Visualization & Optimization Techniques For Analysis and Design of Complex Systems

该项目涉及管理大型互联系统中的复杂性的几种相互关联的方法。该提案涉及的具体应用领域包括分子模型中的相变，以及大型网络模型中的资源分配。理想情况下，我们希望建立一个“显微镜”的集合，使我们能够在不同的尺度上观察这些系统，并过滤掉不相关的信息。PI最近开发了旨在执行这一任务的方法。在拟议的资金期限内，他将开发这些方法和新方法，以解决复杂系统中的可视化和优化问题。将开发的具体分析方法包括：(I)基于快慢时间尺度分离的松弛技术。这包括网络领域的“工作量放松”，以及扩散模型的频谱理论的最新方法。(Ii)马尔可夫过程和扩散模型的一般理论，特别是关于大偏差理论和熵的最新结果。(Iii)包括先验知识的方法，以加快复杂随机模型的模拟和在线学习，以及预测和模拟罕见事件。(Iv)扩展流体模型技术，在现实环境中解决网络优化中的控制和敏感性问题。PI将与伊利诺伊州的研究生、布朗大学的应用数学研究人员以及德国研究机构的计算机科学和物理方面的其他人合作。这项研究的预期影响是深远的，所有这些数学和应用领域之间的交叉影响可能会非常显著。在网络应用中，这些结果将提供新的方法来直观地了解大型网络的行为，新的设计方法，以及有效的模拟和在线调整策略的方法。如果成功，光谱理论和扩散的研究将为分子系统提供另一种建模技术，这些技术可以捕捉基本的动力学，并且可以很容易地模拟。

This project concerns several interrelated approaches to managing complexity in large interconnected systems. Specific application areas addressed in the proposal include phase transitions in molecular models, and resource allocation in large network models. Ideally, we wish to build a collection of 'microscopes' that allow us to view these systems on various scales, and filter away irrelevant information. The PI has recently developed methods that are intended to perform exactly this task. Over the proposed funding period he will develop these and new approaches for addressing visualization and optimization problems in complex systems.Specific analytical approaches to be developed include, (i) Relaxation techniques based upon separation of fast and slow time-scales. This involves 'workload-relaxations' in the network domain, and recent approaches to spectral theory for diffusion models. (ii) General theory of Markov processes and diffusion models, especially recent results concerning large deviations theory and entropy.(iii) Methods for including prior knowledge to speed simulation and on-line learning for complex stochastic models, and for predicting and simulating rare events. (iv) Extend fluid model techniques for addressing control and sensitivity in network optimization in realistic settings.The PI will collaborate with graduate students at Illinois, researchers in applied mathematics at Brown University, and others in computer science and physics at research institutes in Germany. The expected impact of this research is far-reaching, and it is likely that cross-fertilization between all of these mathematical and application areas will be significant. In networks applications, these results will provide new methods for providing intuition about the behavior of large networks, new approaches to design, and efficient approaches to simulation and on-line tuning of policies. If successful, the research on spectral theory and diffusions will provide alternative-modeling techniques for molecular systems that capture essential dynamics and can be easily simulated.