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

非局域能量及其在多粒子系统数据分析和集体行为中的应用

• 批准号: 1211760
• 负责人: Dejan Slepcev
• 金额: $ 13.28万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211760&HistoricalAwards=false
• 关键词: Nonlocal; energies; their; application; data

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.

PI 研究具有非局部效应的系统的两种应用，即点受到远距离影响的系统。一种应用是调查大型数据集。自动数据采集技术（例如数字显微镜）的进步使人们能够收集包含大量信息的大型数据集。 此类数据集通常具有非常高的维度（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。 一种方法是通过低维流形来近似数据集。该项目涉及使用非局部能量通过曲线参数化数据的变分方法。目标是帮助理解哪些函数可以提供良好的数据近似并且适合高效的计算实现。 PI 研究基本问题，例如最小化器的存在性和规律性，以及数值实现和数据集的适用性。 另一条研究路线涉及具有长程相互作用的粒子系统的行为。 这些系统是作为各种生物体（蝗虫、鱼、鸟类、细菌等）集体行为的模型而出现的。这项研究的目标是帮助解释大型、组织良好的生物群体（如群体、群体、群体等）形成的原因和方式。此外，以数学方式描述这些群体的进化，并解释它们为什么稳定，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。新数学工具的发展，例如概率测度空间中的梯度流，提供了可能在这些问题上取得重大进展的技术。能够从大型数据集中提取信息是一项科学挑战，具有重要的实际后果。 PI 的研究可以改进数据参数化和近似的算法。这反过来又可以改进数据聚类、分类、可视化和其他任务。 应用的一个例子是根据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如何形成和行为是一个重要的生物学问题。基于个体之间相互作用的简单规则的数学也可以提供重要的见解。获得的知识可用于预测此类群体（例如蝗虫群）如何受到影响并提供指导。