Homogenisation of dynamical optimal transport on periodic graphs. (English) Zbl 1514.49023
The paper contains convergence results of discrete dynamical optimal transport problems, defined on graphs with a \(\mathbb{Z}^d\)-periodic structure, to suitable continuous optimal transport problems.
The discrete problem is defined in terms of a general convex and lower semicontinuous energy density which enters in a non-trivial way, together with the geometry of the graph, in the form of the limiting continuous problem.
The authors require only mild conditions on the discrete energy density and their results cover a large number of interesting situations.
The discrete problem is defined in terms of a general convex and lower semicontinuous energy density which enters in a non-trivial way, together with the geometry of the graph, in the form of the limiting continuous problem.
The authors require only mild conditions on the discrete energy density and their results cover a large number of interesting situations.
Reviewer: Nicolò De Ponti (Trieste)
MSC:
| 49Q22 | Optimal transportation |
| 49M25 | Discrete approximations in optimal control |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 65K10 | Numerical optimization and variational techniques |
| 74Q10 | Homogenization and oscillations in dynamical problems of solid mechanics |
Keywords:
dynamical optimal transport; homogenisation result; periodic graphs; discrete energy densityReferences:
| [1] | Alicandro, R.; Cicalese, M.; Gloria, A., Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Ration. Mech. Anal., 200, 3, 881-943 (2011) · Zbl 1294.74056 · doi:10.1007/s00205-010-0378-7 |
| [2] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000), Oxford University Press, New York: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York · Zbl 0957.49001 |
| [3] | Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich (2008), Basel: Birkhäuser Verlag, Basel · Zbl 1145.35001 |
| [4] | Alicandro, R.; Cicalese, M., A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal., 36, 1, 1-37 (2004) · Zbl 1070.49009 · doi:10.1137/S0036141003426471 |
| [5] | Bach, A.; Braides, A.; Cicalese, M., Discrete-to-continuum limits of multibody systems with bulk and surface long-range interactions, SIAM J. Math. Anal., 52, 4, 3600-3665 (2020) · Zbl 1446.49009 · doi:10.1137/19M1289212 |
| [6] | Benamou, J-D; Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, 3, 375-393 (2000) · Zbl 0968.76069 · doi:10.1007/s002110050002 |
| [7] | Braides, A.; Fonseca, I.; Leoni, G., A-quasiconvexity: relaxation and homogenization, ESAIM Control Optim. Calculus Var., 5, 539-577 (2000) · Zbl 0971.35010 · doi:10.1051/cocv:2000121 |
| [8] | Bogachev, VI, Measure Theory. II (2007), Berlin: Springer-Verlag, Berlin · Zbl 1120.28001 · doi:10.1007/978-3-540-34514-5 |
| [9] | Braides, A.; Defranceschi, A., Homogenization of Multiple Integrals (1998), Oxford: Oxford University Press, Oxford · Zbl 0911.49010 |
| [10] | Buttazzo, G.; Freddi, L., Functionals defined on measures and applications to non-equi-uniformly elliptic problems, Ann. Mat. Pura Appl., 4, 159, 133-149 (1991) · Zbl 0767.35010 · doi:10.1007/BF01766298 |
| [11] | Chow, S-N; Huang, W.; Li, Y.; Zhou, H., Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203, 3, 969-1008 (2012) · Zbl 1256.35173 · doi:10.1007/s00205-011-0471-6 |
| [12] | Disser, K.; Liero, M., On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Netw. Heterog. Media, 10, 2, 233-253 (2015) · Zbl 1372.35124 · doi:10.3934/nhm.2015.10.233 |
| [13] | Dolbeault, J.; Nazaret, B.; Savaré, G., A new class of transport distances between measures, Calc. Var. Partial Differ. Equ., 34, 2, 193-231 (2009) · Zbl 1157.49042 · doi:10.1007/s00526-008-0182-5 |
| [14] | Eymard, R., Gallouët, T., Herbin, R.: Finite Volume Methods. In: Handbook of Numerical Analysis, Vol. VII, Handb. Numer. Anal., VII, pp. 713-1020. North-Holland, Amsterdam (2000) · Zbl 0981.65095 |
| [15] | Erbar, M.; Henderson, C.; Menz, G.; Tetali, P., Ricci curvature bounds for weakly interacting Markov chains, Electron. J. Probab., 22, 23 (2017) · Zbl 1362.60084 · doi:10.1214/17-EJP49 |
| [16] | Erbar, M.; Maas, J.; Tetali, P., Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models, Ann. Fac. Sci. Toulouse Math., 24, 4, 781-800 (2015) · Zbl 1333.60088 · doi:10.5802/afst.1464 |
