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

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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.