Visualization & Optimization Techniques For Analysis and Design of Complex Systems

这个项目涉及几个相互关联的方法来管理大型互联系统的复杂性。具体的应用领域包括分子模型中的相变和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最近开发了专门用于执行这一任务的方法。在提议的资助期内，他将开发这些和新的方法来解决复杂系统中的可视化和优化问题。需要发展的具体分析方法包括：(i)根据快慢时标分离的松弛技术。这涉及到网络领域的“工作量松弛”，以及扩散模型的谱理论的最新方法。（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）将先验知识纳入复杂随机模型加速模拟和在线学习的方法，以及预测和模拟罕见事件的方法。㈣扩大流体模型技术，以解决现实环境中网络优化的控制和敏感性问题。PI将与伊利诺斯州的研究生、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学研究人员合作。这项研究的预期影响是深远的，很可能所有这些数学和应用领域之间的交叉受精将是显著的。在网络应用中，这些结果将为提供关于大型网络行为的直觉、设计的新方法以及模拟和在线调整策略的有效方法提供新方法。如果成功，对光谱理论和扩散的研究将为分子系统提供替代建模技术，这些技术可以捕获基本动力学并易于模拟。

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.