Variational approach to coarse-graining of generalized gradient flows. (English) Zbl 1444.35107
Summary: In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with non-dissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error.
As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.
As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.
MSC:
| 35K67 | Singular parabolic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 49S99 | Variational principles of physics |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 35K10 | Second-order parabolic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 60F10 | Large deviations |
| 70F40 | Problems involving a system of particles with friction |
Keywords:
variational technique; coarse-graining and passing to a limit; overdamped limit of Vlasov-Fokker-Planck equation; small-noise limit of randomly perturbed Hamiltonian systems; one and many degrees of freedomReferences:
| [1] | Aronson, D., Crandall, M.G., Peletier, L.A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6(10), 1001-1022 (1982) · Zbl 0518.35050 · doi:10.1016/0362-546X(82)90072-4 |
| [2] | Adams, S., Dirr, N., Peletier, M.A., Zimmer, J.: From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307, 791-815 (2011) · Zbl 1267.60106 · doi:10.1007/s00220-011-1328-4 |
| [3] | Adams, S., Dirr, N., Peletier, M.A., Zimmer, J.: Large deviations and gradient flows. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(2005), 20120341 (2013) · Zbl 1292.82023 · doi:10.1098/rsta.2012.0341 |
| [4] | Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser (2008) · Zbl 1145.35001 |
| [5] | Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482 (1992) · Zbl 0770.35005 · doi:10.1137/0523084 |
| [6] | Arnrich, S., Mielke, A., Peletier, M.A., Savaré, G., Veneroni, M.: Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction. Calc. Var. Part. Differ. Eq. 44, 419-454 (2012) · Zbl 1270.35055 · doi:10.1007/s00526-011-0440-9 |
| [7] | Ambrosio, L., Savaré, G., Zambotti, L.: Existence and stability for Fokker-Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145(3), 517-564 (2009) · Zbl 1235.60105 · doi:10.1007/s00440-008-0177-3 |
| [8] | Bonilla, L.L., Carrillo, J.A., Soler, J.: Asymptotic behavior of an initial-boundary value problem for the Vlasov-Poisson-Fokker-Planck system. SIAM J. Appl. Math. 57(5), 1343-1372 (1997) · Zbl 0888.35018 · doi:10.1137/S0036139995291544 |
| [9] | Budhiraja, A., Dupuis, P., Fischer, M.: Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Prob. 40(1), 74-102 (2012) · Zbl 1242.60026 · doi:10.1214/10-AOP616 |
| [10] | Brezis, H., Ekeland, I.: Un principe variationnel associé à certaines equations paraboliques. Le cas indépendant du temps. Comptes Rendus de l’Acad. des Sci. de Paris Série A 282, 971-974 (1976) · Zbl 0332.49032 |
| [11] | Bakry, D., Gentil, I., Ledoux, M., et al.: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften 348. Springer, Cham (2014) · Zbl 1376.60002 |
| [12] | Bertsch, M., Kersner, R., Peletier, L.A.: Positivity versus localization in degenerate diffusion equations. Nonlinear Anal. 9(9), 987-1008 (1985) · Zbl 0596.35073 · doi:10.1016/0362-546X(85)90081-1 |
| [13] | Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker-Planck-Kolmogorov Equations, vol. 207. American Mathematical Soc, Rhode Island, Providence (2015) · Zbl 1342.35002 |
| [14] | Bouchut, F.: Hypoelliptic regularity in kinetic equations. J. de Math. Pures et Appl. 81(11), 1135-1159 (2002) · Zbl 1045.35093 · doi:10.1016/S0021-7824(02)01264-3 |
| [15] | Barret, F., von Renesse, M.: Averaging principle for diffusion processes via Dirichlet forms. Potential Anal. 41(4), 1033-1063 (2014) · Zbl 1317.60095 · doi:10.1007/s11118-014-9405-x |
| [16] | Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. Comptes Rendus Math. 335(1), 99-104 (2002) · Zbl 1001.49016 · doi:10.1016/S1631-073X(02)02429-9 |
| [17] | Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40, 1585 (2008) · Zbl 1167.49013 · doi:10.1137/080713148 |
| [18] | Cerrai, S., Freidlin, M.: On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom. Probab. Theory Relat. Fields 135(3), 363-394 (2006) · Zbl 1093.60036 · doi:10.1007/s00440-005-0465-0 |
| [19] | Crandall, M., Ishii, H., Lions, P.: User’s guide to viscosity solutions of second order partial differential equations. Am. Math. Soc 27, 1-67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 |
