Computational Analysis of Large Dynamical Systems

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)资助的。这项工作包括三个与多自由度动力系统计算分析有关的项目。第一个涉及由波动的外部刺激驱动的耦合生物振荡器和可兴奋元件的网络。其目的是在一类与生物相关的体系结构中阐明网络结构与其响应的再现性或可靠性之间的关系。第二个项目涉及开放系统在其边界与不相等的热库耦合时出现的宏观输运过程。这个项目关注于一类原型模型，它既包括确定性的微观动力学，也包括随机的微观动力学。这里的目的是深入了解一类具体的模型系统中非平衡稳态的性质。第三个项目的目的是开发有效的数值算法来计算统计平均值，例如Lyapunov指数，这些统计平均值经常被用来表征非线性动力系统。要开发的算法是基于利用要计算的量的近似先验知识，例如，系统的不变度量的近似知识。这将通过耦合所讨论的系统的多个模拟来完成，并使用这种耦合来产生具有潜在显著小的方差的无偏估计器。这些算法在包括上述项目在内的各种生物和物理环境中的有效性将被研究。具有许多强非线性自由度的动力系统出现在许多科学和技术问题中。由于它们相互作用的复杂性和它们往往是混沌的动力学，它们的分析和仿真往往是困难的。组成这项工作的项目试图在特定的环境下理解这样的大型动力系统，并开发通用的、有效的数值算法来计算非线性动力系统的相关统计性质。这项研究的预期结果可能会导致对一系列现象的更深层次的洞察，包括生物神经网络编码信息的能力，以及具有复杂微观相互作用的空间扩展系统中宏观能量和物质传输的出现。待开发的算法可能适用于其他应用领域，例如随机化学动力学。预计这些项目将带来跨学科合作，例如与生物科学家的合作，并带来研究生培训的机会。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This work consists of three projects related to the computational analysis of dynamical systems with many degrees of freedom. The first concerns networks of coupled biological oscillators and excitable elements driven by fluctuating external stimuli. The goal is to elucidate, within a class of biologically-relevant architectures, the relation between a network's structure and the reproducibility, or reliability, of its response. The second project concerns the emergence of macroscopic transport processes when an open system is coupled to unequal heat reservoirs at its boundaries. This project focuses on a prototypical class of models that includes both deterministic and stochastic microscopic dynamics. The aim here is to gain insights into the properties of nonequilibrium steady states in a concrete class of model systems. The third project aims to develop efficient numerical algorithms for computing statistical averages, e.g., Lyapunov exponents, that are frequently used to characterize nonlinear dynamical systems. The algorithms to be developed are based on exploiting approximate prior knowledge of the quantity to be computed, e.g., approximate knowledge of the system's invariant measure. This will be done via coupling multiple simulations of the system in question, and to use such couplings to produce unbiased estimators with potentially significantly smaller variance. The efficacy of the algorithms in various biological and physical settings, including the projects outlined above, will be investigated.Dynamical systems with many strongly nonlinear degrees of freedom arise in many scientific and technological problems. Their analysis and simulation is often difficult because of the complexity of their interactions and their often chaotic dynamics. The projects comprising this work seek to understand such large dynamical systems in some specific settings, and to develop general, efficient numerical algorithms for computing relevant statistical properties of nonlinear dynamical systems. The expected outcome of this research may lead to deeper insights into a range of phenomena, including the ability of biological neural networks to encode information and the emergence of macroscopic energy and matter transport in spatially-extended systems with complex microscopic interactions. The algorithms to be developed are potentially applicable to other application domains, e.g., stochastic chemical kinetics. It is expected that the projects will lead to interdisciplinary collaborations, e.g., with biological scientists, and to opportunities for graduate student training.