---
_id: '10030'
abstract:
- lang: eng
text: "This PhD thesis is primarily focused on the study of discrete transport problems,
introduced for the first time in the seminal works of Maas [Maa11] and Mielke
[Mie11] on finite state Markov chains and reaction-diffusion equations, respectively.
More in detail, my research focuses on the study of transport costs on graphs,
in particular the convergence and the stability of such problems in the discrete-to-continuum
limit. This thesis also includes some results concerning\r\nnon-commutative optimal
transport. The first chapter of this thesis consists of a general introduction
to the optimal transport problems, both in the discrete, the continuous, and the
non-commutative setting. Chapters 2 and 3 present the content of two works, obtained
in collaboration with Peter Gladbach, Eva Kopfer, and Jan Maas, where we have
been able to show the convergence of discrete transport costs on periodic graphs
to suitable continuous ones, which can be described by means of a homogenisation
result. We first focus on the particular case of quadratic costs on the real line
and then extending the result to more general costs in arbitrary dimension. Our
results are the first complete characterisation of limits of transport costs on
periodic graphs in arbitrary dimension which do not rely on any additional symmetry.
In Chapter 4 we turn our attention to one of the intriguing connection between
evolution equations and optimal transport, represented by the theory of gradient
flows. We show that discrete gradient flow structures associated to a finite volume
approximation of a certain class of diffusive equations (Fokker–Planck) is stable
in the limit of vanishing meshes, reproving the convergence of the scheme via
the method of evolutionary Γ-convergence and exploiting a more variational point
of view on the problem. This is based on a collaboration with Dominik Forkert
and Jan Maas. Chapter 5 represents a change of perspective, moving away from the
discrete world and reaching the non-commutative one. As in the discrete case,
we discuss how classical tools coming from the commutative optimal transport can
be translated into the setting of density matrices. In particular, in this final
chapter we present a non-commutative version of the Schrödinger problem (or entropic
regularised optimal transport problem) and discuss existence and characterisation
of minimisers, a duality result, and present a non-commutative version of the
well-known Sinkhorn algorithm to compute the above mentioned optimisers. This
is based on a joint work with Dario Feliciangeli and Augusto Gerolin. Finally,
Appendix A and B contain some additional material and discussions, with particular
attention to Harnack inequalities and the regularity of flows on discrete spaces."
acknowledged_ssus:
- _id: M-Shop
- _id: NanoFab
acknowledgement: The author gratefully acknowledges support by the Austrian Science
Fund (FWF), grants No W1245.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Lorenzo
full_name: Portinale, Lorenzo
id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
last_name: Portinale
citation:
ama: Portinale L. Discrete-to-continuum limits of transport problems and gradient
flows in the space of measures. 2021. doi:10.15479/at:ista:10030
apa: Portinale, L. (2021). Discrete-to-continuum limits of transport problems
and gradient flows in the space of measures. Institute of Science and Technology
Austria. https://doi.org/10.15479/at:ista:10030
chicago: Portinale, Lorenzo. “Discrete-to-Continuum Limits of Transport Problems
and Gradient Flows in the Space of Measures.” Institute of Science and Technology
Austria, 2021. https://doi.org/10.15479/at:ista:10030.
ieee: L. Portinale, “Discrete-to-continuum limits of transport problems and gradient
flows in the space of measures,” Institute of Science and Technology Austria,
2021.
ista: Portinale L. 2021. Discrete-to-continuum limits of transport problems and
gradient flows in the space of measures. Institute of Science and Technology Austria.
mla: Portinale, Lorenzo. Discrete-to-Continuum Limits of Transport Problems and
Gradient Flows in the Space of Measures. Institute of Science and Technology
Austria, 2021, doi:10.15479/at:ista:10030.
short: L. Portinale, Discrete-to-Continuum Limits of Transport Problems and Gradient
Flows in the Space of Measures, Institute of Science and Technology Austria, 2021.
date_created: 2021-09-21T09:14:15Z
date_published: 2021-09-22T00:00:00Z
date_updated: 2023-09-07T13:31:06Z
day: '22'
ddc:
- '515'
degree_awarded: PhD
department:
- _id: GradSch
- _id: JaMa
doi: 10.15479/at:ista:10030
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publisher: Institute of Science and Technology Austria
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status: public
supervisor:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
title: Discrete-to-continuum limits of transport problems and gradient flows in the
space of measures
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type: dissertation
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year: '2021'
...
---
_id: '9733'
abstract:
- lang: eng
text: This thesis is the result of the research carried out by the author during
his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich
polaron model, specifically to its regime of strong coupling. This model, which
is rigorously introduced and discussed in the introduction, has been of great
interest in condensed matter physics and field theory for more than eighty years.
It is used to describe an electron interacting with the atoms of a solid material
(the strength of this interaction is modeled by the presence of a coupling constant
α in the Hamiltonian of the system). The particular regime examined here, which
is mathematically described by considering the limit α →∞, displays many interesting
features related to the emergence of classical behavior, which allows for a simplified
effective description of the system under analysis. The properties, the range
of validity and a quantitative analysis of the precision of such classical approximations
are the main object of the present work. We specify our investigation to the study
of the ground state energy of the system, its dynamics and its effective mass.
