---
OA_place: publisher
_id: '20563'
abstract:
- lang: eng
  text: "The theory of optimal transport provides an elegant and powerful description
    of many evolution\r\nequations as gradient flows. The primary objective of this
    thesis is to adapt and extend the\r\ntheory to deal with important equations that
    are not covered by the classical framework,\r\nspecifically boundary value problems
    and kinetic equations. Additionally, we establish new\r\nresults in periodic homogenization
    for discrete dynamical optimal transport and in quantization\r\nof measures.\r\nSection
    1.1 serves as an invitation to the classical theory of optimal transport, including
    the\r\nmain definitions and a selection of well-established theorems. Sections
    1.2-1.5 introduce the\r\nmain results of this thesis, outline the motivations,
    and review the current state of the art.\r\nIn Chapter 2, we consider the Fokker–Planck
    equation on a bounded set with positive Dirichlet\r\nboundary conditions. We construct
    a time-discrete scheme involving a modification of the\r\nWasserstein distance
    and, under weak assumptions, prove its convergence to a solution of this\r\nboundary
    value problem. In dimension 1, we show that this solution is a gradient flow in
    a\r\nsuitable space of measures.\r\nChapter 3 presents joint work with Giovanni
    Brigati and Jan Maas. We introduce a new theory\r\nof optimal transport to describe
    and study particle systems at the mesoscopic scale. We prove\r\nadapted versions
    of some fundamental theorems, including the Benamou–Brenier formula and\r\nthe
    identification of absolutely continuous curves of measures.\r\nChapter 4 presents
    joint work with Lorenzo Portinale. We prove convergence of dynamical\r\ntransportation
    functionals on periodic graphs in the large-scale limit when the cost functional\r\nis
    asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances
    converge\r\nto 1-Wasserstein distances constructed from crystalline norms on R\r\nd\r\n.\r\nChapter
    5 concerns optimal empirical quantization: the problem of approximating a measure\r\nby
    the sum of n equally weighted Dirac deltas, so as to minimize the error in the
    p-Wasserstein\r\ndistance. Our main result is an analog of Zador’s theorem, providing
    asymptotic bounds for\r\nthe minimal error as n tends to infinity.\r\n"
acknowledgement: "The research contained in this thesis has received funding from
  the Austrian Science\r\nFund (FWF) project 10.55776/F65."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Filippo
  full_name: Quattrocchi, Filippo
  id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308
  last_name: Quattrocchi
  orcid: 0009-0000-9773-1931
citation:
  ama: Quattrocchi F. Optimal transport methods for kinetic equations, boundary value
    problems, and discretization of measures. 2025. doi:<a href="https://doi.org/10.15479/AT-ISTA-20563">10.15479/AT-ISTA-20563</a>
  apa: Quattrocchi, F. (2025). <i>Optimal transport methods for kinetic equations,
    boundary value problems, and discretization of measures</i>. Institute of Science
    and Technology Austria. <a href="https://doi.org/10.15479/AT-ISTA-20563">https://doi.org/10.15479/AT-ISTA-20563</a>
  chicago: Quattrocchi, Filippo. “Optimal Transport Methods for Kinetic Equations,
    Boundary Value Problems, and Discretization of Measures.” Institute of Science
    and Technology Austria, 2025. <a href="https://doi.org/10.15479/AT-ISTA-20563">https://doi.org/10.15479/AT-ISTA-20563</a>.
  ieee: F. Quattrocchi, “Optimal transport methods for kinetic equations, boundary
    value problems, and discretization of measures,” Institute of Science and Technology
    Austria, 2025.
  ista: Quattrocchi F. 2025. Optimal transport methods for kinetic equations, boundary
    value problems, and discretization of measures. Institute of Science and Technology
    Austria.
  mla: Quattrocchi, Filippo. <i>Optimal Transport Methods for Kinetic Equations, Boundary
    Value Problems, and Discretization of Measures</i>. Institute of Science and Technology
    Austria, 2025, doi:<a href="https://doi.org/10.15479/AT-ISTA-20563">10.15479/AT-ISTA-20563</a>.
  short: F. Quattrocchi, Optimal Transport Methods for Kinetic Equations, Boundary
    Value Problems, and Discretization of Measures, Institute of Science and Technology
    Austria, 2025.
