---
_id: '12959'
abstract:
- lang: eng
text: "This paper deals with the large-scale behaviour of dynamical optimal transport
on Zd\r\n-periodic graphs with general lower semicontinuous and convex energy
densities. Our main contribution is a homogenisation result that describes the
effective behaviour of the discrete problems in terms of a continuous optimal
transport problem. The effective energy density can be explicitly expressed in
terms of a cell formula, which is a finite-dimensional convex programming problem
that depends non-trivially on the local geometry of the discrete graph and the
discrete energy density. Our homogenisation result is derived from a Γ\r\n-convergence
result for action functionals on curves of measures, which we prove under very
mild growth conditions on the energy density. We investigate the cell formula
in several cases of interest, including finite-volume discretisations of the Wasserstein
distance, where non-trivial limiting behaviour occurs."
acknowledgement: J.M. gratefully acknowledges support by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme
(Grant Agreement No. 716117). J.M and L.P. also acknowledge support from the Austrian
Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support
by the German Research Foundation through the Hausdorff Center for Mathematics and
the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation)—350398276. We thank the
anonymous reviewer for the careful reading and for useful suggestions. Open access
funding provided by Austrian Science Fund (FWF).
article_number: '143'
article_processing_charge: Yes (via OA deal)
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
- first_name: Lorenzo
full_name: Portinale, Lorenzo
id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
last_name: Portinale
citation:
ama: Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of dynamical optimal
transport on periodic graphs. Calculus of Variations and Partial Differential
Equations. 2023;62(5). doi:10.1007/s00526-023-02472-z
apa: Gladbach, P., Kopfer, E., Maas, J., & Portinale, L. (2023). Homogenisation
of dynamical optimal transport on periodic graphs. Calculus of Variations and
Partial Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-023-02472-z
chicago: Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation
of Dynamical Optimal Transport on Periodic Graphs.” Calculus of Variations
and Partial Differential Equations. Springer Nature, 2023. https://doi.org/10.1007/s00526-023-02472-z.
ieee: P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of dynamical
optimal transport on periodic graphs,” Calculus of Variations and Partial Differential
Equations, vol. 62, no. 5. Springer Nature, 2023.
ista: Gladbach P, Kopfer E, Maas J, Portinale L. 2023. Homogenisation of dynamical
optimal transport on periodic graphs. Calculus of Variations and Partial Differential
Equations. 62(5), 143.
mla: Gladbach, Peter, et al. “Homogenisation of Dynamical Optimal Transport on Periodic
Graphs.” Calculus of Variations and Partial Differential Equations, vol.
62, no. 5, 143, Springer Nature, 2023, doi:10.1007/s00526-023-02472-z.
short: P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Calculus of Variations and
Partial Differential Equations 62 (2023).
date_created: 2023-05-14T22:01:00Z
date_published: 2023-04-28T00:00:00Z
date_updated: 2023-10-04T11:34:49Z
day: '28'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-023-02472-z
ec_funded: 1
external_id:
arxiv:
- '2110.15321'
isi:
- '000980588900001'
file:
- access_level: open_access
checksum: 359bee38d94b7e0aa73925063cb8884d
content_type: application/pdf
creator: dernst
date_created: 2023-10-04T11:34:10Z
date_updated: 2023-10-04T11:34:10Z
file_id: '14393'
file_name: 2023_CalculusEquations_Gladbach.pdf
file_size: 1240995
relation: main_file
success: 1
file_date_updated: 2023-10-04T11:34:10Z
has_accepted_license: '1'
intvolume: ' 62'
isi: 1
issue: '5'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
name: Dissipation and Dispersion in Nonlinear Partial Differential Equations
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
eissn:
- 1432-0835
issn:
- 0944-2669
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Homogenisation of dynamical optimal transport on periodic graphs
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 62
year: '2023'
...
---
_id: '11700'
abstract:
- lang: eng
text: This paper contains two contributions in the study of optimal transport on
metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein
distance, which establishes the equivalence of static and dynamical optimal transport.
Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov
equations can be formulated as gradient flow of the free energy in the Wasserstein
space of probability measures. The proofs of these results are based on careful
regularisation arguments to circumvent some of the difficulties arising in metric
graphs, namely, branching of geodesics and the failure of semi-convexity of entropy
functionals in the Wasserstein space.
acknowledgement: "ME acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG),
Grant SFB 1283/2 2021 – 317210226. DF and JM were supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 716117). JM also acknowledges support by the Austrian Science
Fund (FWF), Project SFB F65. The work of DM was partially supported by the Deutsche
Forschungsgemeinschaft\r\n(DFG), Grant 397230547. This article is based upon work
from COST Action\r\n18232 MAT-DYN-NET, supported by COST (European Cooperation in
Science\r\nand Technology), www.cost.eu. We wish to thank Martin Burger and Jan-Frederik\r\nPietschmann
for useful discussions. We are grateful to the anonymous referees for\r\ntheir careful
reading and useful suggestions."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matthias
full_name: Erbar, Matthias
last_name: Erbar
- first_name: Dominik L
full_name: Forkert, Dominik L
id: 35C79D68-F248-11E8-B48F-1D18A9856A87
last_name: Forkert
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Delio
full_name: Mugnolo, Delio
last_name: Mugnolo
citation:
ama: Erbar M, Forkert DL, Maas J, Mugnolo D. Gradient flow formulation of diffusion
equations in the Wasserstein space over a metric graph. Networks and Heterogeneous
Media. 2022;17(5):687-717. doi:10.3934/nhm.2022023
apa: Erbar, M., Forkert, D. L., Maas, J., & Mugnolo, D. (2022). Gradient flow
formulation of diffusion equations in the Wasserstein space over a metric graph.
Networks and Heterogeneous Media. American Institute of Mathematical Sciences.
https://doi.org/10.3934/nhm.2022023
chicago: Erbar, Matthias, Dominik L Forkert, Jan Maas, and Delio Mugnolo. “Gradient
Flow Formulation of Diffusion Equations in the Wasserstein Space over a Metric
Graph.” Networks and Heterogeneous Media. American Institute of Mathematical
Sciences, 2022. https://doi.org/10.3934/nhm.2022023.
ieee: M. Erbar, D. L. Forkert, J. Maas, and D. Mugnolo, “Gradient flow formulation
of diffusion equations in the Wasserstein space over a metric graph,” Networks
and Heterogeneous Media, vol. 17, no. 5. American Institute of Mathematical
Sciences, pp. 687–717, 2022.
ista: Erbar M, Forkert DL, Maas J, Mugnolo D. 2022. Gradient flow formulation of
diffusion equations in the Wasserstein space over a metric graph. Networks and
Heterogeneous Media. 17(5), 687–717.
mla: Erbar, Matthias, et al. “Gradient Flow Formulation of Diffusion Equations in
the Wasserstein Space over a Metric Graph.” Networks and Heterogeneous Media,
vol. 17, no. 5, American Institute of Mathematical Sciences, 2022, pp. 687–717,
doi:10.3934/nhm.2022023.
short: M. Erbar, D.L. Forkert, J. Maas, D. Mugnolo, Networks and Heterogeneous Media
17 (2022) 687–717.
date_created: 2022-07-31T22:01:46Z
date_published: 2022-10-01T00:00:00Z
date_updated: 2023-08-03T12:25:49Z
day: '01'
department:
- _id: JaMa
doi: 10.3934/nhm.2022023
ec_funded: 1
external_id:
arxiv:
- '2105.05677'
isi:
- '000812422100001'
intvolume: ' 17'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2105.05677
month: '10'
oa: 1
oa_version: Preprint
page: 687-717
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication: Networks and Heterogeneous Media
publication_identifier:
eissn:
- 1556-181X
issn:
- 1556-1801
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: Gradient flow formulation of diffusion equations in the Wasserstein space over
a metric graph
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 17
year: '2022'
...
---
_id: '11739'
abstract:
- lang: eng
text: We consider finite-volume approximations of Fokker--Planck equations on bounded
convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures.
