Algorithms and Analysis for Models in Materials Science, Fluids, and Probability

材料科学、流体和概率模型的算法和分析

基本信息

批准号：
1909035
负责人：
Jeremy Marzuola
金额：
$ 28.5万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909035&HistoricalAwards=false
关键词：
Algorithms Analysis Models Materials Science

项目摘要

The project will span applications to problems in physics, chemistry, materials science, and data science; ideas from several mathematical disciplines, including partial differential equations, microlocal analysis, and harmonic analysis, will be used. Mathematics from these areas has made great strides recently on a variety of applications related to molecular structure in quantum mechanics, dynamics of atoms depositing on a surface, partitioning networks into clustered groups, stability of complex phenomena in fluid dynamics, and more. The research here is based largely on newly-developed theories from partial differential equations and probability, though several components of the work will be devoted to implementation of computational algorithms and analysis. Projects will involve training researchers at all levels, including undergraduates, graduate students, and postdoctoral scholars, as well as collaboration with researchers from statistics, physics, computational chemistry, scientific computing, and network science. This research project is aimed at the study of several important interdisciplinary topics ranging from fundamental questions about molecular structure to explorations of algorithms in data analysis. For instance, the principal investigator will continue work on bifurcation theory in models from density functional theory, as well as on finding numerical and analytic tools for studying nonlinear bifurcations of topological lattice models arising in quantum optics. A substantial part of the project will be done in collaboration with students and colleagues developing numerical methods for fast, accurate computation of fluid flows in complicated geometries, with the intention of making experimental predictions. Special attention will be paid to highly nonlinear models and applications of tools from spectral theory. Beyond physical applications, spectral theory can also play important roles in data analysis, specifically in spectral clustering. Along these lines, using ideas from probability, optimization and harmonic analysis, the investigator aims to develop algorithms for sub-graph detection in networks, and more generally Markov chain partitioning.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将涉及物理，化学，材料科学和数据科学的问题；将使用来自几个数学学科的思想，包括部分微分方程，微局部分析和谐波分析。这些区域的数学最近在量子力学中与分子结构，沉积在表面上的原子动力学，将网络分配到聚类组，流体动力学中复杂现象的稳定性等各种应用方面取得了长足的进步。此处的研究主要基于从部分微分方程和概率上新开发的理论，尽管该工作的几个组成部分将致力于实施计算算法和分析。项目将涉及各级培训研究人员，包括本科生，研究生和博士后学者，以及与统计，物理，计算化学，科学计算和网络科学的研究人员的合作。该研究项目旨在研究数据分析中的几个重要的跨学科主题，从有关分子结构的基本问题到算法的探索。例如，主要研究者将继续在密度功能理论的模型中继续研究分叉理论，以及寻找用于研究量子光学中产生的拓扑晶格模型的非线性分叉的数值和分析工具。该项目的很大一部分将与学生和同事合作开发数值方法，以快速，准确地计算复杂几何形状中的流体流，以进行实验预测。将特别注意光谱理论的高度非线性模型和工具的应用。除了物理应用之外，光谱理论还可以在数据分析，特别是光谱聚类中起重要作用。沿着这些思路，使用概率，优化和谐波分析的想法，研究者旨在开发网络中次数检测的算法，更通常是马尔可夫链条分区。该奖项反映了NSF的法定任务，并被视为值得通过评估的支持。利用基金会的知识分子和更广泛的影响审查标准。