Collaborative Research: Computation of instantons in complex nonlinear systems.

合作研究：复杂非线性系统中瞬子的计算。

• 批准号: 1522767
• 负责人: Eric Vanden-Eijnden
• 金额: $ 10万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-02-01 至 2020-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522767&HistoricalAwards=false
• 关键词: Collaborative; Research; Computation; instantons; complex

A wide variety of systems exhibit rare events -- events far from the average system behavior with low probability of occurring. Rare events can have significant consequences, and improved understanding of their occurrence can aid in the design or management of such systems. The characterization of the likelihood of rare events is essential in all systems in which stochasticity plays an important role, as it allows us to take advantage of such events if they are desirable and to avoid them if they present a threat. The outcomes of this project will contribute to the understanding of rare events in complex systems, in particular, fluid dynamics and related geophysical systems. Further applications include the characterization of extreme events in the context of epidemics, population dynamics, and molecular biology. The goal of this project is to develop efficient computational methods to characterize the most likely way rare events occur in complex stochastic systems and to estimate the tails of their probability distributions. For this purpose, the investigators will develop efficient algorithms to compute the so-called "instantons" that are minimizers of the action functional that large deviation theory associates with the stochastic differential equation describing the system's evolution. Numerical methods to calculate instantons will first be developed in the context of turbulence (in particular Burgers equation, magneto-hydrodynamics (MHD), Navier-Stokes equations, and the surface-quasi-geostrophic (SQG) equation) driven by diffusive processes. Then the investigators will extend the methods to stochastic equations that are driven by non-Markovian noise (fractional Brownian motion) or jump-processes, which play an important role in physics, biology, and chemistry. Finally, the investigators will calculate fluctuations around the instantons to get finer estimates of their probability of occurrence via prefactor calculations.

各种各样的系统都表现出罕见的事件 - 事件远非平均系统行为，而发生的可能性较低。 罕见的事件可能会产生重大的后果，并且对其发生的理解有所改善可以帮助对此类系统的设计或管理。在随机性起着重要作用的所有系统中，罕见事件的可能性的表征至关重要，因为它使我们可以利用这些事件是理想的，如果它们构成威胁，则可以避免它们。该项目的结果将有助于理解复杂系统中的罕见事件，特别是流体动力学和相关的地球物理系统。进一步的应用包括在流行病，种群动力学和分子生物学的背景下对极端事件的表征。该项目的目的是开发有效的计算方法，以表征复杂随机系统中罕见事件最有可能发生的方式，并估算其概率分布的尾巴。为此，研究人员将开发有效的算法来计算所谓的“ instantons”，这些“ instantons”是动作功能的最小化，大偏差理论与描述系统演变的随机微分方程相关联。计算Instantons的数值方法将首先在湍流（尤其是Burgers方程，磁性流动力学（MHD），Navier-Stokes方程和表面-quasi-Geostophic（SQG）方程）的情况下开发。然后，研究人员将将方法扩展到由非马克维亚噪声（分数布朗运动）或跳跃过程驱动的随机方程，这些方程在物理，生物学和化学中起着重要作用。最后，研究人员将计算激体周围的波动，以通过先前计算获得更细微的估计。