For each of these problems, we provide in the introduction an overview of the
previously known results and a detailed account of the original contributions
by the author.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Dario
full_name: Feliciangeli, Dario
id: 41A639AA-F248-11E8-B48F-1D18A9856A87
last_name: Feliciangeli
orcid: 0000-0003-0754-8530
citation:
ama: Feliciangeli D. The polaron at strong coupling. 2021. doi:10.15479/at:ista:9733
apa: Feliciangeli, D. (2021). The polaron at strong coupling. Institute of
Science and Technology Austria. https://doi.org/10.15479/at:ista:9733
chicago: Feliciangeli, Dario. “The Polaron at Strong Coupling.” Institute of Science
and Technology Austria, 2021. https://doi.org/10.15479/at:ista:9733.
ieee: D. Feliciangeli, “The polaron at strong coupling,” Institute of Science and
Technology Austria, 2021.
ista: Feliciangeli D. 2021. The polaron at strong coupling. Institute of Science
and Technology Austria.
mla: Feliciangeli, Dario. The Polaron at Strong Coupling. Institute of Science
and Technology Austria, 2021, doi:10.15479/at:ista:9733.
short: D. Feliciangeli, The Polaron at Strong Coupling, Institute of Science and
Technology Austria, 2021.
date_created: 2021-07-27T15:48:30Z
date_published: 2021-08-20T00:00:00Z
date_updated: 2024-03-06T12:30:44Z
day: '20'
ddc:
- '515'
- '519'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
- _id: JaMa
doi: 10.15479/at:ista:9733
ec_funded: 1
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oa: 1
oa_version: Published Version
page: '180'
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '9787'
relation: part_of_dissertation
status: public
- id: '9792'
relation: part_of_dissertation
status: public
- id: '9225'
relation: part_of_dissertation
status: public
- id: '9781'
relation: part_of_dissertation
status: public
- id: '9791'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
title: The polaron at strong coupling
tmp:
image: /image/cc_by_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nd/4.0/legalcode
name: Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)
short: CC BY-ND (4.0)
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2021'
...
---
_id: '7629'
abstract:
- lang: eng
text: "This thesis is based on three main topics: In the first part, we study convergence
of discrete gradient flow structures associated with regular finite-volume discretisations
of Fokker-Planck equations. We show evolutionary I convergence of the discrete
gradient flows to the L2-Wasserstein gradient flow corresponding to the solution
of a Fokker-Planck\r\nequation in arbitrary dimension d >= 1. Along the argument,
we prove Mosco- and I-convergence results for discrete energy functionals, which
are of independent interest for convergence of equivalent gradient flow structures
in Hilbert spaces.\r\nThe second part investigates L2-Wasserstein flows on metric
graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein
distance, which is proved via a regularisation scheme for solutions of the continuity
equation, adapted to the peculiar geometric structure of metric graphs. Based
on those results, we show that the L2-Wasserstein space over a metric graph admits
a gradient flow which may be identified as a solution of a Fokker-Planck equation.\r\nIn
the third part, we focus again on the discrete gradient flows, already encountered
in the first part. We propose a variational structure which extends the gradient
flow structure to Markov chains violating the detailed-balance conditions. Using
this structure, we characterise contraction estimates for the discrete heat flow
in terms of convexity of\r\ncorresponding path-dependent energy functionals. In
addition, we use this approach to derive several functional inequalities for said
functionals."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Dominik L
full_name: Forkert, Dominik L
id: 35C79D68-F248-11E8-B48F-1D18A9856A87
last_name: Forkert
citation:
ama: Forkert DL. Gradient flows in spaces of probability measures for finite-volume
schemes, metric graphs and non-reversible Markov chains. 2020. doi:10.15479/AT:ISTA:7629
apa: Forkert, D. L. (2020). Gradient flows in spaces of probability measures
for finite-volume schemes, metric graphs and non-reversible Markov chains.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7629
chicago: Forkert, Dominik L. “Gradient Flows in Spaces of Probability Measures for
Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains.” Institute
of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7629.
ieee: D. L. Forkert, “Gradient flows in spaces of probability measures for finite-volume
schemes, metric graphs and non-reversible Markov chains,” Institute of Science
and Technology Austria, 2020.
ista: Forkert DL. 2020. Gradient flows in spaces of probability measures for finite-volume
schemes, metric graphs and non-reversible Markov chains. Institute of Science
and Technology Austria.
mla: Forkert, Dominik L. Gradient Flows in Spaces of Probability Measures for
Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains. Institute
of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7629.
short: D.L. Forkert, Gradient Flows in Spaces of Probability Measures for Finite-Volume
Schemes, Metric Graphs and Non-Reversible Markov Chains, Institute of Science
and Technology Austria, 2020.
date_created: 2020-04-02T06:40:23Z
date_published: 2020-03-31T00:00:00Z
date_updated: 2023-09-07T13:03:12Z
day: '31'
ddc:
- '510'
degree_awarded: PhD
department:
- _id: JaMa
doi: 10.15479/AT:ISTA:7629
ec_funded: 1
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month: '03'
oa: 1
oa_version: Published Version
page: '154'
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
status: public
supervisor:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
title: Gradient flows in spaces of probability measures for finite-volume schemes,
metric graphs and non-reversible Markov chains
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2020'
...