Collaborative Research: Revealing the Geometry of Spatio-temporal Chaos with Computational Topology: Theory, Numerics and Experiments

合作研究：用计算拓扑揭示时空混沌的几何：理论、数值和实验

• 批准号: 1622299
• 负责人: Mark Paul
• 金额: $ 10万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1622299&HistoricalAwards=false
• 关键词: Collaborative; Research; Revealing; Geometry; Spatio

The weather we experience is driven by convection, sunlight warms the earth which heats the atmosphere which is cooled by the cold temperatures of outer space. Most people are not interested in microscopic behavior, for example the behavior of the individual molecules in the air, nor macroscopic behavior, such as worldwide average temperature. What is of interest are mesoscopic patterns, for example weather fronts which result in local changes in temperature. This interest in mesoscopic, as opposed to micro- or macroscopic features, of large scale systems occurs in a wide variety of complex large scale physical phenomena such as combustion in engines, dynamics of biomass in the oceans, ventricle fibrillation in a human heart, etc. These mesoscopic patterns take on many different shapes and sizes and change with time, sometimes slowly and sometimes rapidly. The form of these patterns and how they evolve in time is often very dependent on parameters. New technologies are greatly increasing our abilities to measure and simulate these physical phenomena, resulting in enormous data sets, but our ability to extract and quantify this information in a way that leads to understanding, predictability, and control of these systems is not keeping pace. We will explore the use of new mathematical tools to address this problem.The spatial and temporal complexity of Rayleigh-Bénard convection produces high dimensional time series data. A relatively new algebraic topological tool called Persistent Homology will be used to provide new tools for nonlinear dimension reduction. To ensure the applicability of these methods and that physically important mesoscopic features of the dynamics are preserved they will be developed in conjunction with the further development of carefully controlled high precision convection experiments and state-of-the-art, large scale, high-resolution numerical simulations of the Boussinesq equations. This includes the analysis of the geometry of covariant Lyapunov exponents. The new computational tools developed in this work should find broad application in a wide variety of problems involving complex nonequilibrium systems in nature (oceanic and atmospheric flows, climate and weather forecasting) and in technology (nonlinear optical systems, combustion and chemical reactions) where understanding and prediction of complex behavior is desired.

我们体验到的天气是由结构驱动的，阳光温暖了地球，这加热了外太空的寒冷温度所冷却的气氛。大多数人对微观行为不感兴趣，例如单个分子在空气中的行为，也不感兴趣的行为，也不感兴趣的宏观行为，例如全球平均温度。感兴趣的是介观模式，例如天气前线，导致局部温度变化。与微观或宏观特征相反，大规模系统的这种对介观的兴趣出现在各种复杂的大型物理现象中，例如发动机的组合，海洋中的生物质动力学，人类心脏中的通风纤维化动力。这些模式的形式及其在时间上的发展通常非常取决于参数。新技术大大提高了我们测量和模拟这些物理现象的能力，从而产生了巨大的数据集，但是我们以导致理解，可预测性和控制这些系统的方式提取和量化这些信息的能力并不能保持空间。我们将探索使用新的数学工具来解决此问题的使用。Rayleigh-Bénard连接的空间和临时复杂性产生高维时时间序列数据。一种称为持久同源性的相对较新的代数拓扑工具将用于为降低非线性尺寸的新工具。为了确保这些方法的适用性，并保留了在物理上重要的介观特征，并将其与仔细控制的高精度会议实验以及BousSinesQ方程的精心控制的高精度会议实验和最先进的大规模，高分辨率的高分辨率数值模拟一起开发。这包括对协变量Lyapunov指数的几何形状的分析。这项工作中开发的新计算工具应在涉及自然界中复杂非平衡系统（海洋和大气流，气候和天气预报）以及技术（非线性光学系统，组合和化学反应）中的各种问题中找到广泛的应用，其中需要对复杂行为的理解和预测。