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都无法用简洁的特征来描述，因为它们表现出强烈的非线性行为，缺乏所有的对称性，并且在大范围的空间和时间尺度上具有高度的非周期性。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，由此产生的大量数据往往导致很少的洞察力；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。来自代数拓扑的基于计算的新数学工具有可能弥合测量和模型之间的差距；提出的研究将探索使用代数拓扑将数值模拟和实验室实验联系起来，在复杂的情况下，因为所研究的系统被赶出热力学平衡。研究重点是非平衡复杂性的一个杰出范例：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)测量和精确操纵复杂对流流的实验能力；(2)先进、大规模、高分辨率对流流动数值模拟的有效方法；(3)用于计算大型数据集代数拓扑不变量的开源、通用和高效的计算算法和软件。将开发拓扑工具来描述和最小化模型误差，以及比较和量化动态特性，包括李雅普诺夫指数，维度和复杂时空流状态之间的分岔。这项工作应该最终确定基于同构的度量可以用于构建允许预测和可能控制对流的降阶模型的方法。更一般地说，我们期望对流开发的指标应该广泛应用于pde模型问题，从心律失常的控制到天气和气候的预测。我们周围世界中复杂系统的行为现在既可以使用先进的传感技术以高保真度进行测量，也可以使用现代计算机技术以极高的真实感进行模拟。然而，在这些情况下通常产生的大量数据集往往难以解释，因为很少有好的数学工具可以定量地将给定复杂系统的实验测量与同一系统的计算机模拟输出联系起来。提出的研究探索使用拓扑数学将实验室测量与计算机输出联系起来，在一个特定的复杂系统，热对流。这项工作的结果应该会导致新的方法来理解，预测，也许，控制对流，它在自然过程（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）中起着直接作用。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测；海洋生物质的动态；乱流的开始；催化金属表面试剂图案的演变以及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.