| [20] | Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations, vol. 902. John Wiley and & Sons, New York (1997) · Zbl 0904.60001 · doi:10.1002/9781118165904 |
| [21] | Degond, P.: Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in \[1 1\] and \[2 2\] space dimensions. Ann. Sci. de l’École Normale Supérieure 19(4), 519-542 (1986) · Zbl 0619.35087 · doi:10.24033/asens.1516 |
| [22] | Dawson, D.A., Gartner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247-308 (1987) · Zbl 0613.60021 · doi:10.1080/17442508708833446 |
| [23] | Duong, M.H., Lamacz, A., Peletier, M.A., Schlichting, A., Sharma, U.: Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics. (in preparation) · Zbl 1394.35210 |
| [24] | Dirr, N., Laschos, V., Zimmer, J.: Upscaling from particle models to entropic gradient flows. J. Math. Phys. 53(6), 063704 (2012) · Zbl 1280.35009 · doi:10.1063/1.4726509 |
| [25] | de Pablo, J.J., Curtin, W.A.: Multiscale modeling in advanced materials research: challenges, novel methods, and emerging applications. Mrs Bull. 32(11), 905-911 (2007) · doi:10.1557/mrs2007.187 |
| [26] | Duong, M.H., Peletier, M.A., Zimmer, J.: GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles. Nonlinearity 26, 2951-2971 (2013) · Zbl 1288.60029 · doi:10.1088/0951-7715/26/11/2951 |
| [27] | Duong, M.H., Peletier, M.A., Zimmer, J.: Conservative-dissipative approximation schemes for a generalized Kramers equation. Math. Methods Appl. Sci. 37(16), 2517-2540 (2014) · Zbl 1370.35165 · doi:10.1002/mma.2994 |
| [28] | Daneri, S., Savaré, G.: Lecture notes on gradient flows and optimal transport. arXiv preprint arXiv:1009.3737, (2010) · Zbl 1333.49001 |
| [29] | Eidus, D.: The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium. J. Differ. Equ. 84, 309-318 (1990) · Zbl 0707.35074 · doi:10.1016/0022-0396(90)90081-Y |
| [30] | Feller, W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55(3), 468-519 (1952) · Zbl 0047.09303 · doi:10.2307/1969644 |
| [31] | Frank, J., Gottwald, G.A.: The Langevin limit of the Nosé-Hoover-Langevin thermostat. J. Stat. Phys. 143(4), 715-724 (2011) · Zbl 1218.82015 · doi:10.1007/s10955-011-0203-1 |
| [32] | Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes, Volume 131 of Mathematical Surveys and Monographs. American Mathematical Society, Rhode Island, Providence (2006) · Zbl 1113.60002 |
| [33] | Fleming, G., Ratner, M. (eds). Directing Matter and Energy: Five Challenges for Science and the Imagination. Basic Energy Sciences Advisory Committee (2007) · Zbl 0795.60042 |
| [34] | Freidlin, M.: Some remarks on the Smoluchowski-Kramers approximation. J. Stat. Phys. 117(3-4), 617-634 (2004) · Zbl 1113.82055 · doi:10.1007/s10955-004-2273-9 |
| [35] | Funaki, T.: A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrscheinlichkeitstheorie und Verwandte Gebiete 67(3), 331-348 (1984) · Zbl 0546.60081 · doi:10.1007/BF00535008 |
| [36] | Freidlin, M.I., Wentzell, A.D.: Diffusion processes on graphs and the averaging principle. Ann. Prob. 21(4), 2215-2245 (1993) · Zbl 0795.60042 · doi:10.1214/aop/1176989018 |
| [37] | Freidlin, M.I., Wentzell, A.D.: Random perturbations of Hamiltonian systems. Mem. Am. Math. Soc. 109, 523 (1994) · Zbl 0804.60070 |
| [38] | Freidlin, M., Weber, M.: Random perturbations of nonlinear oscillators. Ann. Prob. 26(3), 925-967 (1998) · Zbl 0935.60038 · doi:10.1214/aop/1022855739 |
| [39] | Freidlin, M., Weber, M.: On random perturbations of Hamiltonian systems with many degrees of freedom. Stoch. Process. Appl. 94(2), 199-239 (2001) · Zbl 1053.60058 · doi:10.1016/S0304-4149(01)00083-7 |
| [40] | Freidlin, M.I., Wentzell, A.D.: Diffusion processes on an open book and the averaging principle. Stoch. Process. Appl. 113(1), 101-126 (2004) · Zbl 1075.60100 · doi:10.1016/j.spa.2004.03.009 |
| [41] | Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer, New York (2009) · Zbl 1357.49004 |
| [42] | Grunewald, N., Otto, F., Villani, C., Westdickenberg, M.G.: A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. H. Poincaré Probab. Stat. 45(2), 302-351 (2009) · Zbl 1179.60068 · doi:10.1214/07-AIHP200 |
| [43] | Hottovy, S., Volpe, G., Wehr, J.: Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit. J. Stat. Phys. 146(4), 762-773 (2012) · Zbl 1245.82052 · doi:10.1007/s10955-012-0418-9 |
| [44] | Ishii, H., Souganidis, P.E.: A pde approach to small stochastic perturbations of Hamiltonian flows. J. Differ. Eq. 252(2), 1748-1775 (2012) · Zbl 1287.37036 · doi:10.1016/j.jde.2011.08.036 |
