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

自然界和技术上的许多复杂系统都不能用简明的描述来描述，因为它们表现出强烈的非线性行为，缺乏所有的对称性，并且在广泛的空间和时间尺度上具有高度的非周期性。现在在许多情况下，使用现代测量技术或计算技术，通过详细测量(在实验室实验或直接数值模拟)进行表征是可能的。然而，由此产生的海量数据往往导致洞察力不足；尤其是，往往没有好的方法将特定复杂系统的实验测量与同一系统的模拟/模型的输出定量联系起来。基于代数拓扑学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学将数值模拟和实验室实验联系起来，因为所研究的系统被驱离热力学平衡而产生复杂的情况。这项研究侧重于非平衡复杂性的一个突出范例：由温度梯度(热对流)驱动的流体流动。计划中的工作将三种独特的能力结合在一起：(1)精确测量和操纵复杂对流的实验能力；(2)用于最先进的、大规模、高分辨率对流数值模拟的有效方法；(3)用于计算大数据集上的代数拓扑不变量的开源、通用和高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性质，包括复杂时空流态之间的Lyapunov指数、维度和分叉。这项工作最终应该确定基于同源性的度量可以用于建立降阶模型的方法，这些模型允许预测，也许还可以控制对流。更广泛地说，我们预计为对流开发的度量方法将广泛应用于PDE模拟问题，从控制心脏心律失常到预测天气和气候。现在，我们周围世界的复杂系统的行为既可以使用先进的传感技术进行高保真测量，也可以使用现代计算机技术进行非常逼真的模拟。然而，在这些情况下通常产生的巨大数据集通常很难解释，因为几乎没有好的数学工具来定量地将给定复杂系统的实验测量与该相同系统的计算机模拟的输出联系起来。这项拟议的研究探索了使用拓扑学的数学来将实验室测量结果与计算机输出的特定复杂系统--热对流--联系起来。这项工作的结果应该会导致以新的方式理解、预测和控制对流流动，对流流动在自然过程(例如火山作用、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更普遍地适用于涉及复杂系统的各种其他问题，包括天气和气候预报；海洋中生物量的动态；湍流的开始；催化金属表面试剂模式的演变；以及人体心脏的室颤。

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.