Collaborative Research: Computation of instantons in complex nonlinear systems

各种各样的系统都表现出罕见的事件--远离平均系统行为的事件，发生的概率很低。 罕见事件可能会产生重大后果，对它们的发生有更好的了解可以帮助设计或管理此类系统。在随机性起重要作用的所有系统中，对罕见事件的可能性进行表征是必不可少的，因为它使我们能够利用这些事件（如果它们是可取的），并避免它们（如果它们构成威胁）。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，人口动态和分子生物学的背景下的极端事件的特性。该项目的目标是开发有效的计算方法来描述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由扩散过程驱动的湍流（特别是Burgers方程、磁流体动力学（MHD）、Navier-Stokes方程和表面准地转（SQG）方程）的背景下开发。然后，研究人员将把这些方法扩展到由非马尔可夫噪声（分数布朗运动）或跳跃过程驱动的随机方程，这些随机方程在物理学、生物学和化学中起着重要作用。最后，研究人员将计算瞬子周围的波动，以通过前因子计算获得对其发生概率的更精确估计。

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.