Discrete-to-continuum limits of transport problems and gradient flows in the space of measures

Portinale L. 2021. Discrete-to-continuum limits of transport problems and gradient flows in the space of measures. IST Austria.

Download
Restricted tex_and_pictures.zip 3.88 MB
OA thesis_portinale_Final (1).pdf 2.53 MB

Thesis | Published | English
Series Title
IST Austria Thesis
Abstract
This PhD thesis is primarily focused on the study of discrete transport problems, introduced for the first time in the seminal works of Maas [Maa11] and Mielke [Mie11] on finite state Markov chains and reaction-diffusion equations, respectively. More in detail, my research focuses on the study of transport costs on graphs, in particular the convergence and the stability of such problems in the discrete-to-continuum limit. This thesis also includes some results concerning non-commutative optimal transport. The first chapter of this thesis consists of a general introduction to the optimal transport problems, both in the discrete, the continuous, and the non-commutative setting. Chapters 2 and 3 present the content of two works, obtained in collaboration with Peter Gladbach, Eva Kopfer, and Jan Maas, where we have been able to show the convergence of discrete transport costs on periodic graphs to suitable continuous ones, which can be described by means of a homogenisation result. We first focus on the particular case of quadratic costs on the real line and then extending the result to more general costs in arbitrary dimension. Our results are the first complete characterisation of limits of transport costs on periodic graphs in arbitrary dimension which do not rely on any additional symmetry. In Chapter 4 we turn our attention to one of the intriguing connection between evolution equations and optimal transport, represented by the theory of gradient flows. We show that discrete gradient flow structures associated to a finite volume approximation of a certain class of diffusive equations (Fokker–Planck) is stable in the limit of vanishing meshes, reproving the convergence of the scheme via the method of evolutionary Γ-convergence and exploiting a more variational point of view on the problem. This is based on a collaboration with Dominik Forkert and Jan Maas. Chapter 5 represents a change of perspective, moving away from the discrete world and reaching the non-commutative one. As in the discrete case, we discuss how classical tools coming from the commutative optimal transport can be translated into the setting of density matrices. In particular, in this final chapter we present a non-commutative version of the Schrödinger problem (or entropic regularised optimal transport problem) and discuss existence and characterisation of minimisers, a duality result, and present a non-commutative version of the well-known Sinkhorn algorithm to compute the above mentioned optimisers. This is based on a joint work with Dario Feliciangeli and Augusto Gerolin. Finally, Appendix A and B contain some additional material and discussions, with particular attention to Harnack inequalities and the regularity of flows on discrete spaces.
Publishing Year
Date Published
2021-09-22
Acknowledgement
The author gratefully acknowledges support by the Austrian Science Fund (FWF), grants No W1245.
ISSN
IST-REx-ID

Cite this

Portinale L. Discrete-to-continuum limits of transport problems and gradient flows in the space of measures. 2021. doi:10.15479/at:ista:10030
Portinale, L. (2021). Discrete-to-continuum limits of transport problems and gradient flows in the space of measures. IST Austria. https://doi.org/10.15479/at:ista:10030
Portinale, Lorenzo. “Discrete-to-Continuum Limits of Transport Problems and Gradient Flows in the Space of Measures.” IST Austria, 2021. https://doi.org/10.15479/at:ista:10030.
L. Portinale, “Discrete-to-continuum limits of transport problems and gradient flows in the space of measures,” IST Austria, 2021.
Portinale L. 2021. Discrete-to-continuum limits of transport problems and gradient flows in the space of measures. IST Austria.
Portinale, Lorenzo. Discrete-to-Continuum Limits of Transport Problems and Gradient Flows in the Space of Measures. IST Austria, 2021, doi:10.15479/at:ista:10030.
All files available under the following license(s):
Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
Main File(s)
File Name
tex_and_pictures.zip 3.88 MB
Access Level
Restricted Closed Access
Date Uploaded
2021-09-21
MD5 Checksum
8cd60dcb8762e8f21867e21e8001e183
Access Level
OA Open Access
Date Uploaded
2021-09-27
MD5 Checksum
9789e9d967c853c1503ec7f307170279


Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar