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Adaptive numerical methods for optimal and branched transport problems

dc.contributor.advisorSchmitzer, Bernhard Prof. Dr.
dc.contributor.authorMinevich, Olga
dc.titleAdaptive numerical methods for optimal and branched transport problemsde
dc.contributor.refereeSchmitzer, Bernhard Prof. Dr.
dc.description.abstractengOptimal transport is an area of mathematical research that has been gaining popularity in recent years in various application fields such as economics, statistics or machine learning. Two important factors behind its increasing popularity are its modelling flexibility and the ever-expanding range of available dedicated computational tools. Unbalanced optimal transport is a generalization that allows for the comparison of measures with different mass, which is more appropriate in some applications. In this thesis, we consider the barycenter problem (i.e. finding a weighted average) between several input measures with respect to the unbalanced Hellinger–Kantorovich metric. In particular, we focus on the case with an uncountable number of Dirac input measures. We study existence, uniqueness and stability of the solutions, and demonstrate the intricate behavior of the barycenters with respect to the length scale parameter using analytical and numerical tools. Another important variant is branched transport, where the transport cost encourages the formation of branched transportation networks. We focus in this thesis on its convex relaxation in terms of multimaterial transport. In particular, we study the multimaterial problem in a setting when only a single topology of the solution is admissible and describe the simple structure of the dual solution in this case. We then formulate a problem with 3 sources and 1 sink where two candidate solutions of different topologies give the same transportation cost, study its properties and characterize the solution
dc.contributor.coRefereeWirth, Benedikt Prof. Dr.
dc.subject.engOptimal Transportde
dc.subject.engMultimaterial Transportde
dc.subject.engUnbalanced Optimal Transportde
dc.subject.engNumerical Optimizationde
dc.subject.engHellinger--Kantorovich Barycentersde
dc.affiliation.instituteFakultät für Mathematik und Informatikde
dc.subject.gokfullInformatik (PPN619939052)de
dc.notes.confirmationsentConfirmation sent 2023-11-20T19:45:01de

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