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

Monge-Ampere 型方程的正则性和偏正则性及其在数值中的应用

• 批准号: 1700094
• 负责人: Jun Kitagawa
• 金额: $ 15万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700094&HistoricalAwards=false
• 关键词: Regularity; Partial; Monge; Ampere; Type

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.

这个项目解决了在许多建模问题中出现的一类偏微分方程的理论和计算性质。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是关于解的光滑性，也就是说，这些方程的解是否可以有尖角。平滑度的问题直接影响到作为上述问题模型的这些方程的精度。例如，在设计具有某些折射特性的透镜时，光滑度的失败会导致色差（如棱镜将白色光分成彩虹），这种现象超出了这类方程给出的简化模型的范围。然而，这类特殊的方程是非常方便的建模目的，由于事实上，他们的解决方案往往有简单的几何特征。这在光学仪器的例子中特别重要，因为与其他模型相比，通过应用这些方程产生的透镜或反射镜的制造在逻辑上比通过使用其他模型简单得多，也更有效。光滑性在精确和快速的数值算法的发展中也起着重要的作用。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Monge-Ampere型方程的一个子类，称为生成雅可比方程（GJE）。这些包含许多问题，从几何光学在近场制度，蒙格-康托洛维奇（最佳运输）的问题，和真实的蒙格-安培方程。第一部分的项目集中在正则性和“部分正则性”，这意味着研究奇异行为的弱解。具体目标是系统地分类奇异集的结构（大小，形状，拓扑和可微结构），并定量地描述非齐次数据，这些数据给出了在正则性理论中的标准条件失败时仍然表现出正则行为的解决方案。用于研究这些问题的方法将是一个组合的几何技术开发，以分析退化，完全非线性椭圆方程的规律性，沿着技术起源于变分法和最佳运输理论。该项目的第二部分侧重于GJE的数值求解器的开发。其目的是利用部分正则性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和现有技术无法达到的最佳运输问题，其具体目标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制定解决方案的奇异行为的示意图的有效工具，反过来，对奇异性的理论理解的提高导致了有效的数值算法的发展。

项目成果

${\mathcal {W}}_\infty $-transport with discrete target as a combinatorial matching problem

${mathcal {W}}_infty $-离散目标传输作为组合匹配问题

• DOI: 10.1007/s00013-021-01606-z
• 发表时间: 2021
• 期刊: Archiv der Mathematik
• 影响因子: 0.6
• 作者: Bansil, Mohit;Kitagawa, Jun
• 通讯作者: Kitagawa, Jun

Exponential convergence of parabolic optimal transport on bounded domains

有界域上抛物线最优传输的指数收敛

• DOI: 10.2140/apde.2020.13.2183
• 发表时间: 2020
• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: Abedin, Farhan;Kitagawa, Jun
• 通讯作者: Kitagawa, Jun

An optimal transport problem with storage fees

带仓储费的最优运输问题

• DOI: 10.58997/ejde.2023.22
• 发表时间: 2023
• 期刊: Electronic Journal of Differential Equations
• 影响因子: 0.7
• 作者: Bansil, Mohit;Kitagawa, Jun
• 通讯作者: Kitagawa, Jun

Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

半离散最优输运几何的定量稳定性

• DOI: 10.1093/imrn/rnaa355
• 发表时间: 2020
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Bansil, Mohit;Kitagawa, Jun
• 通讯作者: Kitagawa, Jun

Optimal transport and the Gauss curvature equation

最优传输和高斯曲率方程

• DOI: 10.4310/maa.2020.v27.n4.a5
• 发表时间: 2020
• 期刊: Methods and Applications of Analysis
• 影响因子: 0.3
• 作者: Guillen, Nestor;Kitagawa, Jun
• 通讯作者: Kitagawa, Jun