We reprove the convergence of the discrete to continuous Fokker--Planck equation
via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit
at the level of the gradient flow structures, generalizing the one-dimensional
result obtained by Disser and Liero. The proof is of variational nature and relies
on a Mosco convergence result for functionals in the discrete-to-continuum limit
that is of independent interest. Our results apply to arbitrary regular meshes,
even though the associated discrete transport distances may fail to converge to
the Wasserstein distance in this generality.
acknowledgement: This work was supported by the European Research Council (ERC) under
the European Union's Horizon 2020 Research and Innovation Programme grant 716117
and by the AustrianScience Fund (FWF) through grants F65 and W1245.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Dominik L
full_name: Forkert, Dominik L
id: 35C79D68-F248-11E8-B48F-1D18A9856A87
last_name: Forkert
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Lorenzo
full_name: Portinale, Lorenzo
id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
last_name: Portinale
citation:
ama: Forkert DL, Maas J, Portinale L. Evolutionary $\Gamma$-convergence of entropic
gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM
Journal on Mathematical Analysis. 2022;54(4):4297-4333. doi:10.1137/21M1410968
apa: Forkert, D. L., Maas, J., & Portinale, L. (2022). Evolutionary $\Gamma$-convergence
of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
SIAM Journal on Mathematical Analysis. Society for Industrial and Applied
Mathematics. https://doi.org/10.1137/21M1410968
chicago: Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\Gamma$-Convergence
of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
SIAM Journal on Mathematical Analysis. Society for Industrial and Applied
Mathematics, 2022. https://doi.org/10.1137/21M1410968.
ieee: D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\Gamma$-convergence
of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,”
SIAM Journal on Mathematical Analysis, vol. 54, no. 4. Society for Industrial
and Applied Mathematics, pp. 4297–4333, 2022.
ista: Forkert DL, Maas J, Portinale L. 2022. Evolutionary $\Gamma$-convergence of
entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
SIAM Journal on Mathematical Analysis. 54(4), 4297–4333.
mla: Forkert, Dominik L., et al. “Evolutionary $\Gamma$-Convergence of Entropic
Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
SIAM Journal on Mathematical Analysis, vol. 54, no. 4, Society for Industrial
and Applied Mathematics, 2022, pp. 4297–333, doi:10.1137/21M1410968.
short: D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis
54 (2022) 4297–4333.
date_created: 2022-08-07T22:01:59Z
date_published: 2022-07-18T00:00:00Z
date_updated: 2023-08-03T12:37:21Z
day: '18'
department:
- _id: JaMa
doi: 10.1137/21M1410968
ec_funded: 1
external_id:
arxiv:
- '2008.10962'
isi:
- '000889274600001'
intvolume: ' 54'
isi: 1
issue: '4'
keyword:
- Fokker--Planck equation
- gradient flow
- evolutionary $\Gamma$-convergence
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.2008.10962'
month: '07'
oa: 1
oa_version: Preprint
page: 4297-4333
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
name: Dissipation and Dispersion in Nonlinear Partial Differential Equations
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
eissn:
- 1095-7154
issn:
- 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '10022'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Evolutionary $\Gamma$-convergence of entropic gradient flow structures for
Fokker-Planck equations in multiple dimensions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...
---
_id: '10023'
abstract:
- lang: eng
text: We study the temporal dissipation of variance and relative entropy for ergodic
Markov Chains in continuous time, and compute explicitly the corresponding dissipation
rates. These are identified, as is well known, in the case of the variance in
terms of an appropriate Hilbertian norm; and in the case of the relative entropy,
in terms of a Dirichlet form which morphs into a version of the familiar Fisher
information under conditions of detailed balance. Here we obtain trajectorial
versions of these results, valid along almost every path of the random motion
and most transparent in the backwards direction of time. Martingale arguments
and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer
and Tschiderer for conservative diffusions. Extensions are developed to general
“convex divergences” and to countable state-spaces. The steepest descent and gradient
flow properties for the variance, the relative entropy, and appropriate generalizations,
are studied along with their respective geometries under conditions of detailed
balance, leading to a very direct proof for the HWI inequality of Otto and Villani
in the present context.
acknowledgement: I.K. acknowledges support from the U.S. National Science Foundation
under Grant NSF-DMS-20-04997. J.M. acknowledges support from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 716117) and from the Austrian Science Fund (FWF) through project
F65. W.S. acknowledges support from the Austrian Science Fund (FWF) under grant
P28861 and by the Vienna Science and Technology Fund (WWTF) through projects MA14-008
and MA16-021.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Ioannis
full_name: Karatzas, Ioannis
last_name: Karatzas
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Walter
full_name: Schachermayer, Walter
last_name: Schachermayer
citation:
ama: Karatzas I, Maas J, Schachermayer W. Trajectorial dissipation and gradient
flow for the relative entropy in Markov chains. Communications in Information
and Systems. 2021;21(4):481-536. doi:10.4310/CIS.2021.v21.n4.a1
apa: Karatzas, I., Maas, J., & Schachermayer, W. (2021). Trajectorial dissipation
and gradient flow for the relative entropy in Markov chains. Communications
in Information and Systems. International Press. https://doi.org/10.4310/CIS.2021.v21.n4.a1
chicago: Karatzas, Ioannis, Jan Maas, and Walter Schachermayer. “Trajectorial Dissipation
and Gradient Flow for the Relative Entropy in Markov Chains.” Communications
in Information and Systems. International Press, 2021. https://doi.org/10.4310/CIS.2021.v21.n4.a1.
ieee: I. Karatzas, J. Maas, and W. Schachermayer, “Trajectorial dissipation and
gradient flow for the relative entropy in Markov chains,” Communications in
Information and Systems, vol. 21, no. 4. International Press, pp. 481–536,
2021.
ista: Karatzas I, Maas J, Schachermayer W. 2021. Trajectorial dissipation and gradient
flow for the relative entropy in Markov chains. Communications in Information
and Systems. 21(4), 481–536.
mla: Karatzas, Ioannis, et al. “Trajectorial Dissipation and Gradient Flow for the
Relative Entropy in Markov Chains.” Communications in Information and Systems,
vol. 21, no. 4, International Press, 2021, pp. 481–536, doi:10.4310/CIS.2021.v21.n4.a1.
short: I. Karatzas, J. Maas, W. Schachermayer, Communications in Information and
Systems 21 (2021) 481–536.
date_created: 2021-09-19T08:53:19Z
date_published: 2021-06-04T00:00:00Z
date_updated: 2021-09-20T12:51:18Z
day: '04'
department:
- _id: JaMa
doi: 10.4310/CIS.2021.v21.n4.a1
ec_funded: 1
external_id:
arxiv:
- '2005.14177'
intvolume: ' 21'
issue: '4'
keyword:
- Markov Chain
- relative entropy
- time reversal
- steepest descent
- gradient flow
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2005.14177
month: '06'
oa: 1
oa_version: Preprint
page: 481-536
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication: Communications in Information and Systems
publication_identifier:
issn:
- 1526-7555
publication_status: published
publisher: International Press
quality_controlled: '1'
status: public
title: Trajectorial dissipation and gradient flow for the relative entropy in Markov
chains
type: journal_article
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 21
year: '2021'
...
---
_id: '8758'
abstract:
- lang: eng
text: We consider various modeling levels for spatially homogeneous chemical reaction
systems, namely the chemical master equation, the chemical Langevin dynamics,
and the reaction-rate equation. Throughout we restrict our study to the case where
the microscopic system satisfies the detailed-balance condition. The latter allows
us to enrich the systems with a gradient structure, i.e. the evolution is given
by a gradient-flow equation. We present the arising links between the associated
gradient structures that are driven by the relative entropy of the detailed-balance
steady state. The limit of large volumes is studied in the sense of evolutionary
Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive
hybrid models for coupling different modeling levels.
acknowledgement: The research of A.M. was partially supported by the Deutsche Forschungsgemeinschaft
(DFG) via the Collaborative Research Center SFB 1114 Scaling Cascades in Complex
Systems (Project No. 235221301), through the Subproject C05 Effective models for
materials and interfaces with multiple scales. J.M. gratefully acknowledges support
by the European Research Council (ERC) under the European Union’s Horizon 2020 research
and innovation programme (Grant Agreement No. 716117), and by the Austrian Science
Fund (FWF), Project SFB F65. The authors thank Christof Schütte, Robert I. A. Patterson,
and Stefanie Winkelmann for helpful and stimulating discussions. Open access funding
provided by Austrian Science Fund (FWF).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Alexander
full_name: Mielke, Alexander
last_name: Mielke
citation:
ama: Maas J, Mielke A. Modeling of chemical reaction systems with detailed balance
using gradient structures. Journal of Statistical Physics. 2020;181(6):2257-2303.
doi:10.1007/s10955-020-02663-4
apa: Maas, J., & Mielke, A. (2020). Modeling of chemical reaction systems with
detailed balance using gradient structures. Journal of Statistical Physics.
