---
_id: '71'
abstract:
- lang: eng
  text: "We consider dynamical transport metrics for probability measures on discretisations
    of a bounded convex domain in ℝd. These metrics are natural discrete counterparts
    to the Kantorovich metric \U0001D54E2, defined using a Benamou-Brenier type formula.
    Under mild assumptions we prove an asymptotic upper bound for the discrete transport
    metric Wt in terms of \U0001D54E2, as the size of the mesh T tends to 0. However,
    we show that the corresponding lower bound may fail in general, even on certain
    one-dimensional and symmetric two-dimensional meshes. In addition, we show that
    the asymptotic lower bound holds under an isotropy assumption on the mesh, which
    turns out to be essentially necessary. This assumption is satisfied, e.g., for
    tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence
    of the transport metric."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Peter
  full_name: Gladbach, Peter
  last_name: Gladbach
- first_name: Eva
  full_name: Kopfer, Eva
  last_name: Kopfer
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
citation:
  ama: Gladbach P, Kopfer E, Maas J. Scaling limits of discrete optimal transport.
    <i>SIAM Journal on Mathematical Analysis</i>. 2020;52(3):2759-2802. doi:<a href="https://doi.org/10.1137/19M1243440">10.1137/19M1243440</a>
  apa: Gladbach, P., Kopfer, E., &#38; Maas, J. (2020). Scaling limits of discrete
    optimal transport. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial
    and Applied Mathematics. <a href="https://doi.org/10.1137/19M1243440">https://doi.org/10.1137/19M1243440</a>
  chicago: Gladbach, Peter, Eva Kopfer, and Jan Maas. “Scaling Limits of Discrete
    Optimal Transport.” <i>SIAM Journal on Mathematical Analysis</i>. Society for
    Industrial and Applied Mathematics, 2020. <a href="https://doi.org/10.1137/19M1243440">https://doi.org/10.1137/19M1243440</a>.
  ieee: P. Gladbach, E. Kopfer, and J. Maas, “Scaling limits of discrete optimal transport,”
    <i>SIAM Journal on Mathematical Analysis</i>, vol. 52, no. 3. Society for Industrial
    and Applied Mathematics, pp. 2759–2802, 2020.
  ista: Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport.
    SIAM Journal on Mathematical Analysis. 52(3), 2759–2802.
  mla: Gladbach, Peter, et al. “Scaling Limits of Discrete Optimal Transport.” <i>SIAM
    Journal on Mathematical Analysis</i>, vol. 52, no. 3, Society for Industrial and
    Applied Mathematics, 2020, pp. 2759–802, doi:<a href="https://doi.org/10.1137/19M1243440">10.1137/19M1243440</a>.
  short: P. Gladbach, E. Kopfer, J. Maas, SIAM Journal on Mathematical Analysis 52
    (2020) 2759–2802.
date_created: 2018-12-11T11:44:28Z
date_published: 2020-10-01T00:00:00Z
date_updated: 2023-09-18T08:13:15Z
day: '01'
department:
- _id: JaMa
doi: 10.1137/19M1243440
external_id:
  arxiv:
  - '1809.01092'
  isi:
  - '000546975100017'
intvolume: '        52'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1809.01092
month: '10'
oa: 1
oa_version: Preprint
page: 2759-2802
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  eissn:
  - '10957154'
  issn:
  - '00361410'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
publist_id: '7983'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Scaling limits of discrete optimal transport
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 52
year: '2020'
...
