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

该项目围绕理论工具的发展，这将导致有效的数值方法来解决最佳运输问题，这是一个数学问题，旨在最大限度地减少从一个位置到另一个位置运输质量的总成本。近年来，对这个问题的理论研究取得了很大进展，所获得的结果已成功地应用于数学之外的许多学科，例如物理学中具有特定反射特性的透镜的设计，地质学中地球表面附近的大气建模，以及经济学中的最佳分配等。这一连串的应用程序使得开发有效的计算工具变得更加紧迫，重要的是要有数学上保证表现出出色性能的工具。该项目将开发基于非线性偏微分方程的新型计算方法，并建立关于这些方程的严格数学结果，以保证相应数值算法的理想性能。该项目的工作涉及主要研究者（PI）的个人和合作研究，以及研究生和本科生的研究指导，并为学生确定了适当的问题。PI还将在2024年通过拉斐特学院夏季REU计划共同监督本科团队，并在2025年通过密歇根州立大学的实验数学夏季本科研究所计划进行推广。这两个项目都旨在从本科研究机会有限的学校招收学生，着眼于从数学科学传统上代表性不足的群体中招收学生。该项目为建立一类退化抛物型完全非线性偏微分方程（PDE）在奇异环境下的解的存在性和长时间行为的特征奠定了理论基础。这些偏微分方程是经典的Monge-Ampere方程随时间变化的变体。在过去的几十年里，在光滑数据和Dirichlet边界条件下，时间依赖的Monge-Ampere方程的经典解理论的发展取得了重大进展。相比之下，这种演化方程的广义解理论是严重欠发达，特别是在最佳传输和几何光学的背景下，自然边界条件是一个非标准的。本项目将为一类退化抛物型、完全非线性的Monge-Ampere型斜边界方程的粘性和弱解的研究奠定理论基础。这项工作也将建立定量的收敛率等方程，这将提供一个有前途的数值方法的设计和施工反射器表面所产生的工程问题。这一奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.