Nonlocal energies and their application to data analysis and collective behavior of many-particle systems

PI研究具有非局部效应的系统的两条应用线路，即其中的点受到远距离影响的系统。其中一个应用是对大型数据集的调查。自动化数据获取技术(如数字显微镜)的进步使人们能够收集保存着丰富信息的大型数据集。这样的数据集通常是非常高维的(例如数字图像)。为了能够提取所包含的信息，需要以最小的信息损失来简化它们。一种方法是用低维流形来逼近数据集。这个项目涉及到一种使用非局部能量的变分方法，用于用曲线对数据进行参数化。我们的目标是帮助理解哪些泛函提供了良好的数据逼近，并服从于有效的计算实施。PI研究基本问题，如极小值的存在性和正则性，以及对数据集的数值实现和适用性。另一项研究涉及具有长程相互作用的粒子系统的行为。这种系统作为各种生物(蝗虫、鱼、鸟、细菌等)集体行为的模型出现。这项研究的目的是帮助解释为什么以及如何组织良好的大型群体(如群体、学校、羊群等)。有机体的形式。此外，从数学上描述这些群体的演化，并解释它们为什么是稳定的，并探索它们如何与环境相互作用，特别是在存在环境边界的情况下。新的数学工具的发展，如概率度量空间中的梯度流，提供了在这些问题上取得重大进展的技术。如何从大数据集中提取信息是一个具有重要实际意义的科学挑战。对PI的研究可以改进数据的参数化和逼近算法。这反过来可以改进数据集群、分类、可视化和其他任务。应用程序的一个例子是基于组织样本的图像改进和自动化某些疾病的诊断。了解各种生物体的大群体是如何形成和行为的是一个重要的生物学问题。它也是数学可以提供重要见解的领域，它基于个人之间相互作用的简单规则。所获得的知识可用于预测这类群体(例如蝗虫群)如何受到影响并提供指导。

The PI studies two lines of applications of systems with nonlocal effects, that is systems in which points are subject to far-away influences. One application is to investigation of large data sets. Advances in automated data-acquisition techniques (such as digital microscopy) has enabled one to gather large data sets holding a wealth of information. Such data sets are often of very high dimension (for example digital images). To be able to extract the information contained, it is desirable to simplify them with minimal loss of information. One approach is to approximate the data set by a low-dimensional manifold. This project is concerned with a variational approach using nonlocal energies for parameterizing the data with a curve. The goal is to contribute to understanding of which functionals provide for good data approximation and are amenable to efficient computational implementation. The PI studies the fundamental questions such as existence and regularity of minimizers, as well as numerical implementation and applicability to data sets. The other line of investigation concerns behavior of particle systems with long range interaction. Such systems arise as models of collective behavior of a variety of living organisms (locust, fish, birds, bacteria, and others). Goal of this research is to help explain why and how large, well-organized groups (such as swarms, schools, flocks, etc.) of organisms form. Furthermore to describe mathematically the evolution of these groups, and explain why are they stable and explore how they interact with the environment in particular in the presence of environmental boundaries. The development of the new mathematical tools, such as gradient flows in spaces of probability measures, provides the techniques that make significant progress on these issues likely.Being able to extract information from large data sets is a scientific challenge with important practical consequences. The research of the PI can lead to improved algorithms for data parameterization and approximation. This in turn can improve data clustering, classification, visualization, and other tasks. An example of an application is improving and automating the diagnostics of some diseases, based on images of tissue samples. Understanding how large groups of a variety of organisms form and behave is an important biological question. It is also one where mathematics, based on simple rules of interaction between individuals, can provide important insights. The knowledge obtained can be used to predict and offer guidance on how such groups (for example locust swarms) could be influenced.