PRIMES: Enhancing Capacity for Research in Applied Mathematics at Spelman College

PRIMES：增强斯佩尔曼学院应用数学研究能力

• 批准号: 2331890
• 负责人: Monica Stephens
• 金额: $ 31.89万
• 依托单位: Spelman College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2331890&HistoricalAwards=false
• 关键词: PRIMES; Enhancing; Capacity; Research; Applied

This project aims to enhance research capacity in applied mathematics for undergraduates and faculty in the Department of Mathematics at Spelman College, a historically Black college for women. Through engagement with the Institute for Computational and Experimental Research in Mathematics (ICERM) semester on Numerical Partial Differential Equations, the project will foster research collaborations in numerical PDEs, expose Spelman faculty to ICERM programs, and will offer undergraduate research opportunities and curriculum development in numerical PDEs. The project’s research focus is to develop better computational techniques to solve the equations that model spinodal decomposition, which is the separation of binary mixtures into two phases. The goal is to create higher order numerical approaches using mixed-model methods and to investigate the stability properties of these methods. Additionally, the PI will explore the potential of incorporating machine learning techniques to evolve the model in time. Overall, this project combines cutting-edge research with educational and diversity focused initiatives that will improve research capacity in applied math at Spelman College and will encourage Black women to pursue graduate degrees in applied mathematics. This research project seeks to develop a higher order numerical approach for solving the Cahn-Hilliard equation, a model for spinodal decomposition in binary mixtures. The goal is to investigate the optimal splitting between the implicit and explicit components in an implicit-explicit (IMEX) Runge-Kutta method that yields an accurate and stable solution. The PI firsts extends a semi-implicit approach by Shen to an IMEX implicit midpoint rule. Then the PI will determine if the same splitting is beneficial in a third order diagonally IMEX Runge-Kutta scheme. The research also explores using a mixed-model approach for the variable mobility case that incorporates the constant mobility model. The goal is to choose the optimal splittings to produce variable mobility behavior. Furthermore, the project is incorporating machine learning techniques into the mixed model time evolution. The stability and accuracy of all the proposed methods will be thoroughly investigated. Understanding the dynamics of spinodal decomposition in binary mixtures has significant applications in materials science, chemistry, and engineering. Developing higher order numerical methods and investigating their stability contributes to the advancement of numerical techniques for simulating complex physical phenomena. Moreover, the integration of machine learning into the numerical framework opens avenues for enhancing the accuracy and efficiency of the simulations.The project is funded jointly by the Infrastructure program of the Division of Mathematical Sciences and the HBCU-Excellence in Research Program.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.

该项目旨在通过与数学计算与实验研究所 (ICERM) 的数值偏微分学期合作，提高斯佩尔曼学院数学系本科生和教师的应用数学研究能力，该学院是一所历史悠久的黑人女子学院。该项目将促进数值偏微分方程方面的研究合作，让斯佩尔曼教员参与 ICERM 项目，并将为本科生提供数值偏微分方程方面的研究机会和课程开发。开发更好的计算技术来求解旋节线分解（将二元混合物分离为两相）的方程，目标是使用混合模型方法创建更高阶的数值方法，并研究这些方法的稳定性。 PI 将探索结合机器学习技术来及时改进模型的潜力。总体而言，该项目将尖端研究与教育和多样性重点举措结合起来，这将提高斯佩尔曼学院应用数学的研究能力，并鼓励黑人女性攻读应用数学研究生学位。该项目旨在开发一种高阶数值方法来求解 Cahn-Hilliard 方程，该方程是二元混合中的旋节线分解模型，其目标是研究隐式-显式 (IMEX) 龙格-中隐式组件和显式组件之间的最佳分割。产生准确且稳定的解决方案的库塔方法首先将 Shen 的半隐式方法扩展到 IMEX 隐式中点规则，然后 PI 将确定相同的分裂在三阶中是否有益。该研究还探索了使用混合模型方法来处理可变移动性情况，其目标是选择最佳分裂来产生可变移动性行为。将深入研究所有提出的方法的稳定性和准确性，了解二元混合物中旋节线分解的动力学在材料科学、化学和工程中具有重要的应用。他们对稳定性的研究有助于模拟复杂物理现象的数值技术的进步，此外，将机器学习集成到数值框架中为提高模拟的准确性和效率开辟了途径。该项目由基础设施计划联合资助。数学科学部和 HBCU 卓越研究计划。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。