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

最优运输涉及如何优化从一个地点到另一个地点的质量运输成本的经典问题。最优传输模型已经成功地应用于大气科学、地表匹配、数据聚类和流形学习等领域。最近几年，最优运输的理论研究取得了很大进展，需要改进的数值工具，这些工具在数学上可以保证具有良好的性能。这个项目的目的是开发改进的数值方法，用于最优运输计算和变种，使用偏微分方程技术来产生有严格理论支持的计算工具。该项目为本科生和研究生提供了研究培训机会。PI还将通过该大学的实验数学暑期本科生研究所项目监督一个本科生团队，该项目针对的是处于学习早期阶段的学生，着眼于从数学科学中代表性较低的群体中招收学生。具体地说，该项目旨在利用在最优运输中自然产生的Monge-Ampère型方程的正则性理论中可以识别的几何信息，以便开发具有证明的收敛速度、误差界和计算复杂性的数值算法。该项目还将对奇异行为进行系统研究，当完全规律性不存在时，无法在这种困难的情况下开发快速而准确的数值格式。后一方向的一个意向应用是一种可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NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.