@article{12959, abstract = {This paper deals with the large-scale behaviour of dynamical optimal transport on Zd -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 -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.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {5}, publisher = {Springer Nature}, title = {{Homogenisation of dynamical optimal transport on periodic graphs}}, doi = {10.1007/s00526-023-02472-z}, volume = {62}, year = {2023}, } @article{11739, abstract = {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.}, author = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence}, number = {4}, pages = {4297--4333}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}}, doi = {10.1137/21M1410968}, volume = {54}, year = {2022}, } @phdthesis{10030, 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.}, author = {Portinale, Lorenzo}, issn = {2663-337X}, publisher = {Institute of Science and Technology Austria}, title = {{Discrete-to-continuum limits of transport problems and gradient flows in the space of measures}}, doi = {10.15479/at:ista:10030}, year = {2021}, } @article{10575, abstract = {The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.}, author = {Abbatiello, Anna and Bulíček, Miroslav and Maringová, Erika}, issn = {1793-6314}, journal = {Mathematical Models and Methods in Applied Sciences}, number = {11}, pages = {2165--2212}, publisher = {World Scientific Publishing}, title = {{On the dynamic slip boundary condition for Navier-Stokes-like problems}}, doi = {10.1142/S0218202521500470}, volume = {31}, year = {2021}, } @article{7573, abstract = {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.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo}, issn = {00217824}, journal = {Journal de Mathematiques Pures et Appliquees}, number = {7}, pages = {204--234}, publisher = {Elsevier}, title = {{Homogenisation of one-dimensional discrete optimal transport}}, doi = {10.1016/j.matpur.2020.02.008}, volume = {139}, year = {2020}, }