---
_id: '14703'
abstract:
- lang: eng
  text: We present a discretization of the dynamic optimal transport problem for which
    we can obtain the convergence rate for the value of the transport cost to its
    continuous value when the temporal and spatial stepsize vanish. This convergence
    result does not require any regularity assumption on the measures, though experiments
    suggest that the rate is not sharp. Via an analysis of the duality gap we also
    obtain the convergence rates for the gradient of the optimal potentials and the
    velocity field under mild regularity assumptions. To obtain such rates we discretize
    the dual formulation of the dynamic optimal transport problem and use the mature
    literature related to the error due to discretizing the Hamilton-Jacobi equation.
acknowledgement: "The authors would like to thank Chris Wojtan for his continuous
  support and several interesting discussions. Part of this research was performed
  during two visits: one of SI to the BIDSA research center at Bocconi University,
  and one of HL to the Institute of Science and Technology Austria. Both host institutions
  are warmly acknowledged for the hospital-\r\nity. HL is partially supported by the
  MUR-Prin 2022-202244A7YL “Gradient Flows and Non-Smooth Geometric Structures with
  Applications to Optimization and Machine Learning”, funded by the European Union
  - Next Generation EU. SI is supported in part by ERC Consolidator Grant 101045083
  “CoDiNA” funded by the European Research Council."
article_number: '2312.12213'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
- first_name: Hugo
  full_name: Lavenant, Hugo
  last_name: Lavenant
citation:
  ama: Ishida S, Lavenant H. Quantitative convergence of a discretization of dynamic
    optimal transport using the dual formulation. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2312.12213">10.48550/arXiv.2312.12213</a>
  apa: Ishida, S., &#38; Lavenant, H. (n.d.). Quantitative convergence of a discretization
    of dynamic optimal transport using the dual formulation. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2312.12213">https://doi.org/10.48550/arXiv.2312.12213</a>
  chicago: Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization
    of Dynamic Optimal Transport Using the Dual Formulation.” <i>ArXiv</i>, n.d. <a
    href="https://doi.org/10.48550/arXiv.2312.12213">https://doi.org/10.48550/arXiv.2312.12213</a>.
  ieee: S. Ishida and H. Lavenant, “Quantitative convergence of a discretization of
    dynamic optimal transport using the dual formulation,” <i>arXiv</i>. .
  ista: Ishida S, Lavenant H. Quantitative convergence of a discretization of dynamic
    optimal transport using the dual formulation. arXiv, 2312.12213.
  mla: Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization
    of Dynamic Optimal Transport Using the Dual Formulation.” <i>ArXiv</i>, 2312.12213,
    doi:<a href="https://doi.org/10.48550/arXiv.2312.12213">10.48550/arXiv.2312.12213</a>.
  short: S. Ishida, H. Lavenant, ArXiv (n.d.).
date_created: 2023-12-21T10:14:37Z
date_published: 2023-12-19T00:00:00Z
date_updated: 2023-12-27T13:44:33Z
day: '19'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/arXiv.2312.12213
external_id:
  arxiv:
  - '2312.12213'
keyword:
- Optimal transport
- Hamilton-Jacobi equation
- convex optimization
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2312.12213
month: '12'
oa: 1
oa_version: Preprint
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: arXiv
publication_status: submitted
status: public
title: Quantitative convergence of a discretization of dynamic optimal transport using
  the dual formulation
type: preprint
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '12846'
abstract:
- lang: eng
  text: We present a formula for the signed area of a spherical polygon via prequantization.
    In contrast to the traditional formula based on the Gauss-Bonnet theorem that
    requires measuring angles, the new formula mimics Green's theorem and is applicable
    to a wider range of degenerate spherical curves and polygons.
acknowledgement: The authors acknowledge Chris Wojtan for his continuous support to
  the present work through discussions and advice. The second author thanks Anna Sisak
  for a fruitful discussion on prequantum bundles. This project was funded in part
  by the European Research Council (ERC Consolidator Grant 101045083 CoDiNA).
article_number: '2303.14555'
article_processing_charge: No
arxiv: 1
author:
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
citation:
  ama: Chern A, Ishida S. Area formula for spherical polygons via prequantization.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2303.14555">10.48550/arXiv.2303.14555</a>
  apa: Chern, A., &#38; Ishida, S. (n.d.). Area formula for spherical polygons via
    prequantization. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2303.14555">https://doi.org/10.48550/arXiv.2303.14555</a>
  chicago: Chern, Albert, and Sadashige Ishida. “Area Formula for Spherical Polygons
    via Prequantization.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2303.14555">https://doi.org/10.48550/arXiv.2303.14555</a>.
  ieee: A. Chern and S. Ishida, “Area formula for spherical polygons via prequantization,”
    <i>arXiv</i>. .
  ista: Chern A, Ishida S. Area formula for spherical polygons via prequantization.
    arXiv, 2303.14555.
  mla: Chern, Albert, and Sadashige Ishida. “Area Formula for Spherical Polygons via
    Prequantization.” <i>ArXiv</i>, 2303.14555, doi:<a href="https://doi.org/10.48550/arXiv.2303.14555">10.48550/arXiv.2303.14555</a>.
