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On gradient structures for Markov chains and the passage to Wasserstein gradient flows
1. | Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany |
2. | Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany |
References:
[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. |
show all references
References:
[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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