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

本项目研究一类偏微分方程解的理论和计算性质，这些性质在许多建模问题中出现。这些包括：使用镜子或透镜的光学仪器的设计，经济学中的市场匹配，以及交通网络的优化。一个非常重要的问题涉及到解决方案的流畅性；即，这些方程的解是否有锐角。在上述问题中，平滑性问题直接影响到这些方程作为模型的准确性。例如，在设计具有一定折射特性的透镜时，平滑度的失败可能导致色差（如棱镜将白光分裂成彩虹时），这种现象不在这类方程给出的简化模型的范围之内。然而，这类特殊的方程对于建模来说是非常方便的，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镜子比使用其他模型要简单得多，效率也高得多。平滑性在发展精确、快速的数值算法中也起着重要作用。有效地将理论结果转化为计算语言对于在抽象数学结果和实际结果之间建立桥梁至关重要。更具体地说，该项目将关注蒙日-安培型方程的一个子类，称为生成雅可比方程（GJEs）。这些问题包括近场几何光学、蒙日-坎托洛维奇（最优输运）问题和实蒙日-安培方程。项目的第一部分侧重于正则性和“部分正则性”，这意味着弱解的奇异行为的研究。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，并定量地描述非齐次数据，这些数据给出的解在正则性理论的标准条件失效时仍然表现出规则行为。用于研究这些问题的方法将结合用于分析退化的完全非线性椭圆方程的正则性的几何技术，以及源自变分学和最优输运理论的技术。该项目的第二部分侧重于GJEs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方案，从而产生快速，准确的算法来解决GJE和现有技术无法解决的最佳传输问题，其具体目标是产生具有严格性能界限的算法。项目的两部分是相互关联的，因为改进的数值工具可以有效地可视化和制定关于解的奇异行为的猜想，反过来，改进的对奇点的理论理解导致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.