Springer Nature. https://doi.org/10.1007/s10955-020-02663-4
chicago: Maas, Jan, and Alexander Mielke. “Modeling of Chemical Reaction Systems
with Detailed Balance Using Gradient Structures.” Journal of Statistical Physics.
Springer Nature, 2020. https://doi.org/10.1007/s10955-020-02663-4.
ieee: J. Maas and A. Mielke, “Modeling of chemical reaction systems with detailed
balance using gradient structures,” Journal of Statistical Physics, vol.
181, no. 6. Springer Nature, pp. 2257–2303, 2020.
ista: Maas J, Mielke A. 2020. Modeling of chemical reaction systems with detailed
balance using gradient structures. Journal of Statistical Physics. 181(6), 2257–2303.
mla: Maas, Jan, and Alexander Mielke. “Modeling of Chemical Reaction Systems with
Detailed Balance Using Gradient Structures.” Journal of Statistical Physics,
vol. 181, no. 6, Springer Nature, 2020, pp. 2257–303, doi:10.1007/s10955-020-02663-4.
short: J. Maas, A. Mielke, Journal of Statistical Physics 181 (2020) 2257–2303.
date_created: 2020-11-15T23:01:18Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2023-08-22T13:24:27Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s10955-020-02663-4
ec_funded: 1
external_id:
arxiv:
- '2004.02831'
isi:
- '000587107200002'
file:
- access_level: open_access
checksum: bc2b63a90197b97cbc73eccada4639f5
content_type: application/pdf
creator: dernst
date_created: 2021-02-04T10:29:11Z
date_updated: 2021-02-04T10:29:11Z
file_id: '9087'
file_name: 2020_JourStatPhysics_Maas.pdf
file_size: 753596
relation: main_file
success: 1
file_date_updated: 2021-02-04T10:29:11Z
has_accepted_license: '1'
intvolume: ' 181'
isi: 1
issue: '6'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 2257-2303
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: 260482E2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: ' F06504'
name: Taming Complexity in Partial Di erential Systems
publication: Journal of Statistical Physics
publication_identifier:
eissn:
- '15729613'
issn:
- '00224715'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Modeling of chemical reaction systems with detailed balance using gradient
structures
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: 181
year: '2020'
...
---
_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.
SIAM Journal on Mathematical Analysis. 2020;52(3):2759-2802. doi:10.1137/19M1243440
apa: Gladbach, P., Kopfer, E., & Maas, J. (2020). Scaling limits of discrete
optimal transport. SIAM Journal on Mathematical Analysis. Society for Industrial
and Applied Mathematics. https://doi.org/10.1137/19M1243440
chicago: Gladbach, Peter, Eva Kopfer, and Jan Maas. “Scaling Limits of Discrete
Optimal Transport.” SIAM Journal on Mathematical Analysis. Society for
Industrial and Applied Mathematics, 2020. https://doi.org/10.1137/19M1243440.
ieee: P. Gladbach, E. Kopfer, and J. Maas, “Scaling limits of discrete optimal transport,”
SIAM Journal on Mathematical Analysis, 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.” SIAM
Journal on Mathematical Analysis, vol. 52, no. 3, Society for Industrial and
Applied Mathematics, 2020, pp. 2759–802, doi:10.1137/19M1243440.
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: '7573'
abstract:
- lang: eng
text: This paper deals with dynamical optimal transport metrics defined by spatial
discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such
metrics appear naturally in discretisations of -gradient flow formulations for
dissipative PDE. However, it has recently been shown that these metrics do not
in general converge to , unless strong geometric constraints are imposed on the
discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting,
discrete transport metrics converge to a limiting transport metric with a non-trivial
effective mobility. This mobility depends sensitively on the geometry of the mesh
and on the non-local mobility at the discrete level. Our result quantifies to
what extent discrete transport can make use of microstructure in the mesh to reduce
the cost of transport.
acknowledgement: J.M. gratefully acknowledges support by the European Research Council
(ERC) under the European Union's Horizon 2020 research and innovation programme
(grant agreement No 716117). J.M. and L.P. also acknowledge support from the Austrian
Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support
by the German Research Foundation through the Hausdorff Center for Mathematics and
the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – 350398276.
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
- first_name: Lorenzo
full_name: Portinale, Lorenzo
id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
last_name: Portinale
citation:
ama: Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of one-dimensional
discrete optimal transport. Journal de Mathematiques Pures et Appliquees.
2020;139(7):204-234. doi:10.1016/j.matpur.2020.02.008
apa: Gladbach, P., Kopfer, E., Maas, J., & Portinale, L. (2020). Homogenisation
of one-dimensional discrete optimal transport. Journal de Mathematiques Pures
et Appliquees. Elsevier. https://doi.org/10.1016/j.matpur.2020.02.008
chicago: Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation
of One-Dimensional Discrete Optimal Transport.” Journal de Mathematiques Pures
et Appliquees. Elsevier, 2020. https://doi.org/10.1016/j.matpur.2020.02.008.
ieee: P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of one-dimensional
discrete optimal transport,” Journal de Mathematiques Pures et Appliquees,
vol. 139, no. 7. Elsevier, pp. 204–234, 2020.
ista: Gladbach P, Kopfer E, Maas J, Portinale L. 2020. Homogenisation of one-dimensional
discrete optimal transport. Journal de Mathematiques Pures et Appliquees. 139(7),
204–234.
mla: Gladbach, Peter, et al. “Homogenisation of One-Dimensional Discrete Optimal
Transport.” Journal de Mathematiques Pures et Appliquees, vol. 139, no.
7, Elsevier, 2020, pp. 204–34, doi:10.1016/j.matpur.2020.02.008.
short: P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Journal de Mathematiques Pures
et Appliquees 139 (2020) 204–234.
date_created: 2020-03-08T23:00:47Z
date_published: 2020-07-01T00:00:00Z
date_updated: 2023-09-07T13:31:05Z
day: '01'
department:
- _id: JaMa
doi: 10.1016/j.matpur.2020.02.008
ec_funded: 1
external_id:
arxiv:
- '1905.05757'
isi:
- '000539439400008'
intvolume: ' 139'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1905.05757
month: '07'
oa: 1
oa_version: Preprint
page: 204-234
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: 260482E2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: ' F06504'
name: Taming Complexity in Partial Di erential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
name: Dissipation and Dispersion in Nonlinear Partial Differential Equations
publication: Journal de Mathematiques Pures et Appliquees
publication_identifier:
issn:
- '00217824'
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '10030'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Homogenisation of one-dimensional discrete optimal transport
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 139
year: '2020'
...
---
_id: '6358'
abstract:
- lang: eng
text: We study dynamical optimal transport metrics between density matricesassociated
to symmetric Dirichlet forms on finite-dimensional C∗-algebras. Our settingcovers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative
differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, andspectral
gap estimates.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Eric A.
full_name: Carlen, Eric A.
last_name: Carlen
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: Carlen EA, Maas J. Non-commutative calculus, optimal transport and functional
inequalities in dissipative quantum systems. Journal of Statistical Physics.
2020;178(2):319-378. doi:10.1007/s10955-019-02434-w
apa: Carlen, E. A., & Maas, J. (2020). Non-commutative calculus, optimal transport
and functional inequalities in dissipative quantum systems. Journal of Statistical
Physics. Springer Nature. https://doi.org/10.1007/s10955-019-02434-w
chicago: Carlen, Eric A., and Jan Maas. “Non-Commutative Calculus, Optimal Transport
and Functional Inequalities in Dissipative Quantum Systems.” Journal of Statistical
Physics. Springer Nature, 2020. https://doi.org/10.1007/s10955-019-02434-w.
ieee: E. A. Carlen and J. Maas, “Non-commutative calculus, optimal transport and
functional inequalities in dissipative quantum systems,” Journal of Statistical
Physics, vol. 178, no. 2. Springer Nature, pp. 319–378, 2020.
ista: Carlen EA, Maas J. 2020. Non-commutative calculus, optimal transport and functional
inequalities in dissipative quantum systems. Journal of Statistical Physics.