  short: A. Chern, S. Ishida, ArXiv (n.d.).
date_created: 2023-04-18T19:16:06Z
date_published: 2023-03-25T00:00:00Z
date_updated: 2023-04-25T06:51:21Z
day: '25'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/arXiv.2303.14555
external_id:
  arxiv:
  - '2303.14555'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2303.14555
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: arXiv
publication_status: submitted
status: public
title: Area formula for spherical polygons via prequantization
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '12431'
abstract:
- lang: eng
  text: This paper presents a new representation of curve dynamics, with applications
    to vortex filaments in fluid dynamics. Instead of representing these filaments
    with explicit curve geometry and Lagrangian equations of motion, we represent
    curves implicitly with a new co-dimensional 2 level set description. Our implicit
    representation admits several redundant mathematical degrees of freedom in both
    the configuration and the dynamics of the curves, which can be tailored specifically
    to improve numerical robustness, in contrast to naive approaches for implicit
    curve dynamics that suffer from overwhelming numerical stability problems. Furthermore,
    we note how these hidden degrees of freedom perfectly map to a Clebsch representation
    in fluid dynamics. Motivated by these observations, we introduce untwisted level
    set functions and non-swirling dynamics which successfully regularize sources
    of numerical instability, particularly in the twisting modes around curve filaments.
    A consequence is a novel simulation method which produces stable dynamics for
    large numbers of interacting vortex filaments and effortlessly handles topological
    changes and re-connection events.
acknowledgement: We thank the visual computing group at IST Austria for their valuable
  discussions and feedback. Houdini Education licenses were provided by SideFX software.
  This project was funded in part by the European Research Council (ERC Consolidator
  Grant 101045083 CoDiNA).
article_number: '241'
article_processing_charge: No
article_type: original
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
citation:
  ama: Ishida S, Wojtan C, Chern A. Hidden degrees of freedom in implicit vortex filaments.
    <i>ACM Transactions on Graphics</i>. 2022;41(6). doi:<a href="https://doi.org/10.1145/3550454.3555459">10.1145/3550454.3555459</a>
  apa: Ishida, S., Wojtan, C., &#38; Chern, A. (2022). Hidden degrees of freedom in
    implicit vortex filaments. <i>ACM Transactions on Graphics</i>. Association for
    Computing Machinery. <a href="https://doi.org/10.1145/3550454.3555459">https://doi.org/10.1145/3550454.3555459</a>
  chicago: Ishida, Sadashige, Chris Wojtan, and Albert Chern. “Hidden Degrees of Freedom
    in Implicit Vortex Filaments.” <i>ACM Transactions on Graphics</i>. Association
    for Computing Machinery, 2022. <a href="https://doi.org/10.1145/3550454.3555459">https://doi.org/10.1145/3550454.3555459</a>.
  ieee: S. Ishida, C. Wojtan, and A. Chern, “Hidden degrees of freedom in implicit
    vortex filaments,” <i>ACM Transactions on Graphics</i>, vol. 41, no. 6. Association
    for Computing Machinery, 2022.
  ista: Ishida S, Wojtan C, Chern A. 2022. Hidden degrees of freedom in implicit vortex
    filaments. ACM Transactions on Graphics. 41(6), 241.
  mla: Ishida, Sadashige, et al. “Hidden Degrees of Freedom in Implicit Vortex Filaments.”
    <i>ACM Transactions on Graphics</i>, vol. 41, no. 6, 241, Association for Computing
    Machinery, 2022, doi:<a href="https://doi.org/10.1145/3550454.3555459">10.1145/3550454.3555459</a>.
  short: S. Ishida, C. Wojtan, A. Chern, ACM Transactions on Graphics 41 (2022).
date_created: 2023-01-29T23:00:59Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-04T09:37:23Z
day: '01'
ddc:
- '000'
department:
- _id: ChWo
doi: 10.1145/3550454.3555459
external_id:
  isi:
  - '000891651900061'
file:
- access_level: open_access
  checksum: a2fba257fdefe0e747182be6c0f7c70c
  content_type: application/pdf
  creator: dernst
  date_created: 2023-01-30T07:15:48Z
  date_updated: 2023-01-30T07:15:48Z
  file_id: '12433'
  file_name: 2022_ACM_Ishida.pdf
  file_size: 15551202
  relation: main_file
  success: 1
file_date_updated: 2023-01-30T07:15:48Z
has_accepted_license: '1'
intvolume: '        41'
isi: 1
issue: '6'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
scopus_import: '1'
status: public
title: Hidden degrees of freedom in implicit vortex filaments
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: 41
year: '2022'
...
---
_id: '8384'
abstract:
- lang: eng
  text: Previous research on animations of soap bubbles, films, and foams largely
    focuses on the motion and geometric shape of the bubble surface. These works neglect
    the evolution of the bubble’s thickness, which is normally responsible for visual
    phenomena like surface vortices, Newton’s interference patterns, capillary waves,
    and deformation-dependent rupturing of films in a foam. In this paper, we model
    these natural phenomena by introducing the film thickness as a reduced degree
    of freedom in the Navier-Stokes equations and deriving their equations of motion.
