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

这个项目的核心是理论工具的发展，这将导致有效的数值方法来解决最优运输问题，这是一个数学问题，寻求最小化从一个地方到另一个地方运输质量的总成本。近年来，对这一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以外的许多学科，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大气的建模，以及经济学中创建最佳作业等。这一连串的应用程序使得开发有效的计算工具成为一件更加紧迫的事情，并且拥有能够在数学上保证表现出出色性能的工具非常重要。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。该项目的工作涉及首席研究员的个人和合作研究，以及研究生和本科生的研究指导，并为学生确定适当的问题。这些pi还将在2024年通过拉斐特学院夏季REU项目和2025年通过密歇根州立大学夏季本科生实验数学研究所项目共同监督一个本科生团队，从而参与外展活动。这两个项目都旨在从本科生研究机会有限的学校招收学生，并着眼于从传统上代表性不足的数学科学群体中招收学生。本项目为建立一类退化抛物型全非线性偏微分方程（PDE）在奇异环境下解的存在性和长时间行为特征提供了理论基础。这些偏微分方程是经典蒙日-安培方程的时变变体。在过去的几十年中，在发展具有光滑数据和狄利克雷边界条件的时变蒙日-安培方程的经典解理论方面取得了重大进展。相比之下，这类演化方程的广义解理论严重不发达，特别是在最优输运和几何光学的情况下，其中自然边界条件是非标准的。目前的项目将为一类具有斜边界数据的退化抛物型全非线性蒙日-安培型方程的粘度和弱解建立理论基础。这项工作还将确定这些方程的收敛速度；这将为工程中出现的反射面设计和构造提供一种有前途的数值方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.