178(2), 319–378.
mla: Carlen, Eric A., and Jan Maas. “Non-Commutative Calculus, Optimal Transport
and Functional Inequalities in Dissipative Quantum Systems.” Journal of Statistical
Physics, vol. 178, no. 2, Springer Nature, 2020, pp. 319–78, doi:10.1007/s10955-019-02434-w.
short: E.A. Carlen, J. Maas, Journal of Statistical Physics 178 (2020) 319–378.
date_created: 2019-04-30T07:34:18Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2023-08-17T13:49:40Z
day: '01'
ddc:
- '500'
department:
- _id: JaMa
doi: 10.1007/s10955-019-02434-w
ec_funded: 1
external_id:
arxiv:
- '1811.04572'
isi:
- '000498933300001'
file:
- access_level: open_access
checksum: 7b04befbdc0d4982c0ee945d25d19872
content_type: application/pdf
creator: dernst
date_created: 2019-12-23T12:03:09Z
date_updated: 2020-07-14T12:47:28Z
file_id: '7209'
file_name: 2019_JourStatistPhysics_Carlen.pdf
file_size: 905538
relation: main_file
file_date_updated: 2020-07-14T12:47:28Z
has_accepted_license: '1'
intvolume: ' 178'
isi: 1
issue: '2'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 319-378
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: 260482E2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: ' F06504'
name: Taming Complexity in Partial Di erential Systems
publication: Journal of Statistical Physics
publication_identifier:
eissn:
- '15729613'
issn:
- '00224715'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
link:
- relation: erratum
url: https://doi.org/10.1007/s10955-020-02671-4
scopus_import: '1'
status: public
title: Non-commutative calculus, optimal transport and functional inequalities in
dissipative quantum systems
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: 178
year: '2020'
...
---
_id: '10022'
abstract:
- lang: eng
text: We consider finite-volume approximations of Fokker-Planck equations on bounded
convex domains in R^d and study the corresponding gradient flow structures. We
reprove the convergence of the discrete to continuous Fokker-Planck equation via
the method of Evolutionary Γ-convergence, i.e., we pass to the limit at the level
of the gradient flow structures, generalising the one-dimensional result obtained
by Disser and Liero. The proof is of variational nature and relies on a Mosco
convergence result for functionals in the discrete-to-continuum limit that is
of independent interest. Our results apply to arbitrary regular meshes, even though
the associated discrete transport distances may fail to converge to the Wasserstein
distance in this generality.
acknowledgement: This work is supported by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant agreement
No 716117) and by the Austrian Science Fund (FWF), grants No F65 and W1245.
article_number: '2008.10962'
article_processing_charge: No
arxiv: 1
author:
- first_name: Dominik L
full_name: Forkert, Dominik L
id: 35C79D68-F248-11E8-B48F-1D18A9856A87
last_name: Forkert
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Lorenzo
full_name: Portinale, Lorenzo
id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
last_name: Portinale
citation:
ama: Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient
flow structures for Fokker-Planck equations in multiple dimensions. arXiv.
apa: Forkert, D. L., Maas, J., & Portinale, L. (n.d.). Evolutionary Γ-convergence
of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
arXiv.
chicago: Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary Γ-Convergence
of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
ArXiv, n.d.
ieee: D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary Γ-convergence of entropic
gradient flow structures for Fokker-Planck equations in multiple dimensions,”
arXiv. .
ista: Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient
flow structures for Fokker-Planck equations in multiple dimensions. arXiv, 2008.10962.
mla: Forkert, Dominik L., et al. “Evolutionary Γ-Convergence of Entropic Gradient
Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” ArXiv,
2008.10962.
short: D.L. Forkert, J. Maas, L. Portinale, ArXiv (n.d.).
date_created: 2021-09-17T10:57:27Z
date_published: 2020-08-25T00:00:00Z
date_updated: 2023-09-07T13:31:05Z
day: '25'
department:
- _id: JaMa
ec_funded: 1
external_id:
arxiv:
- '2008.10962'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2008.10962
month: '08'
oa: 1
oa_version: Preprint
page: '33'
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '11739'
relation: later_version
status: public
- id: '10030'
relation: dissertation_contains
status: public
status: public
title: Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck
equations in multiple dimensions
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2020'
...
---
_id: '73'
abstract:
- lang: eng
text: We consider the space of probability measures on a discrete set X, endowed
with a dynamical optimal transport metric. Given two probability measures supported
in a subset Y⊆X, it is natural to ask whether they can be connected by a constant
speed geodesic with support in Y at all times. Our main result answers this question
affirmatively, under a suitable geometric condition on Y introduced in this paper.
The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi
equations, which is of independent interest.
article_number: '19'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Matthias
full_name: Erbar, Matthias
last_name: Erbar
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Melchior
full_name: Wirth, Melchior
last_name: Wirth
citation:
ama: Erbar M, Maas J, Wirth M. On the geometry of geodesics in discrete optimal
transport. Calculus of Variations and Partial Differential Equations. 2019;58(1).
doi:10.1007/s00526-018-1456-1
apa: Erbar, M., Maas, J., & Wirth, M. (2019). On the geometry of geodesics in
discrete optimal transport. Calculus of Variations and Partial Differential
Equations. Springer. https://doi.org/10.1007/s00526-018-1456-1
chicago: Erbar, Matthias, Jan Maas, and Melchior Wirth. “On the Geometry of Geodesics
in Discrete Optimal Transport.” Calculus of Variations and Partial Differential
Equations. Springer, 2019. https://doi.org/10.1007/s00526-018-1456-1.
ieee: M. Erbar, J. Maas, and M. Wirth, “On the geometry of geodesics in discrete
optimal transport,” Calculus of Variations and Partial Differential Equations,
vol. 58, no. 1. Springer, 2019.
ista: Erbar M, Maas J, Wirth M. 2019. On the geometry of geodesics in discrete optimal
transport. Calculus of Variations and Partial Differential Equations. 58(1), 19.
mla: Erbar, Matthias, et al. “On the Geometry of Geodesics in Discrete Optimal Transport.”
Calculus of Variations and Partial Differential Equations, vol. 58, no.
1, 19, Springer, 2019, doi:10.1007/s00526-018-1456-1.
short: M. Erbar, J. Maas, M. Wirth, Calculus of Variations and Partial Differential
Equations 58 (2019).
date_created: 2018-12-11T11:44:29Z
date_published: 2019-02-01T00:00:00Z
date_updated: 2023-09-13T09:12:35Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-018-1456-1
ec_funded: 1
external_id:
arxiv:
- '1805.06040'
isi:
- '000452849400001'
file:
- access_level: open_access
checksum: ba05ac2d69de4c58d2cd338b63512798
content_type: application/pdf
creator: dernst
date_created: 2019-01-28T15:37:11Z
date_updated: 2020-07-14T12:47:55Z
file_id: '5895'
file_name: 2018_Calculus_Erbar.pdf
file_size: 645565
relation: main_file
file_date_updated: 2020-07-14T12:47:55Z
has_accepted_license: '1'
intvolume: ' 58'
isi: 1
issue: '1'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: 260482E2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: ' F06504'
name: Taming Complexity in Partial Di erential Systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- '09442669'
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the geometry of geodesics in discrete optimal transport
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 58
year: '2019'
...