    We discretize the equations on a nonmanifold triangle mesh surface and couple
    it to an existing bubble solver. In doing so, we also introduce an incompressible
    fluid solver for 2.5D films and a novel advection algorithm for convecting fields
    across non-manifold surface junctions. Our simulations enhance state-of-the-art
    bubble solvers with additional effects caused by convection, rippling, draining,
    and evaporation of the thin film.
acknowledged_ssus:
- _id: ScienComp
acknowledgement: "We wish to thank the anonymous reviewers and the members of the
  Visual Computing Group at IST Austria for their valuable feedback, especially Camille
  Schreck for her help in rendering. This research was supported by the Scientific
  Service Units (SSU) of IST Austria through resources provided by Scientific Computing.
  We would like to thank the authors of [Belcour and Barla 2017] for providing their
  implementation, the authors of [Atkins and Elliott 2010] and [Seychelles et al.
  2008] for allowing us to use their results, and Rok Grah for helpful discussions.
  Finally, we thank Ryoichi Ando for many discussions from the beginning of the project
  that resulted in important contents of the paper including our formulation, numerical
  scheme, and initial implementation. This project has received funding from the\r\nEuropean
  Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
  programme under grant agreement No. 638176."
article_number: '31'
article_processing_charge: No
article_type: original
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
- first_name: Peter
  full_name: Synak, Peter
  id: 331776E2-F248-11E8-B48F-1D18A9856A87
  last_name: Synak
- first_name: Fumiya
  full_name: Narita, Fumiya
  last_name: Narita
- first_name: Toshiya
  full_name: Hachisuka, Toshiya
  last_name: Hachisuka
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
citation:
  ama: Ishida S, Synak P, Narita F, Hachisuka T, Wojtan C. A model for soap film dynamics
    with evolving thickness. <i>ACM Transactions on Graphics</i>. 2020;39(4). doi:<a
    href="https://doi.org/10.1145/3386569.3392405">10.1145/3386569.3392405</a>
  apa: Ishida, S., Synak, P., Narita, F., Hachisuka, T., &#38; Wojtan, C. (2020).
    A model for soap film dynamics with evolving thickness. <i>ACM Transactions on
    Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3386569.3392405">https://doi.org/10.1145/3386569.3392405</a>
  chicago: Ishida, Sadashige, Peter Synak, Fumiya Narita, Toshiya Hachisuka, and Chris
    Wojtan. “A Model for Soap Film Dynamics with Evolving Thickness.” <i>ACM Transactions
    on Graphics</i>. Association for Computing Machinery, 2020. <a href="https://doi.org/10.1145/3386569.3392405">https://doi.org/10.1145/3386569.3392405</a>.
  ieee: S. Ishida, P. Synak, F. Narita, T. Hachisuka, and C. Wojtan, “A model for
    soap film dynamics with evolving thickness,” <i>ACM Transactions on Graphics</i>,
    vol. 39, no. 4. Association for Computing Machinery, 2020.
  ista: Ishida S, Synak P, Narita F, Hachisuka T, Wojtan C. 2020. A model for soap
    film dynamics with evolving thickness. ACM Transactions on Graphics. 39(4), 31.
  mla: Ishida, Sadashige, et al. “A Model for Soap Film Dynamics with Evolving Thickness.”
    <i>ACM Transactions on Graphics</i>, vol. 39, no. 4, 31, Association for Computing
    Machinery, 2020, doi:<a href="https://doi.org/10.1145/3386569.3392405">10.1145/3386569.3392405</a>.
  short: S. Ishida, P. Synak, F. Narita, T. Hachisuka, C. Wojtan, ACM Transactions
    on Graphics 39 (2020).
date_created: 2020-09-13T22:01:18Z
date_published: 2020-07-08T00:00:00Z
date_updated: 2024-02-28T12:57:31Z
day: '08'
ddc:
- '000'
department:
- _id: ChWo
doi: 10.1145/3386569.3392405
ec_funded: 1
external_id:
  isi:
  - '000583700300004'
file:
- access_level: open_access
  checksum: 813831ca91319d794d9748c276b24578
  content_type: application/pdf
  creator: dernst
  date_created: 2020-11-23T09:03:19Z
  date_updated: 2020-11-23T09:03:19Z
  file_id: '8795'
  file_name: 2020_soapfilm_submitted.pdf
  file_size: 14935529
  relation: main_file
  success: 1
file_date_updated: 2020-11-23T09:03:19Z
has_accepted_license: '1'
intvolume: '        39'
isi: 1
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1145/3386569.3392405
month: '07'
oa: 1
oa_version: Submitted Version
project:
- _id: 2533E772-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '638176'
  name: Efficient Simulation of Natural Phenomena at Extremely Large Scales
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - '15577368'
  issn:
  - '07300301'
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
scopus_import: '1'
status: public
title: A model for soap film dynamics with evolving thickness
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 39
year: '2020'
...