Collaborative Research: Parabolic Monge-Ampère Equations, Computational Optimal Transport, and Geometric Optics

合作研究：抛物线 Monge-AmpeÌre 方程、计算最优传输和几何光学

• 批准号: 2246611
• 负责人: Farhan Abedin
• 金额: $ 18.46万
• 依托单位: Lafayette 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: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246611&HistoricalAwards=false
• 关键词: Collaborative; Research; Parabolic; Monge; Ampe

This project is centered around the development of theoretical tools which will lead to efficient numerical methods for solving the optimal transport problem, which is a mathematical problem that seeks to minimize the total cost of transporting mass from one location to another. The theoretical study of this problem has advanced greatly in recent years, and the results obtained have been applied successfully to a number of disciplines outside of mathematics, such as the design of lenses with specific reflection properties in Physics, modeling the atmosphere near the earth's surface in Geology, and creating optimal assignments in Economics, among others. This litany of applications makes the development of effective computational tools an ever more urgent matter, and it is important to have tools that can be mathematically guaranteed to exhibit outstanding performance. The project will develop novel computational methods based on nonlinear partial differential equations and establish rigorous mathematical results about these equations in order to guarantee desirable performance of the corresponding numerical algorithms. The work of the project involves individual and collaborative research by the Principal Investigators (PIs), and research mentoring of graduate and undergraduate students, with appropriate problems having been identified for students. The PIs will also engage in outreach through co-supervising an undergraduate team in 2024 through the Lafayette College Summer REU program and in 2025 through the Summer Undergraduate Research Institute in Experimental Mathematics program at Michigan State University. Both programs aim to recruit students from schools with limited opportunities for undergraduate research, with an eye toward recruitment of students from traditionally underrepresented groups in the mathematical sciences. The project develops the theoretical foundations for establishing existence and characterizing long-time behavior of solutions to a class of degenerate-parabolic fully nonlinear partial differential equations (PDE) in singular settings. These PDE are time-dependent variants of the classical Monge-Ampere equation. Significant progress has been made over the last few decades in developing a theory of classical solutions for time-dependent Monge-Ampere equations with smooth data and Dirichlet boundary conditions. By comparison, the theory of generalized solutions for such evolutionary equations is severely underdeveloped, especially in the context of optimal transport and geometric optics, where the natural boundary condition is a non-standard one. The current project will create a theoretical foundation for viscosity and weak solutions of a class of degenerate parabolic, fully nonlinear equations of Monge-Ampere type with oblique boundary data. The work will also establish quantitative rates of convergence for such equations; this will provide a promising numerical method for the design and construction of reflector surfaces arising in engineering problems.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.

该项目围绕理论工具的开发，该工具将导致解决最佳运输问题的有效数值方法，这是一个数学问题，旨在最大程度地减少从一个位置到另一个位置的运输总成本。近年来，对该问题的理论研究已经大大发展，并且获得的结果已成功地应用于数学以外的许多学科，例如在物理学中具有特定反射特性的镜头设计，对地质表面附近的大气建模，并在经济学中创造最佳的经济分配。这种应用程序的开发使有效的计算工具的开发变得更加紧迫，重要的是要有数学上可以保证以表现出杰出性能的工具。该项目将基于非线性偏微分方程开发新的计算方法，并在这些方程式上建立严格的数学结果，以确保相应的数值算法的理想性能。该项目的工作涉及首席研究人员（PIS）的个人和协作研究，以及研究生和本科生的研究指导，为学生发现了适当的问题。 PIS还将在2024年通过Lafayette College Summer REU计划和2025年，直到密歇根州立大学实验数学课程的夏季本科研究所，并在2024年通过共同探索一支本科团队进行宣传。这两种计划旨在招募来自学校本科研究机会有限的学校的学生，并着眼于招募数学科学中传统代表性不足的群体的学生。该项目为建立存在和表征解决方案的长期行为的理论基础开发了对单数环境中的一类简并促谁非线性偏微分方程（PDE）的长期行为。这些PDE是经典Monge-Ampere方程的时间依赖性变体。在过去的几十年中，在开发具有平滑数据和差异边界条件的时间依赖性的蒙格 - 安培方程的经典解决方案的理论中取得了重大进展。相比之下，这种进化方程的通用解决方案的理论严重欠发达，尤其是在最佳传输和几何光学元件的背景下，自然边界条件是非标准的。当前的项目将为粘度和弱抛物线词的粘度和弱解决方案创造一个理论基础，并具有斜边界数据的Monge-Ampere类型的完全非线性方程。这项工作还将建立此类方程的融合定量率；这将提供一种有希望的数值方法，用于设计和构建工程问题中出现的反射器表面。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。