<div class="csl-bib-body">
<div class="csl-entry">Forkert, D. L. (2014). <i>Gradient flows in general metric and Wasserstein spaces</i> [Diploma Thesis, Technische Universität Wien]. reposiTUm. https://doi.org/10.34726/hss.2014.25503</div>
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dc.identifier.uri
https://doi.org/10.34726/hss.2014.25503
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/3084
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dc.description.abstract
In this thesis we approach gradient flows in metric spaces which are not necessarily endowed with a natural linear structure. The first part is an introduction to the theory in abstract metric spaces: We present three possible generalizations of gradient flows to the metric setting and investigate uniqueness and contraction properties of the various notions. However, an existence theory may not easily be archived at this level of generality. Therefore, we content ourselves by presenting only the principal definitions of the so-called minimizing movements scheme which provides a variational interpolation technique for gradient flows in the metric setting. In the second part of this thesis we deal with gradient flows in Wasserstein spaces. We introduce the theory of optimal transportation. We focus on the Kantorovich problem which is a relaxation of the Monge problem. Specifically, we show that the Kantorovich problem admits an optimal solution under very general assumptions. Moreover, one may utilize the theory of optimal transportation to define a family of metrics on a certain subset in the class of probability measures on a metric space. The resulting family of metric spaces, called Wasserstein spaces, inherit to a great extend the topological and geometrical structure of the underlying metric space. We give proof to an existence result of gradient flows in the quadratic Wasserstein space, employing the aforementioned minimizing movements scheme. This result allows comprehensive application of gradient flows in the field of partial differential equations. In a concluding example we show the main ideas applied to the heat equation.
en
dc.language
English
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dc.language.iso
en
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dc.rights.uri
http://rightsstatements.org/vocab/InC/1.0/
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dc.subject
Wasserstein Space
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dc.subject
Length Space
en
dc.subject
Optimal Transport
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dc.subject
Gradient Flow
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dc.title
Gradient flows in general metric and Wasserstein spaces
en
dc.type
Thesis
en
dc.type
Hochschulschrift
de
dc.rights.license
In Copyright
en
dc.rights.license
Urheberrechtsschutz
de
dc.identifier.doi
10.34726/hss.2014.25503
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dc.contributor.affiliation
TU Wien, Österreich
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dc.rights.holder
Dominik Leopold Forkert
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tuw.version
vor
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tuw.thesisinformation
Technische Universität Wien
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dc.contributor.assistant
Maresch, Gabriel
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tuw.publication.orgunit
E104 - Institut für Diskrete Mathematik und Geometrie
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dc.type.qualificationlevel
Diploma
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dc.identifier.libraryid
AC12164687
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dc.description.numberOfPages
80
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dc.identifier.urn
urn:nbn:at:at-ubtuw:1-68659
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dc.thesistype
Diplomarbeit
de
dc.thesistype
Diploma Thesis
en
dc.rights.identifier
In Copyright
en
dc.rights.identifier
Urheberrechtsschutz
de
tuw.advisor.staffStatus
staff
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tuw.assistant.staffStatus
staff
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tuw.advisor.orcid
0000-0002-7389-6720
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item.openaccessfulltext
Open Access
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item.grantfulltext
open
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item.cerifentitytype
Publications
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item.mimetype
application/pdf
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item.openairecristype
http://purl.org/coar/resource_type/c_bdcc
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item.languageiso639-1
en
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item.openairetype
master thesis
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item.fulltext
with Fulltext
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crisitem.author.dept
E101 - Institut für Analysis und Scientific Computing