A Transfer Operator Approach to Modeling Deterministic and Stochastic Transport, with Applications in the Physical Sciences

提议：DMS-0404778PI：Erik M Bollt机构：Clarkson University标题：一种转移算子方法来模拟确定性和随机运输，及其在物理科学中的应用摘要虽然随机常微分方程(SDE)和随机偏微分方程(SPDE)非常成功，但需要学习的背景材料开销很大。传统的方法在处理实际问题时变得特别复杂，这些问题通常是高维的，具有一般的噪声分布，并且具有非线性相互作用。直接逼近随机Frobenius-Perron算子的作用提供了一种实用的工具。研究人员开发了计算方法来模拟和识别确定性和随机扰动动力系统中的传输活动。研究人员对转移算符的分析超出了它们通常的理论用途，发展成一套广泛而统一的计算工具，用于解决实际问题。研究人员正在开发计算方法的重要应用，包括：1)映射与高传输流量活动相对应的区域的相空间，例如导致多稳态系统中的噪声诱导的突发；2)开发传输算法的控制，以通过低能量控制输入放大或减少突发活动；3)从实验数据开发高效和低维的传输操作员模型，以在传输操作员的全局视角内进行系统识别和参数估计；4)开发低维、非参数统计假设检验以识别非平稳性和显著的系统变化或系统“损伤”，在仅通过测量数据已知的高维数据集中。尽管在我们周围的物理世界中看到的复杂的振荡已经成为几乎每一个科学和工程分支的密集研究的主题，但当涉及到噪音时，用于分析它们的传统工具仍然具有一定的技术性，无法学习和应用于实际问题。在许多例子中，决定论和一个小噪声分量之间的相互作用可以引起复杂的运动，如果没有这两种机制的相互作用，就不会发生这种运动。噪声诱发猝发的映射机制和时间具有重要的社会影响。例如，研究人员正在研究疾病传播的种群动力学，其中外部噪声激励可以导致复杂的振荡，以及与流行病相对应的大爆发。了解这些流行病背后的时间和机制，以及控制算法，导致提出一种全新的疫苗接种方案，在该方案中，适时但相对非侵入性的干预将导致避免问题。同样，在机械光束结构、非线性光学(噪声激光器)、电子电路、空气污染动力学和化学振荡器的反应扩散系统中的噪声诱导跃迁的研究表明，这些开发工具具有广泛的应用和重要意义。具有全局视角的非平稳性损伤检测测试的发展表明，当系统发生根本变化时，可以采用一种强大的新方法来观察，并为新一类更好的传感器提供算法，以避免危险。

Proposal: DMS-0404778PI: Erik M BolltInstitution: Clarkson UniversityTitle: A Transfer Operator Approach to Modeling Deterministic and Stochastic Transport, with Applications in the Physical SciencesABSTRACTWhile highly successful, the stochastic ordinary differential equations (SDEs) and stochastic partial differential equations (SPDEs), have a high overhead of background material to learn. Traditional methods become particularly complex when coping with real problems, which may generally be high dimensional, with general noise profiles, and have nonlinear interactions. Directly approximating the action of the stochastic Frobenius-Perron operator offers a practical and accessible tool. The investigator develops computational methods to model and identify transport activity in both deterministic and stochastically perturbed dynamical systems. The investigator develops an analysis of the transfer operator beyond their usual theoretical use, into a broad and unified suite of computational tools for solving practical problems. The investigator is developing important applications of the computational methods including: 1) mapping the phase space for regions corresponding to high transport flux activity, leading for example to noise-induced bursting in multi-stable systems, 2) developing control of transport algorithms, to either amplify or decrease bursting activity through low-energy control inputs, 3) developing efficient and low-dimensional models of the transfer operator from experimental data, to do system identification and parameter estimation within the global perspective of the transfer operator, 4) developing low-dimensional, nonparametric statistical hypothesis tests to identify nonstationarity and significant system changes, or system "damage," in high dimensional data sets known only through measured data. Even though the complex oscillations seen in the physical world around us have been a subject of intense study in practically every branch of science and engineering, the traditional tools for their analysis remain somewhat technical to learn and apply to practical problems when noise is involved. There are many examples in which the interaction between determinism and a small noise component can give rise to complicated motions that would not occur without the interaction of both mechanisms. Mapping mechanism and timing of noise induced bursting has implications of important social impact. For example, the investigator is studying population dynamics of disease spread, in which external noise excitation can lead to complicated oscillations, and large bursts corresponding to epidemic. Understanding the timing and mechanisms behind these epidemics together with control algorithms leads to suggesting a radically new vaccination protocal in which well timed but relatively noninvasive intervention will result in averting the problem. Similarly, investigations of noise induced transitions of a mechanical beam structure, in nonlinear optics (a noisy lasers), in electronic circuits, in the dynamics of air-pollution, and a reaction diffusion system of a chemical oscillator shows that these developing tools have wide ranging application and importance. Development of damage detection tests of nonstationarity with global perspective suggests a powerful new way to observe when a system has been radically changed, suggesting algorithms for a new class of better sensors to avert hazards.