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

提案：DMS-0404778 PI：Erik M Bollt机构：克拉克森大学标题：一种传递算子方法来建模确定性和随机运输，在物理科学中的应用摘要虽然非常成功，随机常微分方程（SDEs）和随机偏微分方程（SPDEs），有一个高开销的背景材料学习。 传统方法在处理真实的问题时变得特别复杂，这些问题通常可能是高维的，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个实用和方便的工具。 研究人员开发计算方法来建模和识别确定性和随机扰动动力系统中的运输活动。研究人员开发了一个分析的转移算子超出了他们通常的理论用途，成为一个广泛的和统一的计算工具套件，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）映射对应于高传输通量活动的区域的相空间，导致例如多稳态系统中的噪声诱导的爆发，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爆发活动，3）从实验数据开发转移算子的有效和低维模型，以在转移算子的全局视角内进行系统识别和参数估计，4）开发低维模型，非参数统计假设检验，用于在仅通过测量数据已知的高维数据集中识别非平稳性和显著的系统变化或系统“损坏”。 尽管在我们周围的物理世界中看到的复杂振荡实际上已经成为科学和工程的每一个分支的深入研究的主题，但当涉及噪声时，用于分析它们的传统工具仍然有些技术性，难以学习和应用于实际问题。有许多例子表明，决定论和一个小的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，这种运动就不会发生。噪声诱发爆发的映射机制和时间具有重要的社会影响意义。 例如，研究人员正在研究疾病传播的群体动力学，其中外部噪声激励可以导致复杂的振荡，以及对应于流行病的大爆发。 了解这些流行病背后的时间和机制，以及控制算法，导致提出一个全新的疫苗接种方案，其中及时但相对非侵入性的干预将导致避免问题。 类似地，对机械束结构、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性。 开发具有全局视角的非平稳性损伤检测测试表明了一种强大的新方法来观察系统何时发生根本性变化，并提出了一类新的更好的传感器来避免危险的算法。

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.