Collaborative Research: Computation of instantons in complex nonlinear systems

合作研究：复杂非线性系统中瞬时子的计算

• 批准号: 1522737
• 负责人: Tobias Schaefer
• 金额: $ 9.96万
• 依托单位: CUNY College of Staten Island
• 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=1522737&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.

各种各样的系统都会出现罕见事件--这些事件与系统的平均行为相去甚远，发生的概率很低。罕见事件可能会产生重大后果，而对其发生的更好理解有助于此类系统的设计或管理。在随机性发挥重要作用的所有系统中，描述罕见事件的可能性是至关重要的，因为它使我们能够在此类事件是合乎需要的情况下利用它们，在它们构成威胁时避免它们。该项目的成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、人口动力学和分子生物学的背景下描述极端事件的特征。这个项目的目标是开发有效的计算方法来表征复杂随机系统中罕见事件发生的最可能方式，并估计其概率分布的尾部。为此，研究人员将开发有效的算法来计算所谓的“瞬子”，即大偏差理论与描述系统演化的随机微分方程相关联的作用泛函的最小化。计算瞬子的数值方法将首先在扩散过程驱动的湍流(特别是Burgers方程、磁流体动力学(MHD)、Navier-Stokes方程和地表准地转(SQG)方程)的背景下发展。然后，研究人员将把这些方法扩展到由非马尔科夫噪声(分数布朗运动)或跳跃过程驱动的随机方程，这些噪声或跳跃过程在物理、生物和化学中发挥着重要作用。最后，研究人员将计算瞬子周围的波动，通过预因子计算获得对它们发生概率的更精确估计。