---
_id: '649'
abstract:
- lang: eng
text: We give a short overview on a recently developed notion of Ricci curvature
for discrete spaces. This notion relies on geodesic convexity properties of the
relative entropy along geodesics in the space of probability densities, for a
metric which is similar to (but different from) the 2-Wasserstein metric. The
theory can be considered as a discrete counterpart to the theory of Ricci curvature
for geodesic measure spaces developed by Lott–Sturm–Villani.
article_processing_charge: No
author:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: 'Maas J. Entropic Ricci curvature for discrete spaces. In: Najman L, Romon
P, eds. Modern Approaches to Discrete Curvature. Vol 2184. Lecture Notes
in Mathematics. Springer; 2017:159-174. doi:10.1007/978-3-319-58002-9_5'
apa: Maas, J. (2017). Entropic Ricci curvature for discrete spaces. In L. Najman
& P. Romon (Eds.), Modern Approaches to Discrete Curvature (Vol. 2184,
pp. 159–174). Springer. https://doi.org/10.1007/978-3-319-58002-9_5
chicago: Maas, Jan. “Entropic Ricci Curvature for Discrete Spaces.” In Modern
Approaches to Discrete Curvature, edited by Laurent Najman and Pascal Romon,
2184:159–74. Lecture Notes in Mathematics. Springer, 2017. https://doi.org/10.1007/978-3-319-58002-9_5.
ieee: J. Maas, “Entropic Ricci curvature for discrete spaces,” in Modern Approaches
to Discrete Curvature, vol. 2184, L. Najman and P. Romon, Eds. Springer, 2017,
pp. 159–174.
ista: 'Maas J. 2017.Entropic Ricci curvature for discrete spaces. In: Modern Approaches
to Discrete Curvature. vol. 2184, 159–174.'
mla: Maas, Jan. “Entropic Ricci Curvature for Discrete Spaces.” Modern Approaches
to Discrete Curvature, edited by Laurent Najman and Pascal Romon, vol. 2184,
Springer, 2017, pp. 159–74, doi:10.1007/978-3-319-58002-9_5.
short: J. Maas, in:, L. Najman, P. Romon (Eds.), Modern Approaches to Discrete Curvature,
Springer, 2017, pp. 159–174.
date_created: 2018-12-11T11:47:42Z
date_published: 2017-10-05T00:00:00Z
date_updated: 2022-05-24T07:01:33Z
day: '05'
department:
- _id: JaMa
doi: 10.1007/978-3-319-58002-9_5
editor:
- first_name: Laurent
full_name: Najman, Laurent
last_name: Najman
- first_name: Pascal
full_name: Romon, Pascal
last_name: Romon
intvolume: ' 2184'
language:
- iso: eng
month: '10'
oa_version: None
page: 159 - 174
publication: Modern Approaches to Discrete Curvature
publication_identifier:
eissn:
- 978-3-319-58002-9
isbn:
- 978-3-319-58001-2
publication_status: published
publisher: Springer
publist_id: '7123'
quality_controlled: '1'
scopus_import: '1'
series_title: Lecture Notes in Mathematics
status: public
title: Entropic Ricci curvature for discrete spaces
type: book_chapter
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2184
year: '2017'
...
---
_id: '956'
abstract:
- lang: eng
text: We study a class of ergodic quantum Markov semigroups on finite-dimensional
unital C⁎-algebras. These semigroups have a unique stationary state σ, and we
are concerned with those that satisfy a quantum detailed balance condition with
respect to σ. We show that the evolution on the set of states that is given by
such a quantum Markov semigroup is gradient flow for the relative entropy with
respect to σ in a particular Riemannian metric on the set of states. This metric
is a non-commutative analog of the 2-Wasserstein metric, and in several interesting
cases we are able to show, in analogy with work of Otto on gradient flows with
respect to the classical 2-Wasserstein metric, that the relative entropy is strictly
and uniformly convex with respect to the Riemannian metric introduced here. As
a consequence, we obtain a number of new inequalities for the decay of relative
entropy for ergodic quantum Markov semigroups with detailed balance.
article_processing_charge: No
author:
- first_name: Eric
full_name: Carlen, Eric
last_name: Carlen
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: Carlen E, Maas J. Gradient flow and entropy inequalities for quantum Markov
semigroups with detailed balance. Journal of Functional Analysis. 2017;273(5):1810-1869.
doi:10.1016/j.jfa.2017.05.003
apa: Carlen, E., & Maas, J. (2017). Gradient flow and entropy inequalities for
quantum Markov semigroups with detailed balance. Journal of Functional Analysis.
Academic Press. https://doi.org/10.1016/j.jfa.2017.05.003
chicago: Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for
Quantum Markov Semigroups with Detailed Balance.” Journal of Functional Analysis.
Academic Press, 2017. https://doi.org/10.1016/j.jfa.2017.05.003.
ieee: E. Carlen and J. Maas, “Gradient flow and entropy inequalities for quantum
Markov semigroups with detailed balance,” Journal of Functional Analysis,
vol. 273, no. 5. Academic Press, pp. 1810–1869, 2017.
ista: Carlen E, Maas J. 2017. Gradient flow and entropy inequalities for quantum
Markov semigroups with detailed balance. Journal of Functional Analysis. 273(5),
1810–1869.
mla: Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for Quantum
Markov Semigroups with Detailed Balance.” Journal of Functional Analysis,
vol. 273, no. 5, Academic Press, 2017, pp. 1810–69, doi:10.1016/j.jfa.2017.05.003.
short: E. Carlen, J. Maas, Journal of Functional Analysis 273 (2017) 1810–1869.
date_created: 2018-12-11T11:49:24Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2023-09-22T10:00:18Z
day: '01'
department:
- _id: JaMa
doi: 10.1016/j.jfa.2017.05.003
external_id:
isi:
- '000406082300005'
intvolume: ' 273'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1609.01254
month: '09'
oa: 1
oa_version: Submitted Version
page: 1810 - 1869
publication: Journal of Functional Analysis
publication_identifier:
issn:
- '00221236'
publication_status: published
publisher: Academic Press
publist_id: '6452'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed
balance
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 273
year: '2017'
...
---
_id: '989'
abstract:
- lang: eng
text: We present a generalized optimal transport model in which the mass-preserving
constraint for the L2-Wasserstein distance is relaxed by introducing a source
term in the continuity equation. The source term is also incorporated in the path
energy by means of its squared L2-norm in time of a functional with linear growth
in space. This extension of the original transport model enables local density
modulations, which is a desirable feature in applications such as image warping
and blending. A key advantage of the use of a functional with linear growth in
space is that it allows for singular sources and sinks, which can be supported
on points or lines. On a technical level, the L2-norm in time ensures a disintegration
of the source in time, which we use to obtain the well-posedness of the model
and the existence of geodesic paths. The numerical discretization is based on
the proximal splitting approach [18] and selected numerical test cases show the
potential of the proposed approach. Furthermore, the approach is applied to the
warping and blending of textures.
alternative_title:
- LNCS
article_processing_charge: No
author:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Martin
full_name: Rumpf, Martin
last_name: Rumpf
- first_name: Stefan
full_name: Simon, Stefan
last_name: Simon
citation:
ama: 'Maas J, Rumpf M, Simon S. Transport based image morphing with intensity modulation.
In: Lauze F, Dong Y, Bjorholm Dahl A, eds. Vol 10302. Springer; 2017:563-577.
doi:10.1007/978-3-319-58771-4_45'
apa: 'Maas, J., Rumpf, M., & Simon, S. (2017). Transport based image morphing
with intensity modulation. In F. Lauze, Y. Dong, & A. Bjorholm Dahl (Eds.)
(Vol. 10302, pp. 563–577). Presented at the SSVM: Scale Space and Variational
Methods in Computer Vision, Kolding, Denmark: Springer. https://doi.org/10.1007/978-3-319-58771-4_45'
chicago: Maas, Jan, Martin Rumpf, and Stefan Simon. “Transport Based Image Morphing
with Intensity Modulation.” edited by François Lauze, Yiqiu Dong, and Anders Bjorholm
Dahl, 10302:563–77. Springer, 2017. https://doi.org/10.1007/978-3-319-58771-4_45.
ieee: J. Maas, M. Rumpf, and S. Simon, “Transport based image morphing with intensity
modulation,” presented at the SSVM: Scale Space and Variational Methods in Computer
Vision, Kolding, Denmark, 2017, vol. 10302, pp. 563–577.
ista: Maas J, Rumpf M, Simon S. 2017. Transport based image morphing with intensity
modulation. SSVM: Scale Space and Variational Methods in Computer Vision, LNCS,
vol. 10302, 563–577.
mla: Maas, Jan, et al. Transport Based Image Morphing with Intensity Modulation.
