INSPIRE: Nonlinear Data Reduction applied to Dense Granular Media

该INSPIRE奖的部分资金来自数学与物理科学局数学科学部的应用数学和拓扑项目，以及工程局化学、生物工程、环境和运输系统部的颗粒材料项目。它将支持将最先进的实验技术、大规模分子动力学模拟和拓扑数据分析的前沿理论发展相结合的工作，以理解和预测致密颗粒材料（DGM）的行为。特别是，它基于使用持久性图的新命题，持久性图是应用代数拓扑中的一个相对较新的概念，作为DGM的基本建模工具。核心任务包括以下内容。(1)开发一个有效的计算框架，用于处理来自时空系统的持久性图数据。(2)利用离散元模拟（DEM）和实验，对DGM中的力和应力进行精确、明确和完整的表征，包括给定系统的空间和时间变异，以及模拟和实验之间的差异的量化。(3)制定用于分析给定系统的实验和模拟的最佳组合，从而对描述包括应力、体积和剪切模量在内的系统的宏观量的性质和演变进行可靠的预测。(4)使用持久性图来表征空间和时间波动，包括对系统大小、物理维度（如2D与3D）和粒子特性（如形状或摩擦）的依赖。(5)研究持久图空间的拓扑结构。利用纯拓扑技术研究时间序列重构动力学，表征DGM的动力学特性。颗粒材料在密集的流体状和固体状态提出了我们这个时代最重要的建模挑战之一。这些材料出现在从重工业到制药业的广泛实际环境中。因此，它们引起了工程界的强烈兴趣。对于软凝聚态物质和统计物理学界来说，在DGM中颗粒堆积和力传递的方式是一个重大的难题。与力和包装复杂性相关的复杂高维多尺度时空数据分析是具有挑战性的数学问题。DGM本质上是高维系统，几何在其中起着基本的作用。然而，由于它们的颗粒性质，它们不能用解析连续统模型有效地近似，因此缺乏一种明确的简化为可处理问题的方法。我们的目标是证明与计算拓扑相关的新思想为DGM空间结构的时间演化提供了一种高效、可靠和连贯的方法。正在开发的拓扑方法的抽象性质意味着在这种情况下开发的工具将适用于展示复杂时空结构的广泛系统。

This INSPIRE award is partially funded by the Applied Mathematics and the Topology program in the Division of Mathematical Sciences in the Directorate for Mathematical and Physical Sciences and the Granular Materials Program in the Division of Chemical, Bioengineering, Environmental, and Transport Systems in the Engineering Directorate. It will support work that uniquely combines state of the art experimental techniques, large scale molecular dynamics simulations, and cutting edge theoretical developments in topological data analysis for the purpose of understanding and predicting the behavior of dense granular materials (DGM). In particular, it is based on the novel proposition of using persistence diagrams, a relatively new concept in applied algebraic topology, as the fundamental modeling tool for DGM. The core tasks include the following. (1) Develop an efficient computational framework for working with persistence diagram data arising from spatiotemporal systems. (2) Using discrete element simulations (DEM) and experiments, carry out precise, well-defined, and complete characterizations of forces and stresses in DGM, including spatial and temporal variability of a given system and quantification of the differences between simulations and experiments. (3) Formulation of an optimum combination of experiments and simulations to be used to analyze a given system, leading to reliable predictions for the nature and evolution of macroscopic quantities describing a system including stresses and bulk and shear moduli. (4) Use of persistence diagrams to characterize spatial and temporal fluctuations, including the dependence on system size, on the physical dimensions (e.g. 2D versus 3D), and particle properties such as shape or friction. (5) Study the topology of the space of persistence diagrams. Characterize the dynamics of DGM by using purely topological techniques to study time series reconstructed dynamics.Granular materials in the dense fluid-like and solid states present one of the most significant modeling challenges of our times. These materials appear in a broad spectrum of practical settings from heavy industry to pharmaceuticals. As such, they are of intense interest to the engineering world. The way in which grains pack and in which forces are carried in DGM presents a significant puzzle for the soft condensed matter and statistical physics communities. Associated with force and packing complexity are challenging mathematical issues related to the analysis of complex high dimensional multiscale spatiotemporal data. DGM are inherently high dimensional systems in which geometry plays a fundamental role. However because of their granular nature they cannot be usefully approximated by analytic continuum models and thus a clear method of reduction to a tractable problem is lacking. Our goal is to demonstrate that new ideas associated with computational topology provide an efficient, faithful and coherent approach to providing tractable models for the temporal evolution of spatial structures in DGM. The abstract nature of the topological methods being developed imply that the tools developed in this context will be applicable to a wide range of systems demonstrating complex spatiotemporal structures.