Homogenisation of dynamical optimal transport on periodic graphs

Abstract

This paper deals with the large-scale behaviour of dynamical optimal transport on Z d -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. Contents 42 9. Analysis of the cell problem 57 Appendix A. The Kantorovich-Rubinstein metric on signed measures 66 Appendix B. Norms on curves in the space of measures 68 Appendix C. Domain property of convex functions 68 Appendix D. Notation 68 References 70

On a class of nonlocal continuity equations on graphs

“…In this manuscript, we resume the analysis of partial differential equations (PDEs) on graphs started in our previous work [13], focusing on a larger class of nonlocal continuity equations (NCE), including different dynamics. The main motivation for this study comes from data science, as graphs represent a relevant ambient space for data representation and classification [4,15,16,22,23,28]. However, most of the results obtained so far in the literature are concerned with static problems rather than time-dependent ones.…”