---
_id: '9039'
abstract:
- lang: eng
  text: We give a short and self-contained proof for rates of convergence of the Allen--Cahn
    equation towards mean curvature flow, assuming that a classical (smooth) solution
    to the latter exists and starting from well-prepared initial data. Our approach
    is based on a relative entropy technique. In particular, it does not require a
    stability analysis for the linearized Allen--Cahn operator. As our analysis also
    does not rely on the comparison principle, we expect it to be applicable to more
    complex equations and systems.
acknowledgement: "This work was supported by the European Union's Horizon 2020 Research
  and Innovation\r\nProgramme under Marie Sklodowska-Curie grant agreement 665385
  and by the Deutsche\r\nForschungsgemeinschaft (DFG, German Research Foundation)
  under Germany's Excellence Strategy, EXC-2047/1--390685813."
article_processing_charge: No
article_type: original
author:
- first_name: Julian L
  full_name: Fischer, Julian L
  id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
  last_name: Fischer
  orcid: 0000-0002-0479-558X
- first_name: Tim
  full_name: Laux, Tim
  last_name: Laux
- first_name: Theresa M.
  full_name: Simon, Theresa M.
  last_name: Simon
citation:
  ama: 'Fischer JL, Laux T, Simon TM. Convergence rates of the Allen-Cahn equation
    to mean curvature flow: A short proof based on relative entropies. <i>SIAM Journal
    on Mathematical Analysis</i>. 2020;52(6):6222-6233. doi:<a href="https://doi.org/10.1137/20M1322182">10.1137/20M1322182</a>'
  apa: 'Fischer, J. L., Laux, T., &#38; Simon, T. M. (2020). Convergence rates of
    the Allen-Cahn equation to mean curvature flow: A short proof based on relative
    entropies. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial
    and Applied Mathematics. <a href="https://doi.org/10.1137/20M1322182">https://doi.org/10.1137/20M1322182</a>'
  chicago: 'Fischer, Julian L, Tim Laux, and Theresa M. Simon. “Convergence Rates
    of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative
    Entropies.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial
    and Applied Mathematics, 2020. <a href="https://doi.org/10.1137/20M1322182">https://doi.org/10.1137/20M1322182</a>.'
  ieee: 'J. L. Fischer, T. Laux, and T. M. Simon, “Convergence rates of the Allen-Cahn
    equation to mean curvature flow: A short proof based on relative entropies,” <i>SIAM
    Journal on Mathematical Analysis</i>, vol. 52, no. 6. Society for Industrial and
    Applied Mathematics, pp. 6222–6233, 2020.'
  ista: 'Fischer JL, Laux T, Simon TM. 2020. Convergence rates of the Allen-Cahn equation
    to mean curvature flow: A short proof based on relative entropies. SIAM Journal
    on Mathematical Analysis. 52(6), 6222–6233.'
  mla: 'Fischer, Julian L., et al. “Convergence Rates of the Allen-Cahn Equation to
    Mean Curvature Flow: A Short Proof Based on Relative Entropies.” <i>SIAM Journal
    on Mathematical Analysis</i>, vol. 52, no. 6, Society for Industrial and Applied
    Mathematics, 2020, pp. 6222–33, doi:<a href="https://doi.org/10.1137/20M1322182">10.1137/20M1322182</a>.'
  short: J.L. Fischer, T. Laux, T.M. Simon, SIAM Journal on Mathematical Analysis
    52 (2020) 6222–6233.
date_created: 2021-01-24T23:01:09Z
date_published: 2020-12-15T00:00:00Z
date_updated: 2023-08-24T11:15:16Z
day: '15'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1137/20M1322182
ec_funded: 1
external_id:
  isi:
  - '000600695200027'
file:
- access_level: open_access
  checksum: 21aa1cf4c30a86a00cae15a984819b5d
  content_type: application/pdf
  creator: dernst
  date_created: 2021-01-25T07:48:39Z
  date_updated: 2021-01-25T07:48:39Z
  file_id: '9041'
  file_name: 2020_SIAM_Fischer.pdf
  file_size: 310655
  relation: main_file
  success: 1
file_date_updated: 2021-01-25T07:48:39Z
has_accepted_license: '1'
intvolume: '        52'
isi: 1
issue: '6'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 6222-6233
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  eissn:
  - '10957154'
  issn:
  - '00361410'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Convergence rates of the Allen-Cahn equation to mean curvature flow: A short
  proof based on relative entropies'
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: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 52
year: '2020'
...
