A Wasserstein-like Distance on Vector Fields
by Vincent Sommer
Date of Examination:2023-02-14
Date of issue:2023-10-06
Advisor:Prof. Dr. Max Wardetzky
Referee:Prof. Dr. Max Wardetzky
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Bernhard Schmitzer
Referee:Prof. Dr. Axel Munk
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Christoph Lehrenfeld
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Abstract
English
We introduce a new distance on the space of vector fields over a Riemannian manifold that is motivated by the construction and properties of the Wasserstein distance. The construction relies on a particular metric on the unit circle bundle which leads to a relation between the distance and parallelism of moving vector fields. Moreover, we investigate properties of this distance and related concepts like the curve energy, a dual formulation and a Benamou-Brenier formula. We also consider possible discretizations of the unit circle bundle that could be useful for numerical implementations of the distance. Furthermore, this work contains a piecewise linear discretization of the Wasserstein distance as was already done by Lavenant et al. but we give the construction with significantly more detail and add some original remarks to the theory.
Keywords: Wasserstein geometry; Riemannian geometry; Optimal transport; Discrete differential geometry; Piecewise-linear discretization