Collaborative Research: Topological Methods for the Study of Nonlinear Infinite Dimensional Systems

合作研究：研究非线性无限维系统的拓扑方法

• 批准号: 0511115
• 负责人: Konstantin Mischaikow
• 金额: $ 14.06万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2006-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0511115&HistoricalAwards=false
• 关键词: Collaborative; Research; Topological; Methods; Study

We are surrounded by patterns that change their shape and structure with time -- particular examples include: chemical concentrations in materials, velocity profiles in fluids, population densities in ecosystems, etc. With modern information technologies it is relatively easy to collect enormous data on these patterns either through experimentation or numerical simulation. One goal of this project is to develop and employ computational topological techniques to use the observed patterns to identify, quantify and classify the time-dependent properties of the underlying systems. Mathematical models for spatially dependent systems that evolve with time are typically extremely difficult to analyze using traditional mathematical techniques, and thus much of what we know about the detailed evolution of these systems comes from numerical simulations. However, the process of performing these numerical simulations introduces errors, which can potentially grow and propagate. In principle, algebraic topological properties remain invariant under small perturbations. With this in mind, another goal of this project is to combine the newly developed computational topological tools with standard numerical methods to verify that the solutions obtained through numerical simulation are indeed valid results for the systems being studied.

我们周围的模式，改变他们的形状和结构随着时间的推移-特别的例子包括：在材料中的化学浓度，在流体中的速度分布，在生态系统中的人口密度等与现代信息技术，它是相对容易收集这些模式的大量数据，无论是通过实验或数值模拟。 该项目的一个目标是开发和采用计算拓扑技术，使用观察到的模式来识别，量化和分类的基础系统的时间依赖性。随着时间演化的空间相关系统的数学模型通常很难使用传统的数学技术进行分析，因此我们对这些系统详细演化的了解大部分来自数值模拟。 然而，执行这些数值模拟的过程会引入误差，这些误差可能会增长和传播。 原则上，代数拓扑性质在小扰动下保持不变。 考虑到这一点，这个项目的另一个目标是联合收割机结合新开发的计算拓扑工具与标准的数值方法，以验证通过数值模拟得到的解决方案确实是有效的结果，为系统的研究。