings: one can construct those by defining the second process as a functional of the entire trajectory of the first. However, in the context of modelling in the sciences, it is natural to focus on co-adapted couplings, namely couplings whereby the second process at a given time can only be constructed based on the trajectory of the first upto and including the present time (i.e. no information about the future trajectory of the first process can be taken into account). The difficulty here is that it is hard to obtain optimal (called "extremal") couplings. In fact it's difficult to know how good any given co-adapted coupling is. This proposal is about taking any co-adapted coupling and providing a method of improving it. Not just locally, but proving mathematically that the sequential improvements we propose yield a co-adapted coupling that is as good as it can get. Essentially we are looking to solve a stochastic optimisation problem under the additional constraint of co-adaptivity. In this proposal, the main method for improving a co-adapted coupling to achieve optimality is via the application of control theory. We aim to use the Policy Improvement Algorithm, a tool from control theory that works in discrete time, and develop its application in continuous time. In the application part of the project, we aim to develop applications of the PIA in the theory of non-linear PDEs and Multi-Level Monte Carlo (MLMC) algorithms for processes with jumps. The areas of non-linear PDEs and MLMC simulation have applications with vast societal and economic impact: the former has applications in biology, physics, engineering to name a few, and the latter is of crucial importance in Uncertainty Quantification in engineering and science. When the uncertainty is high-dimensional and strongly nonlinear, Monte Carlo simulation remains the preferred approach, with applications in areas as diverse as biochemical reactions and plasma physics.

随机性在自然世界中无处不在，理解和建模随机事件的进步对于在自然和社会科学，工程，统计学中的许多问题中取得进步至关重要，仅举几例。耦合是一种概率的基本范例，可以通过“刻度方向”比较将随机数量（随机变量，随机过程）的概率分布（随机变量，随机过程）进行比较。它产生了分析随机系统的强大技术。马尔可夫过程是一个随机过程，以当前的条件，其未来和过去是独立的。也就是说，如果我们知道过程的当前状态，我们将通过更多地了解其过去来获得有关其未来演变的其他信息。该范式描述了自然科学和社会科学中用作模型的许多随机过程。在耦合中，我们正在研究从不同位置开始并共同发展的两个马尔可夫过程。我们有兴趣符合许多标准，例如尽快遇到的两个过程，尽可能长时间地保持亲密关系，或其他标准（例如，耦合时间的较大偏差行为，即耦合时间的衰减指数率是多少）。除了是一个有趣的数学问题外，这个问题还具有重要的潜在应用。例如，收敛速率与随机平衡的速度（许多应用中的关键问题）受到耦合发生的速率的控制。 “最快”的耦合，即在任何给定时间尚未达到两个过程的概率最小的概率的耦合，称为“最大”耦合：一个人可以通过将第二个过程定义为第一个轨迹的第二个过程来构建这些过程。但是，在科学中建模的背景下，专注于共同适应的耦合是很自然的，即，在给定时间的第二个过程只能基于第一个和包括当前时间的轨迹来构建第二个过程（即，关于第一个过程的未来轨迹没有任何信息）。这里的困难是很难获得最佳（称为“极端”）耦合。实际上，很难知道任何给定的共同适应的耦合有多好。该建议是关于采用任何共同适应的耦合并提供改进它的方法。不仅在本地，而且在数学上证明我们提出的顺序改进产生了同时适应的耦合，这是尽可能出色的。本质上，我们正在寻求在共同适应性的其他约束下解决随机优化问题。在此提案中，通过应用控制理论的应用，改进共同适应的耦合以实现最佳性的主要方法。我们旨在使用策略改进算法，这是一种在离散时间工作的控制理论的工具，并在连续的时间内开发其应用。在项目的应用部分中，我们旨在开发PIA在非线性PDE和多级蒙特卡洛（MLMC）算法中的应用中的应用，以进行跳跃的过程。非线性PDE和MLMC模拟的领域具有巨大的社会和经济影响：前者在生物学，物理学，工程学中都有应用，而后者在工程和科学中的不确定性量化中至关重要。当不确定性高维且强烈非线性时，蒙特卡洛模拟仍然是首选方法，在像生化反应和血浆物理学一样多样化的区域中应用。