---
res:
  bibo_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\r\nnon-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.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Lorenzo
      foaf_name: Portinale, Lorenzo
      foaf_surname: Portinale
      foaf_workInfoHomepage: http://www.librecat.org/personId=30AD2CBC-F248-11E8-B48F-1D18A9856A87
  bibo_doi: 10.15479/at:ista:10030
  dct_date: 2021^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2663-337X
  dct_language: eng
  dct_publisher: Institute of Science and Technology Austria@
  dct_title: Discrete-to-continuum limits of transport problems and gradient flows
    in the space of measures@
...