CDI-TYPE II--COLLABORATIVE RESEARCH: Using Algebraic Topology to Connect Models with Measurements in Complex Nonequilibrium Systems

CDI-TYPE II——协作研究：使用代数拓扑将模型与复杂非平衡系统中的测量联系起来

• 批准号: 1125234
• 负责人: Mark Paul
• 金额: $ 38.04万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1125234&HistoricalAwards=false
• 关键词: CDI; TYPE; II; COLLABORATIVE; RESEARCH

Numerous complex systems in nature and in technology defy concise characterization because they exhibit strongly nonlinear behaviors that lack all symmetries and are highly non-periodic on a wide range of spatial and temporal scales. Characterization by detailed measurement (in lab experiments or direct numerical simulations) is now possible in many cases using modern measurement technologies or computational techniques. However, the resulting deluge of data often leads to little insight; in particular, there is frequently no good way to connect quantitatively experimental measurements of a particular complex system with the output from simulations/models of the same system. New, computationally-based, mathematical tools from algebraic topology have the potential to bridge the gap between measurements and models; the proposed research will explore the use of algebraic topology to link numerical simulations and laboratory experiments in situations where complexity arises because the system under study is driven out of thermodynamic equilibrium. The research focuses on an outstanding paradigm for nonequilibrium complexity: fluid flow driven by temperature gradients (thermal convection). The planned work brings three unique capabilities together in a single effort: (1) the experimental ability both to measure and to manipulate precisely complex, convective flows; (2) efficient methods for state-of-the-art, large scale, high-resolution numerical simulations of convective flow; (3) open source, general purpose, and efficient computational algorithms and software for computing algebraic topological invariants on large data sets. Topological tools will be developed both to characterize and to minimize model error as well as to compare and to quantify dynamical properties including Lyapunov exponents, dimensionality and bifurcations between complex spatiotemporal flow states. This effort should ultimately identify ways in which homology-based metrics can be used for building reduced order models that permit prediction and, perhaps, control of convective flow. More generally, we expect the metrics developed for convection should find broad application to PDE-modeled problems ranging from the control of cardiac arrythmias to the prediction of weather and climate.The behaviors of complex systems in the world around us can now both be measured with high fidelity using advanced sensing technologies and simulated with great realism using modern computer techniques. However, the enormous data sets typically produced in these cases are often difficult to interpret because there exist few good mathematical tools to connect quantitatively the experimental measurements of a given complex system with the output of computer simulations of that same system. The proposed research explores the use of the mathematics of topology to relate lab measurements to computer outputs in a particular complex system, thermal convection. The results of this work should lead to new ways to understand, to predict, and, perhaps, to control convective flow, which plays a direct role in natural processes (e.g., volcanism, earthquake dynamics, continential drift) and industrial applications (e.g., thermal regulation of many devices, the growth of semiconductor materials). Moreover, the topological tools developed for thermal convection should apply more generally to a wide variety of other problems involving complex systems including the forecasting of weather and climate; the dynamics of the biomass in the oceans; the onset of turbulence; the evolution of reagent patterns on a catalytic metal surface; and ventricular fibrillation in a human heart.

自然界中的许多复杂系统和技术违背了简洁的表征，因为它们表现出强烈的非线性行为，这些行为缺乏所有对称性，并且在广泛的空间和时间尺度上是高度非周期性的。在许多情况下，使用现代测量技术或计算技术，可以通过详细测量（在实验室实验或直接数值模拟中）进行表征。但是，由此产生的数据通常会导致几乎没有洞察力。特别是，通常没有很好的方法将特定复合系统的定量实验测量与来自同一系统的模拟/模型的输出相关联。 来自代数拓扑的新的，基于计算的数学工具有可能弥合测量和模型之间的差距。拟议的研究将探讨使用代数拓扑来将数值模拟和实验室实验联系起来的情况，因为研究系统被驱逐出热力学平衡，因此出现了复杂性。 该研究重点是针对非平衡复杂性的出色范式：由温度梯度驱动的流体流动（热对流）。计划的工作将三个独特的能力融合在一起：（1）测量和操纵精确复杂，对流流的实验能力； （2）对流流的最先进的，大规模的高分辨率数值模拟的有效方法； （3）用于计算大型数据集上代数拓扑不变的开源，通用和有效的计算算法和软件。 将开发拓扑工具来表征和最小化模型误差，以及比较和量化动力学特性，包括Lyapunov指数，维度和复杂时空流量状态之间的分叉。 这项工作最终应确定可以使用基于同源的指标来构建允许预测并控制对流流量的减少订单模型的方法。 更普遍地，我们预计为对流开发的指标应该在PDE模型的问题上找到广泛的应用，这些问题从控制心脏和气候的预测到天气和气候的预测。现在，我们周围世界上复杂系统的行为都可以使用高级富裕技术和使用现代计算机技术进行高级忠诚度来衡量。但是，通常在这些情况下通常产生的庞大数据集通常很难解释，因为很少有良好的数学工具可以定量地将给定复杂系统与同一系统的计算机模拟输出进行定量连接。 拟议的研究探讨了拓扑数学的使用将实验室测量与特定复杂系统中的计算机输出相关联。这项工作的结果应导致新的方法来理解，预测并可能控制对流流，这在自然过程（例如火山主义，地震动力学，远处漂移）和工业应用中起着直接作用（例如，许多设备的热调控，半导体材料的生长）。 此外，开发用于热对流的拓扑工具应更普遍地应用于涉及复杂系统的其他各种问题，包括对天气和气候的预测；海洋生物质的动力学；湍流的发作；试剂模式在催化金属表面上的演变；和人心中的心室纤颤。