| [17] | Erbar, M.; Fathi, M., Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature, J. Funct. Anal., 274, 11, 3056-3089 (2018) · Zbl 1386.53020 · doi:10.1016/j.jfa.2018.03.011 |
| [18] | Erbar, M.; Maas, J., Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206, 3, 997-1038 (2012) · Zbl 1256.53028 · doi:10.1007/s00205-012-0554-z |
| [19] | Erbar, M.; Maas, J., Gradient flow structures for discrete porous medium equations, Discrete Contin. Dyn. Syst., 34, 4, 1355-1374 (2014) · Zbl 1275.49084 · doi:10.3934/dcds.2014.34.1355 |
| [20] | Fathi, M.; Maas, J., Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Probab., 26, 3, 1774-1806 (2016) · Zbl 1345.60076 · doi:10.1214/15-AAP1133 |
| [21] | Forkert, D.; Maas, J.; Portinale, L., Evolutionary \(\Gamma \)-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions, SIAM J. Math. Anal., 54, 4, 4297-4333 (2022) · Zbl 1498.35543 · doi:10.1137/21M1410968 |
| [22] | Ford, LR Jr; Fulkerson, DR, Flows in Networks (1962), Princeton: Princeton University Press, Princeton · Zbl 0106.34802 |
| [23] | García Trillos, N.: Gromov-Hausdorff limit of Wasserstein spaces on point clouds. Calc. Var. Partial Differ. Equ. 59(2), 43 (2020) · Zbl 1436.49017 |
| [24] | Gigli, N.; Maas, J., Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45, 2, 879-899 (2013) · Zbl 1268.49054 · doi:10.1137/120886315 |
| [25] | Gladbach, P.; Kopfer, E.; Maas, J.; Portinale, L., Homogenisation of one-dimensional discrete optimal transport, J. Math. Pures Appl., 9, 139, 204-234 (2020) · Zbl 1440.49051 · doi:10.1016/j.matpur.2020.02.008 |
| [26] | Gladbach, P.; Kopfer, E.; Maas, J., Scaling limits of discrete optimal transport, SIAM J. Math. Anal., 52, 3, 2759-2802 (2020) · Zbl 1447.49062 · doi:10.1137/19M1243440 |
| [27] | Gladbach, P., Kopfer, E.: Limits of density-constrained optimal transport. Calc. Var. Partial Differential Equations. 61(2), (2022). doi:10.1007/s00526-021-02163-7 · Zbl 1491.49036 |
| [28] | Heida, M.; Patterson, RIA; Renger, DRM, Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space, J. Evol. Equ., 19, 1, 111-152 (2019) · Zbl 1508.46016 · doi:10.1007/s00028-018-0471-1 |
| [29] | Lisini, S.; Marigonda, A., On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals, Manuscr. Math., 133, 1-2, 197-224 (2010) · Zbl 1196.49034 · doi:10.1007/s00229-010-0371-3 |
| [30] | Maas, J., Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261, 8, 2250-2292 (2011) · Zbl 1237.60058 · doi:10.1016/j.jfa.2011.06.009 |
| [31] | Marcellini, P., Periodic solutions and homogenization of non linear variational problems, Annali di Matematica, 117, 1, 139-152 (1978) · Zbl 0395.49007 · doi:10.1007/BF02417888 |
| [32] | Mielke, A., A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24, 4, 1329-1346 (2011) · Zbl 1227.35161 · doi:10.1088/0951-7715/24/4/016 |
| [33] | Mielke, A., Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differ. Equ., 48, 1-2, 1-31 (2013) · Zbl 1282.60072 · doi:10.1007/s00526-012-0538-8 |
| [34] | Mielke, A., Roubíček, T.: Rate-Independent Systems. Applied Mathematical Sciences, vol. 193. Springer, New York (2015) · Zbl 1339.35006 |
| [35] | Peyré, G., Cuturi, M.: Computational optimal transport. Found. Trends® Mach. Learn. 11(5-6), 355-607 (2019) · Zbl 1475.68011 |
| [36] | Piatnitski, A.; Remy, E., Homogenization of elliptic difference operators, SIAM J. Math. Anal., 33, 1, 53-83 (2001) · Zbl 1097.39501 · doi:10.1137/S003614100033808X |
| [37] | Peletier, M., Rossi, R., Savaré, G., Tse, O.: Jump processes as generalized gradient flows. Calc. Var. Partial Differential Equations. 61(1), (2022). doi:10.1007/s00526-021-02130-2 · Zbl 1490.60022 |
| [38] | Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer-Verlag, Berlin (1998) · Zbl 0888.49001 |
| [39] | Santambrogio, F.: Optimal transport for applied mathematicians. Progress in Nonlinear Differential Equations and their Applications, vol. 87. Birkhäuser/Springer, Cham (2015) · Zbl 1401.49002 |
| [40] | Schmitzer, B.; Wirth, B., Dynamic models of Wasserstein-1-type unbalanced transport, ESAIM Control Optim. Calc. Var., 25, 54 (2019) · Zbl 1510.90206 · doi:10.1051/cocv/2018017 |
| [41] | Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003) · Zbl 1106.90001 |
| [42] | Villani, C.: Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (2009) · Zbl 1156.53003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
