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On gradient structures for Markov chains and the passage to Wasserstein gradient flows

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  • We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.
    Mathematics Subject Classification: 35K10, 35K20, 37L05, 49M25, 60F99, 65M08, 60J27, 70G75, 82B35.

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  • [1]

    S. Adams, N. Dirr, M. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: A new micro-macro passage, Communications in Mathematical Physics, 307 (2011), 791-815.doi: 10.1007/s00220-011-1328-4.

    [2]

    L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

    [3]

    S. Arnrich, A. Mielke, M. A. Peletier, G. Savaré and M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calc. Var. Part. Diff. Eqns., 44 (2012), 419-454.doi: 10.1007/s00526-011-0440-9.

    [4]

    J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.doi: 10.1007/s002110050002.

    [5]

    M. Bessemoulin-Chatard, A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme, Numer. Math., 121 (2012), 637-670.doi: 10.1007/s00211-012-0448-x.

    [6]

    A. Bradji and J. Fuhrmann, Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Appl. Math., 58 (2013), 1-38.doi: 10.1007/s10492-013-0001-y.

    [7]

    C. Chainais-Hillairet, M. Gisclon and A. Jüngel, A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors, Numer. Methods Partial Differential Equations, 27 (2011), 1483-1510.doi: 10.1002/num.20592.

    [8]

    S.-N. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Rational Mech. Anal., 203 (2012), 969-1008.doi: 10.1007/s00205-011-0471-6.

    [9]

    M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Rational Mech. Anal., 206 (2012), 997-1038.doi: 10.1007/s00205-012-0554-z.

    [10]

    M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, Discrete Contin. Dyn. Syst., 34 (2014), 1355-1374.doi: 10.3934/dcds.2014.34.1355.

    [11]

    R. Eymard, T. Gallouët and R. Herbin, The finite volume method, in Handbook of Numerical Analysis. Vol. VII, North Holland, Amsterdam, 2000, 713-1022.

    [12]

    R. Eymard and J.-M. Hérard, eds., Finite Volumes for Complex Applications V, ISTE, London, 2008.

    [13]

    J. Fořt, J. Fürst Jiří, H. R. Herbin and F. Hubert, eds., Finite Volumes for Complex Applications. VI. Problems & perspectives. Volume 1, 2, Springer Proceedings in Mathematics, 4, Springer, Heidelberg, 2011.doi: 10.1007/978-3-642-20671-9.

    [14]

    J. Fuhrmann, A. Linke and H. Langmach, A numerical method for mass conservative coupling between fluid flow and solute transport, Appl. Numer. Math., 61 (2011), 530-553.doi: 10.1016/j.apnum.2010.11.015.

    [15]

    K. Gärtner, Charge transport in semiconductors and a finite volume scheme, in Finite volumes for complex applications. VI. Problems & perspectives. Volume 1, 2, Springer Proc. Math., 4, Springer, Heidelberg, 2011, 513-522.doi: 10.1007/978-3-642-20671-9_54.

    [16]

    N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39 (2010), 101-120.doi: 10.1007/s00526-009-0303-9.

    [17]

    N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899.doi: 10.1137/120886315.

    [18]

    D. Hilhorst, H. C. V. Do and Y. Wang, A finite volume method for density driven flows in porous media, in CEMRACS'11: Multiscale Coupling of Complex Models in Scientific Computing, ESAIM Proc., 38, EDP Sci., Les Ulis, 2012, 376-386.doi: 10.1051/proc/201238021.

    [19]

    R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17.doi: 10.1137/S0036141096303359.

    [20]

    M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 919-942.doi: 10.1007/s00030-012-0189-7.

    [21]

    M. Liero and A. Mielke, Gradient structures and geodesic convexity for reaction-diffusion systems, Phil. Trans. Royal Soc. A, 371 (2013), 20120346, 28pp.doi: 10.1098/rsta.2012.0346.

    [22]

    M. Liero and U. Stefanelli, Weighted inertia-dissipation-energy functionals for semilinear equations, Boll. Unione Mat. Ital. (9), 6 (2013), 1-27.

    [23]

    J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292.doi: 10.1016/j.jfa.2011.06.009.

    [24]

    P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986.doi: 10.1007/978-3-7091-3678-2.

    [25]

    D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726.doi: 10.1051/m2an/2013126.

    [26]

    A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.doi: 10.1088/0951-7715/24/4/016.

    [27]

    A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Part. Diff. Eqns., 48 (2013), 1-31.doi: 10.1007/s00526-012-0538-8.

    [28]

    A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479-499.doi: 10.3934/dcdss.2013.6.479.

    [29]

    A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Part. Diff. Eqns., 31 (2008), 387-416.doi: 10.1007/s00526-007-0119-4.

    [30]

    A. Mielke and U. Stefanelli, Weighted energy-dissipation functionals for gradient flows, ESAIM Control Optim. Calc. Var., 17 (2011), 52-85.doi: 10.1051/cocv/2009043.

    [31]

    L. Onsager, Reciprocal relations in irreversible processes, I+II, Physical Review, 37 (1931), 405-426; Part II, 38, 2265-2267.

    [32]

    F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.doi: 10.1007/s002050050073.

    [33]

    F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.doi: 10.1081/PDE-100002243.

    [34]

    E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.doi: 10.1002/cpa.20046.

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