Set-valued numerical analysis for critical transitions

关键转变的集值数值分析

• 批准号: EP/L00187X/1
• 负责人: Martin Rasmussen
• 金额: $ 22.4万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL00187X%2F1
• 关键词: Set; valued; numerical; analysis; critical

Complex systems research has been on the forefront of scientific priorities of many national and international research councils for more than a decade. Many very interesting phenomena havebeen identified and explored, but the development of the underpinning mathematical theory has been lagging behind. The proposed research builds on an emerging movement in appliedmathematics which aims to provide proper mathematical bifurcation theory for the existence of early-warning signals for sudden changes in dynamical behaviour. These sudden changes are commonlyreferred to as critical transitions, and have been reported by applied scientists in various contexts. Practical implications for the existence of such early-warning signals are far reaching, since these would enable the development of better control strategies to avoid or diminish the effect of catastrophes.In this project, techniques for the numerical study of critical transitions will be developed. In particular, it will provide new techniques for the approximation of invariant sets. The research will be based on very recent results that make a representation of such invariant sets as functions in a Banach space possible. One of the main advantages of this new approach is that it can be used to study bifurcations, in contrast to grid-cell discretisations, which is the current state-of-the-art for the computation of invariant objects. The specific numerical studies on random systems with bounded noise will lead to insights into how an early-warning can be given should a dynamical system approach a bifurcation point.

十多年来，复杂系统研究一直是许多国家和国际研究委员会的科学重点。许多非常有趣的现象已经被发现和探索，但基础数学理论的发展一直落后。提出的研究建立在应用数学中的一个新兴运动，旨在为动态行为突然变化的预警信号的存在提供适当的数学分岔理论。这些突然的变化通常被称为临界转变，并且已经被应用科学家在各种情况下报道过。这种早期预警信号的存在所产生的实际影响是深远的，因为这将有助于制定更好的控制战略，以避免或减少灾难的影响。在这个项目中，关键转变的数值研究技术将得到发展。特别是，它将为不变量集的逼近提供新的技术。这项研究将基于最近的结果，这些结果使得在巴拿赫空间中将这些不变集表示为函数成为可能。这种新方法的主要优点之一是，它可以用来研究分岔，而不是网格单元离散，这是目前计算不变对象的最新技术。对具有有界噪声的随机系统的具体数值研究将有助于了解当动力系统接近分岔点时如何给出预警。

项目成果

Approximation of reachable sets using optimal control and support vector machines

• DOI: 10.1016/j.cam.2016.06.015
• 发表时间: 2014-12
• 期刊: J. Comput. Appl. Math.
• 作者: M. Rasmussen;J. Rieger;K. Webster

A reinterpretation of set differential equations as differential equations in a Banach space

将集合微分方程重新解释为 Banach 空间中的微分方程

• DOI: 10.1017/s0308210517000178
• 发表时间: 2017
• 期刊: Section A Mathematics
• 作者: Rasmussen M

Using Lin's method to solve Bykov's problems

用Lin的方法解决Bykov的问题

• DOI: 10.1016/j.jde.2014.06.006
• 发表时间: 2014
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Knobloch J