| [45] | Kramers, H.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284-304 (1940) · Zbl 0061.46405 · doi:10.1016/S0031-8914(40)90098-2 |
| [46] | Kružkov, S.N.: First order quasilinear equations in several independent variables. Mat. USSR Sb. 10(2), 217-243 (1970) · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156 |
| [47] | Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Classics in Applied Mathematics (2000) · Zbl 0988.49003 · doi:10.1137/1.9780898719451 |
| [48] | Lions, J.L.: Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer, Berlin (1961) · Zbl 0098.31101 |
| [49] | Legoll, F., Lelièvre, T.: Effective dynamics using conditional expectations. Nonlinearity 23(9), 2131 (2010) · Zbl 1209.60036 · doi:10.1088/0951-7715/23/9/006 |
| [50] | Mandl, P.: Analytical Treatment of One-Dimensional Markov Processes. Academia, Publishing House of the Czechoslovak Academy of Sciences (1968) · Zbl 0179.47802 |
| [51] | Mielke, A.: On evolutionary gamma-convergence for gradient systems. Technical Report 1915, WIAS, Berlin (2014) · Zbl 1288.60029 |
| [52] | Mielke, A., Peletier, M.A., Renger, D.R.M.: On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion. Potential Anal. 41(4), 1293-1327 (2014) · Zbl 1304.35692 · doi:10.1007/s11118-014-9418-5 |
| [53] | Mielke, A., Roubíček, T., Stefanelli, \[U.: \Gamma\] Γ-limits and relaxations for rate-independent evolutionary problems. Cal. Var. Partial Differ. Equ. 31(3), 387-416 (2008) · Zbl 1302.49013 · doi:10.1007/s00526-007-0119-4 |
| [54] | Mielke, A., Rossi, R., Savaré, G.: Variational convergence of gradient flows and rate-independent evolutions in metric spaces. Milan J. Math. 80(2), 381-410 (2012) · Zbl 1255.49078 · doi:10.1007/s00032-012-0190-y |
| [55] | Murat, F.: A survey on compensated compactness. Contrib. Modern Cal. Var. 148, 145-183 (1987) |
| [56] | Narita, K.: Asymptotic behavior of fluctuation and deviation from limit system in the Smoluchowski-Kramers approximation for SDE. Yokohama Math. J. 42(1), 41-76 (1994) · Zbl 0814.60054 |
| [57] | Nayroles, B.: Deux théoremes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Ser. A B 282, A1035-A1038 (1976) · Zbl 0345.73037 |
| [58] | Nelson, E.: Dynamical Theories of Brownian Motion, vol. 17. Princeton University Press, Princeton (1967) · Zbl 0165.58502 |
| [59] | Nicolis, G., Nicolis, C.: Foundations of Complex Systems: Emergence, Information and Predicition. World Scientific, Singapore (2012) · Zbl 1260.37001 · doi:10.1142/8260 |
| [60] | Oelschläger, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Prob. 12(2), 458-479 (1984) · Zbl 0544.60097 · doi:10.1214/aop/1176993301 |
| [61] | Ottobre, M., Pavliotis, G.A.: Asymptotic analysis for the generalized Langevin equation. Nonlinearity 24(5), 1629-1653 (2011) · Zbl 1221.82076 · doi:10.1088/0951-7715/24/5/013 |
| [62] | Öttinger, H.: Beyond Equilibrium Thermodynamics. Wiley, Hoboken (2005) · doi:10.1002/0471727903 |
| [63] | Peletier, M.A., Duong, M.H., Sharma, U.: Coarse-graining and fluctuations: two birds with one stone. In: Oberwolfach Reports, vol. 10(4) (2013) · Zbl 1093.60036 |
| [64] | Pennacchio, M., Savaré, G., Colli Franzone, P.: Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37(4), 1333-1370 (2005) · Zbl 1113.35019 · doi:10.1137/040615249 |
| [65] | Rosenau, P., Kamin, S.: Non-linear diffusion in a finite mass medium. Commun. Pure Appl. Math. 35, 113-127 (1982) · Zbl 0469.35060 · doi:10.1002/cpa.3160350106 |
| [66] | Stainforth, D.A., Allen, M.R., Tredger, E.R., Smith, L.A.: Confidence, uncertainty and decision-support relevance in climate predictions. Philos. Trans. A 365(1857), 2145 (2007) · doi:10.1098/rsta.2007.2074 |
| [67] | Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. A 31(4), 1427-1451 (2011) · Zbl 1239.35015 · doi:10.3934/dcds.2011.31.1427 |
| [68] | Sharma, U.: Coarse-Graining of Fokker-Planck Equations. Ph.D. thesis, Eindhoven University of Technology (2017) · Zbl 1065.49011 |
| [69] | Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1994) · Zbl 0807.35002 · doi:10.1007/978-1-4612-0873-0 |
| [70] | Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. 57(12), 1627-1672 (2004) · Zbl 1065.49011 · doi:10.1002/cpa.20046 |
| [71] | Stefanelli, U.: The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47, 1615 (2008) · Zbl 1194.35214 · doi:10.1137/070684574 |
| [72] | Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Anal. Mech. Heriot Watt Symp. 4, 136-212 (1979) · Zbl 0437.35004 |
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