---
_id: '1014'
abstract:
- lang: eng
  text: 'We consider the large-scale regularity of solutions to second-order linear
    elliptic equations with random coefficient fields. In contrast to previous works
    on regularity theory for random elliptic operators, our interest is in the regularity
    at the boundary: We consider problems posed on the half-space with homogeneous
    Dirichlet boundary conditions and derive an associated C1,α-type large-scale regularity
    theory in the form of a corresponding decay estimate for the homogenization-adapted
    tilt-excess. This regularity theory entails an associated Liouville-type theorem.
    The results are based on the existence of homogenization correctors adapted to
    the half-space setting, which we construct-by an entirely deterministic argument-as
    a modification of the homogenization corrector on the whole space. This adaption
    procedure is carried out inductively on larger scales, crucially relying on the
    regularity theory already established on smaller scales.'
article_processing_charge: No
author:
- first_name: Julian L
  full_name: Fischer, Julian L
  id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
  last_name: Fischer
  orcid: 0000-0002-0479-558X
- first_name: Claudia
  full_name: Raithel, Claudia
  last_name: Raithel
citation:
  ama: Fischer JL, Raithel C. Liouville principles and a large-scale regularity theory
    for random elliptic operators on the half-space. <i>SIAM Journal on Mathematical
    Analysis</i>. 2017;49(1):82-114. doi:<a href="https://doi.org/10.1137/16M1070384">10.1137/16M1070384</a>
  apa: Fischer, J. L., &#38; Raithel, C. (2017). Liouville principles and a large-scale
    regularity theory for random elliptic operators on the half-space. <i>SIAM Journal
    on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics .
    <a href="https://doi.org/10.1137/16M1070384">https://doi.org/10.1137/16M1070384</a>
  chicago: Fischer, Julian L, and Claudia Raithel. “Liouville Principles and a Large-Scale
    Regularity Theory for Random Elliptic Operators on the Half-Space.” <i>SIAM Journal
    on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics ,
    2017. <a href="https://doi.org/10.1137/16M1070384">https://doi.org/10.1137/16M1070384</a>.
  ieee: J. L. Fischer and C. Raithel, “Liouville principles and a large-scale regularity
    theory for random elliptic operators on the half-space,” <i>SIAM Journal on Mathematical
    Analysis</i>, vol. 49, no. 1. Society for Industrial and Applied Mathematics ,
    pp. 82–114, 2017.
  ista: Fischer JL, Raithel C. 2017. Liouville principles and a large-scale regularity
    theory for random elliptic operators on the half-space. SIAM Journal on Mathematical
    Analysis. 49(1), 82–114.
  mla: Fischer, Julian L., and Claudia Raithel. “Liouville Principles and a Large-Scale
    Regularity Theory for Random Elliptic Operators on the Half-Space.” <i>SIAM Journal
    on Mathematical Analysis</i>, vol. 49, no. 1, Society for Industrial and Applied
    Mathematics , 2017, pp. 82–114, doi:<a href="https://doi.org/10.1137/16M1070384">10.1137/16M1070384</a>.
  short: J.L. Fischer, C. Raithel, SIAM Journal on Mathematical Analysis 49 (2017)
    82–114.
date_created: 2018-12-11T11:49:41Z
date_published: 2017-01-12T00:00:00Z
date_updated: 2023-09-22T09:43:36Z
day: '12'
doi: 10.1137/16M1070384
extern: '1'
external_id:
  isi:
  - '000396681800004'
intvolume: '        49'
isi: 1
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1703.04328
month: '01'
oa: 1
oa_version: Submitted Version
page: 82 - 114
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  issn:
  - '00361410'
publication_status: published
publisher: 'Society for Industrial and Applied Mathematics '
publist_id: '6381'
quality_controlled: '1'
status: public
title: Liouville principles and a large-scale regularity theory for random elliptic
  operators on the half-space
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 49
year: '2017'
...