corr_author: '1'
date_created: 2025-10-28T13:10:49Z
date_published: 2025-11-03T00:00:00Z
date_updated: 2026-06-18T18:02:13Z
day: '03'
ddc:
- '515'
- '519'
degree_awarded: PhD
department:
- _id: GradSch
- _id: JaMa
doi: 10.15479/AT-ISTA-20563
file:
- access_level: open_access
  checksum: 6f55275bdf99992be3a6457d949dd664
  content_type: application/pdf
  creator: fquattro
  date_created: 2025-11-17T21:04:15Z
  date_updated: 2026-01-01T23:30:03Z
  embargo: 2026-01-01
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  file_name: 2025_quattrocchi_filippo_thesis.pdf
  file_size: 4326411
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  date_created: 2025-11-17T21:05:43Z
  date_updated: 2026-01-01T23:30:03Z
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  file_size: 11726509
  relation: source_file
file_date_updated: 2026-01-01T23:30:03Z
has_accepted_license: '1'
keyword:
- optimal transport
- kinetic equations
- boundary value problems
- quantization
- gradient flows
- homogenization
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '11'
oa: 1
oa_version: Published Version
page: '240'
project:
- _id: 260482E2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: F06504
  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: '20569'
    relation: part_of_dissertation
    status: public
  - id: '20571'
    relation: part_of_dissertation
    status: public
  - id: '20570'
    relation: part_of_dissertation
    status: public
  - id: '18706'
    relation: part_of_dissertation
    status: public
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: Optimal transport methods for kinetic equations, boundary value problems, and
  discretization of measures
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20570'
abstract:
- lang: eng
  text: "We investigate the minimal error in approximating a general probability\r\nmeasure
    $\\mu$ on $\\mathbb{R}^d$ by the uniform measure on a finite set with\r\nprescribed
    cardinality $n$. The error is measured in the $p$-Wasserstein\r\ndistance. In
    particular, when $1\\le p<d$, we establish asymptotic upper and\r\nlower bounds
    as $n \\to \\infty$ on the rescaled minimal error that have the\r\nsame, explicit
    dependency on $\\mu$.\r\n  In some instances, we prove that the rescaled minimal
    error has a limit.\r\nThese include general measures in dimension $d = 2$ with
    $1 \\le p < 2$, and\r\nuniform measures in arbitrary dimension with $1 \\le p
    < d$. For some uniform\r\nmeasures, we prove the limit existence for $p \\ge d$
    as well.\r\n  For a class of compactly supported measures with H\\\"older densities,
    we\r\ndetermine the convergence speed of the minimal error for every $p \\ge 1$.\r\n
    \ Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper\r\nestimate
    of the minimal error when $1 \\le p < d$.\r\n  In the initial sections, we survey
    the state of the art and draw connections\r\nwith similar problems, such as classical
    and random quantization."
acknowledgement: "The author is thankful to Nicolas Clozeau, Lorenzo Dello Schiavo,
  Jan Maas, Dejan Slepčev,\r\nand Dario Trevisan for many fruitful discussions and
  comments. The author gratefully acknowledges support from the Austrian Science Fund
  (FWF) project 10.55776/F65."
article_number: '2408.12924'
article_processing_charge: No
arxiv: 1
author:
- first_name: Filippo
  full_name: Quattrocchi, Filippo
  id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308
  last_name: Quattrocchi
  orcid: 0009-0000-9773-1931
citation:
  ama: Quattrocchi F. Asymptotics for optimal empirical quantization of measures.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2408.12924">10.48550/arXiv.2408.12924</a>
  apa: Quattrocchi, F. (n.d.). Asymptotics for optimal empirical quantization of measures.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2408.12924">https://doi.org/10.48550/arXiv.2408.12924</a>
  chicago: Quattrocchi, Filippo. “Asymptotics for Optimal Empirical Quantization of
    Measures.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2408.12924">https://doi.org/10.48550/arXiv.2408.12924</a>.
  ieee: F. Quattrocchi, “Asymptotics for optimal empirical quantization of measures,”
    <i>arXiv</i>. .
  ista: Quattrocchi F. Asymptotics for optimal empirical quantization of measures.
    arXiv, 2408.12924.
  mla: Quattrocchi, Filippo. “Asymptotics for Optimal Empirical Quantization of Measures.”
    <i>ArXiv</i>, 2408.12924, doi:<a href="https://doi.org/10.48550/arXiv.2408.12924">10.48550/arXiv.2408.12924</a>.
  short: F. Quattrocchi, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-10-28T13:12:22Z
date_published: 2024-08-23T00:00:00Z
date_updated: 2026-07-07T22:30:39Z
day: '23'
department:
- _id: GradSch
- _id: JaMa
doi: 10.48550/arXiv.2408.12924
external_id:
  arxiv:
  - '2408.12924'
keyword:
- optimal empirical quantization
- vector quantization
- Wasserstein distance
- semidiscrete optimal transport
- Zador’s Theorem
- Pierce’s Lemma
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2408.12924
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: 260482E2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: F06504
  name: Taming Complexity in Partial Differential Systems
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20563'
    relation: dissertation_contains
    status: public
status: public
title: Asymptotics for optimal empirical quantization of measures
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...