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

自然界和技术中的许多复杂系统无法进行简洁的表征，因为它们表现出强烈的非线性行为，缺乏所有对称性，并且在广泛的空间和时间尺度上高度非周期性。现在，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。然而，由此产生的海量数据往往导致洞察力有限；特别是，通常没有好的方法将特定复杂系统的定量实验测量与同一系统的模拟/模型的输出联系起来。 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；拟议的研究将探索使用代数拓扑在由于所研究的系统脱离热力学平衡而出现复杂性的情况下将数值模拟和实验室实验联系起来。 该研究重点关注非平衡复杂性的一个突出范例：温度梯度驱动的流体流动（热对流）。计划中的工作将三种独特的能力结合在一起：（1）测量和操纵精确复杂的对流流的实验能力； (2) 最先进、大规模、高分辨率的对流数值模拟的有效方法； (3) 开源、通用、高效的计算算法和软件，用于计算大数据集上的代数拓扑不变量。 将开发拓扑工具来表征和最小化模型误差，以及比较和量化动态特性，包括李雅普诺夫指数、复杂时空流状态之间的维数和分岔。 这项工作最终应该确定基于同源性的度量可用于构建降阶模型的方法，该模型允许预测，或许还可以控制对流。 更一般地说，我们预计为对流开发的指标应该广泛应用于偏微分方程模型的问题，从心律失常的控制到天气和气候的预测。我们周围世界的复杂系统的行为现在既可以使用先进的传感技术进行高保真度测量，也可以使用现代计算机技术进行非常真实的模拟。然而，在这些情况下通常产生的大量数据集通常难以解释，因为很少有好的数学工具可以将给定复杂系统的实验测量与同一系统的计算机模拟的输出定量地联系起来。 拟议的研究探索了如何利用拓扑数学将实验室测量结果与特定复杂系统（热对流）中的计算机输出联系起来。这项工作的结果应该会带来理解、预测、或许还有控制对流的新方法，对流在自然过程（例如火山活动、地震动力学、大陆漂移）和工业应用（例如许多设备的热调节、半导体材料的生长）中发挥着直接作用。 此外，为热对流开发的拓扑工具应该更广泛地应用于涉及复杂系统的各种其他问题，包括天气和气候的预报；海洋生物量的动态；湍流的开始；催化金属表面上试剂图案的演变；和人类心脏的心室颤动。

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.