Edited by François Lauze et al., vol. 10302, Springer, 2017, pp. 563–77, doi:10.1007/978-3-319-58771-4_45.
short: J. Maas, M. Rumpf, S. Simon, in:, F. Lauze, Y. Dong, A. Bjorholm Dahl (Eds.),
Springer, 2017, pp. 563–577.
conference:
end_date: 2017-06-08
location: Kolding, Denmark
name: 'SSVM: Scale Space and Variational Methods in Computer Vision'
start_date: 2017-06-04
date_created: 2018-12-11T11:49:34Z
date_published: 2017-05-18T00:00:00Z
date_updated: 2023-09-22T09:55:50Z
day: '18'
department:
- _id: JaMa
doi: 10.1007/978-3-319-58771-4_45
editor:
- first_name: François
full_name: Lauze, François
last_name: Lauze
- first_name: Yiqiu
full_name: Dong, Yiqiu
last_name: Dong
- first_name: Anders
full_name: Bjorholm Dahl, Anders
last_name: Bjorholm Dahl
external_id:
isi:
- '000432210900045'
intvolume: ' 10302'
isi: 1
language:
- iso: eng
month: '05'
oa_version: None
page: 563 - 577
publication_identifier:
issn:
- '03029743'
publication_status: published
publisher: Springer
publist_id: '6410'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Transport based image morphing with intensity modulation
type: conference
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 10302
year: '2017'
...
---
_id: '1448'
abstract:
- lang: eng
text: We develop a new and systematic method for proving entropic Ricci curvature
lower bounds for Markov chains on discrete sets. Using different methods, such
bounds have recently been obtained in several examples (e.g., 1-dimensional birth
and death chains, product chains, Bernoulli–Laplace models, and random transposition
models). However, a general method to obtain discrete Ricci bounds had been lacking.
Our method covers all of the examples above. In addition we obtain new Ricci curvature
bounds for zero-range processes on the complete graph. The method is inspired
by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.
acknowledgement: "Supported by the German Research Foundation through the Collaborative
Research Center 1060\r\nThe Mathematics of Emergent Effects and the Hausdorff Center
for Mathematics. Part of this work has been done while M. Fathi visited J. Maas
at the University of Bonn in July 2014.We would like to thank the referees for their
careful reading of the manuscript. "
author:
- first_name: Max
full_name: Fathi, Max
last_name: Fathi
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: Fathi M, Maas J. Entropic Ricci curvature bounds for discrete interacting systems.
The Annals of Applied Probability. 2016;26(3):1774-1806. doi:10.1214/15-AAP1133
apa: Fathi, M., & Maas, J. (2016). Entropic Ricci curvature bounds for discrete
interacting systems. The Annals of Applied Probability. Institute of Mathematical
Statistics. https://doi.org/10.1214/15-AAP1133
chicago: Fathi, Max, and Jan Maas. “Entropic Ricci Curvature Bounds for Discrete
Interacting Systems.” The Annals of Applied Probability. Institute of Mathematical
Statistics, 2016. https://doi.org/10.1214/15-AAP1133.
ieee: M. Fathi and J. Maas, “Entropic Ricci curvature bounds for discrete interacting
systems,” The Annals of Applied Probability, vol. 26, no. 3. Institute
of Mathematical Statistics, pp. 1774–1806, 2016.
ista: Fathi M, Maas J. 2016. Entropic Ricci curvature bounds for discrete interacting
systems. The Annals of Applied Probability. 26(3), 1774–1806.
mla: Fathi, Max, and Jan Maas. “Entropic Ricci Curvature Bounds for Discrete Interacting
Systems.” The Annals of Applied Probability, vol. 26, no. 3, Institute
of Mathematical Statistics, 2016, pp. 1774–806, doi:10.1214/15-AAP1133.
short: M. Fathi, J. Maas, The Annals of Applied Probability 26 (2016) 1774–1806.
date_created: 2018-12-11T11:52:05Z
date_published: 2016-06-01T00:00:00Z
date_updated: 2021-01-12T06:50:49Z
day: '01'
department:
- _id: JaMa
doi: 10.1214/15-AAP1133
intvolume: ' 26'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1501.00562
month: '06'
oa: 1
oa_version: Preprint
page: 1774 - 1806
publication: The Annals of Applied Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5748'
quality_controlled: '1'
scopus_import: 1
status: public
title: Entropic Ricci curvature bounds for discrete interacting systems
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 26
year: '2016'
...
---
_id: '1261'
abstract:
- lang: eng
text: 'We consider a non-standard finite-volume discretization of a strongly non-linear
fourth order diffusion equation on the d-dimensional cube, for arbitrary . The
scheme preserves two important structural properties of the equation: the first
is the interpretation as a gradient flow in a mass transportation metric, and
the second is an intimate relation to a linear Fokker-Planck equation. Thanks
to these structural properties, the scheme possesses two discrete Lyapunov functionals.
These functionals approximate the entropy and the Fisher information, respectively,
and their dissipation rates converge to the optimal ones in the discrete-to-continuous
limit. Using the dissipation, we derive estimates on the long-time asymptotics
of the discrete solutions. Finally, we present results from numerical experiments
which indicate that our discretization is able to capture significant features
of the complex original dynamics, even with a rather coarse spatial resolution.'
acknowledgement: This research was supported by the DFG Collaborative Research Centers TRR 109, ‘
Discretization in Geometry and Dynamics ’ and 1060 ‘ The Mathematics of Emergent
Effects ’ .
author:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Daniel
full_name: Matthes, Daniel
last_name: Matthes
citation:
ama: Maas J, Matthes D. Long-time behavior of a finite volume discretization for
a fourth order diffusion equation. Nonlinearity. 2016;29(7):1992-2023.
doi:10.1088/0951-7715/29/7/1992
apa: Maas, J., & Matthes, D. (2016). Long-time behavior of a finite volume discretization
for a fourth order diffusion equation. Nonlinearity. IOP Publishing Ltd.
https://doi.org/10.1088/0951-7715/29/7/1992
chicago: Maas, Jan, and Daniel Matthes. “Long-Time Behavior of a Finite Volume Discretization
for a Fourth Order Diffusion Equation.” Nonlinearity. IOP Publishing Ltd.,
2016. https://doi.org/10.1088/0951-7715/29/7/1992.
ieee: J. Maas and D. Matthes, “Long-time behavior of a finite volume discretization
for a fourth order diffusion equation,” Nonlinearity, vol. 29, no. 7. IOP
Publishing Ltd., pp. 1992–2023, 2016.
ista: Maas J, Matthes D. 2016. Long-time behavior of a finite volume discretization
for a fourth order diffusion equation. Nonlinearity. 29(7), 1992–2023.
mla: Maas, Jan, and Daniel Matthes. “Long-Time Behavior of a Finite Volume Discretization
for a Fourth Order Diffusion Equation.” Nonlinearity, vol. 29, no. 7, IOP
Publishing Ltd., 2016, pp. 1992–2023, doi:10.1088/0951-7715/29/7/1992.
short: J. Maas, D. Matthes, Nonlinearity 29 (2016) 1992–2023.
date_created: 2018-12-11T11:51:00Z
date_published: 2016-06-10T00:00:00Z
date_updated: 2021-01-12T06:49:28Z
day: '10'
department:
- _id: JaMa
doi: 10.1088/0951-7715/29/7/1992
intvolume: ' 29'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1505.03178
month: '06'
oa: 1
oa_version: Preprint
page: 1992 - 2023
publication: Nonlinearity
publication_status: published
publisher: IOP Publishing Ltd.
publist_id: '6062'
quality_controlled: '1'
scopus_import: 1
status: public
title: Long-time behavior of a finite volume discretization for a fourth order diffusion
equation
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 29
year: '2016'
...
