Numerical Methods for Optimal Transport with Applications to Manifold Learning on Singular Spaces

最优传输的数值方法及其在奇异空间流形学习中的应用

• 批准号: 2000128
• 负责人: Jun Kitagawa
• 金额: $ 18万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000128&HistoricalAwards=false
• 关键词: Numerical; Methods; Optimal; Transport; Applications

Optimal transport concerns the classical question of how to optimize the cost of transporting mass from one location to another. Optimal transport models have been successfully applied in fields as diverse as atmospheric sciences, surface matching, data clustering, and manifold learning, among others. The theoretical study of optimal transport has greatly advanced in recent years and calls for improved numerical tools that can be mathematically guaranteed to have good performance. This project is aimed at the development of improved numerical methods for optimal transport calculations and variants, employing partial-differential-equation techniques to produce computational tools backed by rigorous theory. The project provides research training opportunities for undergraduate and graduate students. The PI will also engage in outreach by supervising an undergraduate team through the university's Summer Undergraduate Research Institute in Experimental Mathematics program, aimed at students who are at an earlier stage of study, with an eye toward recruitment of students from groups underrepresented in the mathematical sciences. Specifically, the project aims to exploit the geometric information that can be discerned from the regularity theory of the Monge-Ampère type equation that arises naturally in optimal transport, in order to develop numerical algorithms with proven convergence rates, error bounds, and computational complexity. The project will also undertake a systematic study of singular behavior when full regularity is unavailable to develop fast and accurate numerical schemes in such difficult cases. One intended application of this latter direction is toward a theory of manifold learning that can be applied to data sets coming from singular geometries.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.

最优运输涉及如何优化从一个地点到另一个地点的运输成本的经典问题。最优输运模型已成功应用于大气科学、地表匹配、数据聚类和流形学习等多个领域。近年来，最优输运的理论研究有了很大的进展，这就要求改进的数值工具能够在数学上保证其良好的性能。该项目旨在发展改进的最优输运计算和变量的数值方法，采用偏微分方程技术来产生严格理论支持的计算工具。本项目为本科生和研究生提供研究训练机会。PI还将通过该大学的实验数学暑期本科生研究所（Summer undergraduate Research Institute in Experimental Mathematics）项目监督一个本科生团队，以招收数学科学领域代表性不足的学生为目标。具体来说，该项目旨在利用可从最优传输中自然产生的monge - ampantere型方程的正则理论中识别出的几何信息，以开发具有已证明的收敛率、误差界限和计算复杂性的数值算法。该项目还将进行系统的奇异行为研究，当完全正则性无法获得时，在这种困难的情况下开发快速和准确的数值方案。后一个方向的一个预期应用是可以应用于来自奇异几何的数据集的流形学习理论。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。

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

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