Regularity and Partial Regularity for Monge-Ampere-Type Equations, with Applications to Numerics

这个项目研究了一类偏微分方程解的理论和计算性质，这类偏微分方程解出现在许多建模问题中。其中包括：使用反射镜或透镜的光学仪器的设计，经济上的市场匹配，以及交通网络的优化。一个非常重要的问题与解的光滑性有关；即，这些方程的解是否可以有尖角。光滑性问题直接影响到这些方程作为上述问题中的模型的准确性。例如，在设计具有某些折射特性的透镜时，光滑度的失效会导致色差(就像棱镜将白光分裂成彩虹一样)，这种现象超出了这类方程所给出的简化模型的范围。然而，这类特殊的方程对于建模非常方便，因为它们的解通常具有简单的几何特征。这在光学仪器的例子中尤其重要，因为与其他模型相比，应用这些方程制造透镜或反射镜在逻辑上要比使用其他模型简单和高效得多。光滑性在开发准确和快速的数值算法中也起着很大的作用。有效地将理论结果转化为计算语言是在抽象数学结果和实际结果之间建立桥梁的关键。更具体地说，该项目将关注Monge-Ampere类型方程的一个子类，即生成的Jacobian方程(GJE)。这些问题包括近场几何光学、Monge-Kantorovich(最优输运)问题和真实Monge-Ampere方程。该项目的第一部分侧重于正则性和“部分正则性”，即研究弱解的奇异行为。具体目标是系统地对奇异集的结构(大小、形状、拓扑和可微结构)进行分类，并对非齐次数据进行定量表征，这些数据给出了当标准条件在正则性理论中失效时仍表现出规则行为的解。用来研究这些问题的方法将是用来分析退化的、完全非线性的椭圆型方程的正则性的几何技术，以及起源于变分和最优传输理论的技术的组合。该项目的第二部分侧重于GJE数值求解器的开发。其目的是利用部分正则性所提供的理解来开发避免退化奇异结构的数值格式，从而产生快速、准确的算法来解决GJE和超出现有技术范围的最优传输问题，具体目的是产生具有严格性能界限的算法。该项目的两个部分是相互关联的，因为改进的数值工具可以成为可视化和形成关于解的奇异行为的猜想的有效工具，反过来，对奇点的改进的理论理解导致了有效的数值算法的发展。

This project addresses theoretical and computational properties of solutions to a certain class of partial differential equations, which arise in many modeling problems. These include the following: the design of optical instruments using mirrors or lenses, market matching in economics, and the optimization of transportation networks. One very important question concerns the smoothness of solutions; namely, if solutions of these equations can have sharp corners or not. The question of smoothness has direct implications in the accuracy of these equations as models in the aforementioned problems. For example, in designing a lens with certain refractive properties, failure of smoothness can cause chromatic aberration (as when a prism splits white light into a rainbow), a phenomenon that lies outside the scope of the simplified model given by this class of equations. Nevertheless, this special class of equations is extremely convenient for modeling purposes, owing to the fact that their solutions often have simple geometric characterizations. This is especially important in the example of optical instruments, since in comparison with other models, it is logistically far simpler and more efficient to manufacture lenses or mirrors that arise by application of these equations than by using other models. Smoothness also plays a large role in the development of accurate and fast numerical algorithms. Effectively translating theoretical results into the language of computation is vital in creating a bridge between abstract mathematical results and practical outcomes.In more specific terms, the project will be concerned with a subclass of Monge-Ampere-type equations known as generated Jacobian equations (GJEs). These contain many problems from geometric optics in the near-field regime, the Monge-Kantorovich (optimal transport) problem, and the real Monge-Ampere equation. The first portion of the project focuses on regularity and "partial regularity," by which is meant the study of singular behavior of weak solutions. Specific goals are to categorize systematically the structure (size, shape, topological, and differentiable structure) of singular sets, and to characterize quantitatively the inhomogeneous data that give solutions that still exhibit regular behavior when standard conditions in regularity theory fail. The methods used to study these problems will be a combination of geometric techniques developed to analyze regularity of degenerate, fully nonlinear elliptic equations, along with techniques originating in the calculus of variations and optimal transport theory. The second part of the project focuses on the development of numerical solvers for GJEs. The intent is to leverage the understanding afforded by partial regularity to develop numerical schemes that avoid degenerate singular structures, resulting in fast, accurate algorithms to solve GJE and optimal transport problems beyond the reach of existing techniques, with a specific aim of producing algorithms backed with rigorous performance bounds. The two halves of the project are interrelated, as improved numerical tools can be an effective tool for visualizing and formulating conjectures on the singular behavior of solutions, and in turn, improved theoretical understanding of singularities leads to the development of effective numerical algorithms.