---
_id: '1635'
abstract:
- lang: eng
text: We calculate a Ricci curvature lower bound for some classical examples of
random walks, namely, a chain on a slice of the n-dimensional discrete cube (the
so-called Bernoulli-Laplace model) and the random transposition shuffle of the
symmetric group of permutations on n letters.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matthias
full_name: Erbar, Matthias
last_name: Erbar
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Prasad
full_name: Tetali, Prasad
last_name: Tetali
citation:
ama: Erbar M, Maas J, Tetali P. Discrete Ricci curvature bounds for Bernoulli-Laplace
and random transposition models. Annales de la faculté des sciences de Toulouse.
2015;24(4):781-800. doi:10.5802/afst.1464
apa: Erbar, M., Maas, J., & Tetali, P. (2015). Discrete Ricci curvature bounds
for Bernoulli-Laplace and random transposition models. Annales de La Faculté
Des Sciences de Toulouse. Faculté des sciences de Toulouse. https://doi.org/10.5802/afst.1464
chicago: Erbar, Matthias, Jan Maas, and Prasad Tetali. “Discrete Ricci Curvature
Bounds for Bernoulli-Laplace and Random Transposition Models.” Annales de La
Faculté Des Sciences de Toulouse. Faculté des sciences de Toulouse, 2015.
https://doi.org/10.5802/afst.1464.
ieee: M. Erbar, J. Maas, and P. Tetali, “Discrete Ricci curvature bounds for Bernoulli-Laplace
and random transposition models,” Annales de la faculté des sciences de Toulouse,
vol. 24, no. 4. Faculté des sciences de Toulouse, pp. 781–800, 2015.
ista: Erbar M, Maas J, Tetali P. 2015. Discrete Ricci curvature bounds for Bernoulli-Laplace
and random transposition models. Annales de la faculté des sciences de Toulouse.
24(4), 781–800.
mla: Erbar, Matthias, et al. “Discrete Ricci Curvature Bounds for Bernoulli-Laplace
and Random Transposition Models.” Annales de La Faculté Des Sciences de Toulouse,
vol. 24, no. 4, Faculté des sciences de Toulouse, 2015, pp. 781–800, doi:10.5802/afst.1464.
short: M. Erbar, J. Maas, P. Tetali, Annales de La Faculté Des Sciences de Toulouse
24 (2015) 781–800.
date_created: 2018-12-11T11:53:10Z
date_published: 2015-01-01T00:00:00Z
date_updated: 2023-10-18T07:48:28Z
day: '01'
department:
- _id: JaMa
doi: 10.5802/afst.1464
external_id:
arxiv:
- '1409.8605'
intvolume: ' 24'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1409.8605
month: '01'
oa: 1
oa_version: Preprint
page: 781 - 800
publication: Annales de la faculté des sciences de Toulouse
publication_status: published
publisher: Faculté des sciences de Toulouse
publist_id: '5520'
quality_controlled: '1'
status: public
title: Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition
models
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2015'
...
---
_id: '1639'
abstract:
- lang: eng
text: In this paper the optimal transport and the metamorphosis perspectives are
combined. For a pair of given input images geodesic paths in the space of images
are defined as minimizers of a resulting path energy. To this end, the underlying
Riemannian metric measures the rate of transport cost and the rate of viscous
dissipation. Furthermore, the model is capable to deal with strongly varying image
contrast and explicitly allows for sources and sinks in the transport equations
which are incorporated in the metric related to the metamorphosis approach by
Trouvé and Younes. In the non-viscous case with source term existence of geodesic
paths is proven in the space of measures. The proposed model is explored on the
range from merely optimal transport to strongly dissipative dynamics. For this
model a robust and effective variational time discretization of geodesic paths
is proposed. This requires to minimize a discrete path energy consisting of a
sum of consecutive image matching functionals. These functionals are defined on
corresponding pairs of intensity functions and on associated pairwise matching
deformations. Existence of time discrete geodesics is demonstrated. Furthermore,
a finite element implementation is proposed and applied to instructive test cases
and to real images. In the non-viscous case this is compared to the algorithm
proposed by Benamou and Brenier including a discretization of the source term.
Finally, the model is generalized to define discrete weighted barycentres with
applications to textures and objects.
acknowledgement: The authors acknowledge support of the Collaborative Research Centre
1060 funded by the German Science foundation. This work is further supported by
the King Abdullah University for Science and Technology (KAUST) Award No. KUK-I1-007-43
and the EPSRC grant Nr. EP/M00483X/1.
arxiv: 1
author:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Martin
full_name: Rumpf, Martin
last_name: Rumpf
- first_name: Carola
full_name: Schönlieb, Carola
last_name: Schönlieb
- first_name: Stefan
full_name: Simon, Stefan
last_name: Simon
citation:
ama: 'Maas J, Rumpf M, Schönlieb C, Simon S. A generalized model for optimal transport
of images including dissipation and density modulation. ESAIM: Mathematical
Modelling and Numerical Analysis. 2015;49(6):1745-1769. doi:10.1051/m2an/2015043'
apa: 'Maas, J., Rumpf, M., Schönlieb, C., & Simon, S. (2015). A generalized
model for optimal transport of images including dissipation and density modulation.
ESAIM: Mathematical Modelling and Numerical Analysis. EDP Sciences. https://doi.org/10.1051/m2an/2015043'
chicago: 'Maas, Jan, Martin Rumpf, Carola Schönlieb, and Stefan Simon. “A Generalized
Model for Optimal Transport of Images Including Dissipation and Density Modulation.”
ESAIM: Mathematical Modelling and Numerical Analysis. EDP Sciences, 2015.
https://doi.org/10.1051/m2an/2015043.'
ieee: 'J. Maas, M. Rumpf, C. Schönlieb, and S. Simon, “A generalized model for optimal
transport of images including dissipation and density modulation,” ESAIM: Mathematical
Modelling and Numerical Analysis, vol. 49, no. 6. EDP Sciences, pp. 1745–1769,
2015.'
ista: 'Maas J, Rumpf M, Schönlieb C, Simon S. 2015. A generalized model for optimal
transport of images including dissipation and density modulation. ESAIM: Mathematical
Modelling and Numerical Analysis. 49(6), 1745–1769.'
mla: 'Maas, Jan, et al. “A Generalized Model for Optimal Transport of Images Including
Dissipation and Density Modulation.” ESAIM: Mathematical Modelling and Numerical
Analysis, vol. 49, no. 6, EDP Sciences, 2015, pp. 1745–69, doi:10.1051/m2an/2015043.'
short: 'J. Maas, M. Rumpf, C. Schönlieb, S. Simon, ESAIM: Mathematical Modelling
and Numerical Analysis 49 (2015) 1745–1769.'
date_created: 2018-12-11T11:53:11Z
date_published: 2015-11-01T00:00:00Z
date_updated: 2021-01-12T06:52:10Z
day: '01'
department:
- _id: JaMa
doi: 10.1051/m2an/2015043
external_id:
arxiv:
- '1504.01988'
intvolume: ' 49'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1504.01988
month: '11'
oa: 1
oa_version: Preprint
page: 1745 - 1769
publication: 'ESAIM: Mathematical Modelling and Numerical Analysis'
publication_status: published
publisher: EDP Sciences
publist_id: '5514'
quality_controlled: '1'
scopus_import: 1
status: public
title: A generalized model for optimal transport of images including dissipation and
density modulation
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 49
year: '2015'
...
---
_id: '1517'
abstract:
- lang: eng
text: "We study the large deviation rate functional for the empirical distribution
of independent Brownian particles with drift. In one dimension, it has been shown
by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent
(in the sense of Γ-convergence) to the Jordan-Kinderlehrer-Otto functional arising
in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher
dimensions, part of this statement (the lower bound) has been recently proved
by Duong, Laschos and Renger, but the upper bound remained open, since the proof
of Duong et al relies on regularity properties of optimal transport maps that
are restricted to one dimension. In this note we present a new proof of the upper
bound, thereby generalising the result of Adams et al to arbitrary dimensions.\r\n"
article_number: '89'
author:
- first_name: Matthias
full_name: Erbar, Matthias
last_name: Erbar
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Michiel
full_name: Renger, Michiel
last_name: Renger
citation:
ama: Erbar M, Maas J, Renger M. From large deviations to Wasserstein gradient flows
in multiple dimensions. Electronic Communications in Probability. 2015;20.
doi:10.1214/ECP.v20-4315
apa: Erbar, M., Maas, J., & Renger, M. (2015). From large deviations to Wasserstein
gradient flows in multiple dimensions. Electronic Communications in Probability.
