---
_id: '642'
abstract:
- lang: eng
text: Cauchy problems with SPDEs on the whole space are localized to Cauchy problems
on a ball of radius R. This localization reduces various kinds of spatial approximation
schemes to finite dimensional problems. The error is shown to be exponentially
small. As an application, a numerical scheme is presented which combines the localization
and the space and time discretization, and thus is fully implementable.
author:
- first_name: Mate
full_name: Gerencser, Mate
id: 44ECEDF2-F248-11E8-B48F-1D18A9856A87
last_name: Gerencser
- first_name: István
full_name: Gyöngy, István
last_name: Gyöngy
citation:
ama: Gerencser M, Gyöngy I. Localization errors in solving stochastic partial differential
equations in the whole space. Mathematics of Computation. 2017;86(307):2373-2397.
doi:10.1090/mcom/3201
apa: Gerencser, M., & Gyöngy, I. (2017). Localization errors in solving stochastic
partial differential equations in the whole space. Mathematics of Computation.
American Mathematical Society. https://doi.org/10.1090/mcom/3201
chicago: Gerencser, Mate, and István Gyöngy. “Localization Errors in Solving Stochastic
Partial Differential Equations in the Whole Space.” Mathematics of Computation.
American Mathematical Society, 2017. https://doi.org/10.1090/mcom/3201.
ieee: M. Gerencser and I. Gyöngy, “Localization errors in solving stochastic partial
differential equations in the whole space,” Mathematics of Computation,
vol. 86, no. 307. American Mathematical Society, pp. 2373–2397, 2017.
ista: Gerencser M, Gyöngy I. 2017. Localization errors in solving stochastic partial
differential equations in the whole space. Mathematics of Computation. 86(307),
2373–2397.
mla: Gerencser, Mate, and István Gyöngy. “Localization Errors in Solving Stochastic
Partial Differential Equations in the Whole Space.” Mathematics of Computation,
vol. 86, no. 307, American Mathematical Society, 2017, pp. 2373–97, doi:10.1090/mcom/3201.
short: M. Gerencser, I. Gyöngy, Mathematics of Computation 86 (2017) 2373–2397.
date_created: 2018-12-11T11:47:40Z
date_published: 2017-01-01T00:00:00Z
date_updated: 2021-01-12T08:07:26Z
day: '01'
department:
- _id: JaMa
doi: 10.1090/mcom/3201
intvolume: ' 86'
issue: '307'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1508.05535
month: '01'
oa: 1
oa_version: Submitted Version
page: 2373 - 2397
publication: Mathematics of Computation
publication_identifier:
issn:
- '00255718'
publication_status: published
publisher: American Mathematical Society
publist_id: '7144'
quality_controlled: '1'
scopus_import: 1
status: public
title: Localization errors in solving stochastic partial differential equations in
the whole space
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 86
year: '2017'
...
---
_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: '447'
abstract:
- lang: eng
text: We consider last passage percolation (LPP) models with exponentially distributed
random variables, which are linked to the totally asymmetric simple exclusion
process (TASEP). The competition interface for LPP was introduced and studied
in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process
had a rarefaction fan. Here we consider situations with a shock and determine
the law of the fluctuations of the competition interface around its deter- ministic
law of large number position. We also study the multipoint distribution of the
LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).
article_processing_charge: No
article_type: original
author:
- first_name: Patrik
full_name: Ferrari, Patrik
last_name: Ferrari
- first_name: Peter
full_name: Nejjar, Peter
id: 4BF426E2-F248-11E8-B48F-1D18A9856A87
last_name: Nejjar
citation:
ama: Ferrari P, Nejjar P. Fluctuations of the competition interface in presence
of shocks. Revista Latino-Americana de Probabilidade e Estatística. 2017;9:299-325.
doi:10.30757/ALEA.v14-17
apa: Ferrari, P., & Nejjar, P. (2017). Fluctuations of the competition interface
in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística.
Instituto Nacional de Matematica Pura e Aplicada. https://doi.org/10.30757/ALEA.v14-17
chicago: Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface
in Presence of Shocks.” Revista Latino-Americana de Probabilidade e Estatística.
Instituto Nacional de Matematica Pura e Aplicada, 2017. https://doi.org/10.30757/ALEA.v14-17.
ieee: P. Ferrari and P. Nejjar, “Fluctuations of the competition interface in presence
of shocks,” Revista Latino-Americana de Probabilidade e Estatística, vol.
9. Instituto Nacional de Matematica Pura e Aplicada, pp. 299–325, 2017.
ista: Ferrari P, Nejjar P. 2017. Fluctuations of the competition interface in presence
of shocks. Revista Latino-Americana de Probabilidade e Estatística. 9, 299–325.
mla: Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface
in Presence of Shocks.” Revista Latino-Americana de Probabilidade e Estatística,
vol. 9, Instituto Nacional de Matematica Pura e Aplicada, 2017, pp. 299–325, doi:10.30757/ALEA.v14-17.
short: P. Ferrari, P. Nejjar, Revista Latino-Americana de Probabilidade e Estatística
9 (2017) 299–325.
date_created: 2018-12-11T11:46:31Z
date_published: 2017-03-23T00:00:00Z
date_updated: 2023-10-10T13:10:32Z
day: '23'
department:
- _id: LaEr
- _id: JaMa
doi: 10.30757/ALEA.v14-17
ec_funded: 1
intvolume: ' 9'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://alea.impa.br/articles/v14/14-17.pdf
month: '03'
oa: 1
oa_version: Submitted Version
page: 299 - 325
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Revista Latino-Americana de Probabilidade e Estatística
publication_status: published
publisher: Instituto Nacional de Matematica Pura e Aplicada
publist_id: '7376'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Fluctuations of the competition interface in presence of shocks
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 9
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
file:
- access_level: open_access
checksum: 135741c17d3e1547ca696b6fbdcd559c
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:39Z
date_updated: 2020-07-14T12:45:00Z
file_id: '4828'
file_name: IST-2016-494-v1+1_4315-23820-1-PB.pdf
file_size: 230525
relation: main_file
file_date_updated: 2020-07-14T12:45:00Z
has_accepted_license: '1'
intvolume: ' 20'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
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
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...