Institute of Mathematical Statistics. https://doi.org/10.1214/ECP.v20-4315
chicago: Erbar, Matthias, Jan Maas, and Michiel Renger. “From Large Deviations to
Wasserstein Gradient Flows in Multiple Dimensions.” Electronic Communications
in Probability. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/ECP.v20-4315.
ieee: M. Erbar, J. Maas, and M. Renger, “From large deviations to Wasserstein gradient
flows in multiple dimensions,” Electronic Communications in Probability,
vol. 20. Institute of Mathematical Statistics, 2015.
ista: Erbar M, Maas J, Renger M. 2015. From large deviations to Wasserstein gradient
flows in multiple dimensions. Electronic Communications in Probability. 20, 89.
mla: Erbar, Matthias, et al. “From Large Deviations to Wasserstein Gradient Flows
in Multiple Dimensions.” Electronic Communications in Probability, vol.
20, 89, Institute of Mathematical Statistics, 2015, doi:10.1214/ECP.v20-4315.
short: M. Erbar, J. Maas, M. Renger, Electronic Communications in Probability 20
(2015).
date_created: 2018-12-11T11:52:29Z
date_published: 2015-11-29T00:00:00Z
date_updated: 2021-01-12T06:51:19Z
day: '29'
ddc:
- '519'
department:
- _id: JaMa
doi: 10.1214/ECP.v20-4315
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creator: system
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date_updated: 2020-07-14T12:45:00Z
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has_accepted_license: '1'
intvolume: ' 20'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
publication: Electronic Communications in Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5660'
pubrep_id: '494'
quality_controlled: '1'
scopus_import: 1
status: public
title: From large deviations to Wasserstein gradient flows in multiple dimensions
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: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 20
year: '2015'
...
---
_id: '2131'
abstract:
- lang: eng
text: We study approximations to a class of vector-valued equations of Burgers type
driven by a multiplicative space-time white noise. A solution theory for this
class of equations has been developed recently in Probability Theory Related Fields
by Hairer and Weber. The key idea was to use the theory of controlled rough paths
to give definitions of weak/mild solutions and to set up a Picard iteration argument.
In this article the limiting behavior of a rather large class of (spatial) approximations
to these equations is studied. These approximations are shown to converge and
convergence rates are given, but the limit may depend on the particular choice
of approximation. This effect is a spatial analogue to the Itô-Stratonovich correction
in the theory of stochastic ordinary differential equations, where it is well
known that different approximation schemes may converge to different solutions.
acknowledgement: JM is supported by Rubicon grant 680-50-0901 of the Netherlands Organisation
for Scientific Research (NWO). MH is supported by EPSRC grant EP/D071593/1 and by
the Royal Society through a Wolfson Research Merit Award. Both MH and HW are supported
by the Le
author:
- first_name: Martin
full_name: Hairer, Martin M
last_name: Hairer
- first_name: Jan
full_name: Jan Maas
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Hendrik
full_name: Weber, Hendrik
last_name: Weber
citation:
ama: Hairer M, Maas J, Weber H. Approximating Rough Stochastic PDEs. Communications
on Pure and Applied Mathematics. 2014;67(5):776-870. doi:10.1002/cpa.21495
apa: Hairer, M., Maas, J., & Weber, H. (2014). Approximating Rough Stochastic
PDEs. Communications on Pure and Applied Mathematics. Wiley-Blackwell.
https://doi.org/10.1002/cpa.21495
chicago: Hairer, Martin, Jan Maas, and Hendrik Weber. “Approximating Rough Stochastic
PDEs.” Communications on Pure and Applied Mathematics. Wiley-Blackwell,
2014. https://doi.org/10.1002/cpa.21495.
ieee: M. Hairer, J. Maas, and H. Weber, “Approximating Rough Stochastic PDEs,” Communications
on Pure and Applied Mathematics, vol. 67, no. 5. Wiley-Blackwell, pp. 776–870,
2014.
ista: Hairer M, Maas J, Weber H. 2014. Approximating Rough Stochastic PDEs. Communications
on Pure and Applied Mathematics. 67(5), 776–870.
mla: Hairer, Martin, et al. “Approximating Rough Stochastic PDEs.” Communications
on Pure and Applied Mathematics, vol. 67, no. 5, Wiley-Blackwell, 2014, pp.
776–870, doi:10.1002/cpa.21495.
short: M. Hairer, J. Maas, H. Weber, Communications on Pure and Applied Mathematics
67 (2014) 776–870.
date_created: 2018-12-11T11:55:53Z
date_published: 2014-05-01T00:00:00Z
date_updated: 2021-01-12T06:55:30Z
day: '01'
doi: 10.1002/cpa.21495
extern: 1
intvolume: ' 67'
issue: '5'
main_file_link:
- open_access: '1'
url: 'http://arxiv.org/abs/1202.3094 '
month: '05'
oa: 1
page: 776 - 870
publication: Communications on Pure and Applied Mathematics
publication_status: published
publisher: Wiley-Blackwell
publist_id: '4902'
quality_controlled: 0
status: public
title: Approximating Rough Stochastic PDEs
type: journal_article
volume: 67
year: '2014'
...
---
_id: '2132'
abstract:
- lang: eng
text: We consider discrete porous medium equations of the form ∂tρt=Δϕ(ρt), where
Δ is the generator of a reversible continuous time Markov chain on a finite set
χ, and ϕ is an increasing function. We show that these equations arise as gradient
flows of certain entropy functionals with respect to suitable non-local transportation
metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow
structure for porous medium equations in ℝn discovered by Otto. We present a one-dimensional
counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence
to the Wasserstein metric.
author:
- first_name: Matthias
full_name: Erbar, Matthias
last_name: Erbar
- first_name: Jan
full_name: Jan Maas
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: Erbar M, Maas J. Gradient flow structures for discrete porous medium equations.
Discrete and Continuous Dynamical Systems- Series A. 2014;34(4):1355-1374.
doi:10.3934/dcds.2014.34.1355
apa: Erbar, M., & Maas, J. (2014). Gradient flow structures for discrete porous
medium equations. Discrete and Continuous Dynamical Systems- Series A.
Southwest Missouri State University. https://doi.org/10.3934/dcds.2014.34.1355
chicago: Erbar, Matthias, and Jan Maas. “Gradient Flow Structures for Discrete Porous
Medium Equations.” Discrete and Continuous Dynamical Systems- Series A.
Southwest Missouri State University, 2014. https://doi.org/10.3934/dcds.2014.34.1355 .
ieee: M. Erbar and J. Maas, “Gradient flow structures for discrete porous medium
equations,” Discrete and Continuous Dynamical Systems- Series A, vol. 34,
no. 4. Southwest Missouri State University, pp. 1355–1374, 2014.
ista: Erbar M, Maas J. 2014. Gradient flow structures for discrete porous medium
equations. Discrete and Continuous Dynamical Systems- Series A. 34(4), 1355–1374.
mla: Erbar, Matthias, and Jan Maas. “Gradient Flow Structures for Discrete Porous
Medium Equations.” Discrete and Continuous Dynamical Systems- Series A,
vol. 34, no. 4, Southwest Missouri State University, 2014, pp. 1355–74, doi:10.3934/dcds.2014.34.1355
.
short: M. Erbar, J. Maas, Discrete and Continuous Dynamical Systems- Series A 34
(2014) 1355–1374.
date_created: 2018-12-11T11:55:54Z
date_published: 2014-04-01T00:00:00Z
date_updated: 2021-01-12T06:55:30Z
day: '01'
doi: '10.3934/dcds.2014.34.1355 '
extern: 1
intvolume: ' 34'
issue: '4'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1212.1129
month: '04'
oa: 1
page: 1355 - 1374
publication: Discrete and Continuous Dynamical Systems- Series A
publication_status: published
publisher: Southwest Missouri State University
publist_id: '4903'
quality_controlled: 0
status: public
title: Gradient flow structures for discrete porous medium equations
type: journal_article
volume: 34
year: '2014'
...