American Institute of Mathematical Sciences

Advanced
April  2022, 17(2): 227-254. doi: 10.3934/nhm.2022004

Stochastic two-scale convergence and Young measures

Martin Heida 1,2, , Stefan Neukamm 3,, and  Mario Varga 3,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
2. 

Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching bei München, Germany
3. 

Fakultät Mathematik, Technische Universität Dresden, 01062 Dresden, Germany

*Corresponding author: Stefan Neukamm

Received  June 2021 Revised  January 2022 Published  April 2022 Early access  February 2022

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.

Keywords: stochastic homogenization, unfolding, two-scale convergence, Young measures, weak convergence.
Mathematics Subject Classification: Primary: 74Q05, 47J30; Secondary: 49J45.
Citation: Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17 (2) : 227-254. doi: 10.3934/nhm.2022004
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

K. T. Andrews and S. Wright, Stochastic homogenization of elliptic boundary-value problems with $L^p$-data, Asymptot. Anal., 17 (1998), 165-184. 

[3]

T. Arbogast, J. Douglas, Jr and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.

[4]

E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), 570-598.  doi: 10.1137/0322035.

[5]

A. Bourgeat, S. Luckhaus and A. Mikelić, A rigorous result for a double porosity model of immiscible two-phase flow, Comptes Rendusa l'Académie des Sciences, 320 (1994), 1289–1294.

[6]

A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. 

[7]

D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math., 339 (2004), 77-82.  doi: 10.1016/j.crma.2004.03.028.

[8]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.

[9]

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math., 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[10]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.

[11]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization., Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.

[12]

D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics. Springer-Verlag, New York, 1988.

[13]

J. Fischer and S. Neukamm, Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems, Arch. Ration. Mech. Anal., 242 (2021), 343-452.  doi: 10.1007/s00205-021-01686-9.

[14]

G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286. 

[15]

H. Hanke, Homogenization in gradient plasticity, Math. Models Methods Appl. Sci., 21 (2011), 1651-1684.  doi: 10.1142/S0218202511005520.

[16]

M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.  doi: 10.3233/ASY-2010-1022.

[17]

M. Heida, Stochastic homogenization of rate-independent systems and applications, Contin. Mech. Thermodyn., 29 (2017), 853-894.  doi: 10.1007/s00161-017-0564-z.

[18]

M. Heida and S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptot. Anal., 112 (2019), 185-212.  doi: 10.3233/ASY-181502.

[19]

M. Heida, S. Neukamm and M. Varga, Stochastic homogenization of $\Lambda$-convex gradient flows, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 427-453.  doi: 10.3934/dcdss.2020328.

[20]

H. Hoppe, Homogenization of Rapidly Oscillating Riemannian Manifolds, Dissertation, TU Dresden, 2020, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-743766.

[21]

H. Hoppe, S. Neukamm and M. Schäffner, Stochastic homogenization of non-convex integral functionals with degenerate growth, (in preparation), 2021.

[22]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[23]

S. M. Kozlov, Averaging of random operators, Mat. Sb., 109 (1979), 188-202. 

[24]

M. Liero and S. Reichelt, Homogenization of Cahn–Hilliard-type equations via evolutionary $\Gamma$-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.

[25]

D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. 

[26]

A. Mielke, S. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.  doi: 10.3934/nhm.2014.9.353.

[27]

A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.  doi: 10.1137/060672790.

[28]

S. Neukamm, Homogenization, linearization and dimension reduction in elasticity with variational methods, Technische Universität München, 2010.

[29]

S. Neukamm, M. Schäffner and A. Schlömerkemper, Stochastic homogenization of nonconvex discrete energies with degenerate growth, SIAM J. Math. Anal., 49 (2017), 1761-1809.  doi: 10.1137/16M1097705.

[30]

S. Neukamm and M. Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.  doi: 10.1137/17M1141230.

[31]

S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, Asymptot. Anal., 125 (2021), 247-287. 

[32]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.

[33]

G. C. Papanicolaou and S. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, 1 (1979), 835-873. 

[34]

M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342.

[35]

A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.  doi: 10.1051/cocv:2006012.

[36]

C. Vogt, A homogenization theorem leading to a Volterra-integrodifferential equation for permeation chromotography, Preprint No 155, SFB 123, Heidelberg, 1982.

[37]

V. V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sb. Math., 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

[38]

V. V. Zhikov and A. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.  doi: 10.1070/IM2006v070n01ABEH002302.

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

K. T. Andrews and S. Wright, Stochastic homogenization of elliptic boundary-value problems with $L^p$-data, Asymptot. Anal., 17 (1998), 165-184. 

[3]

T. Arbogast, J. Douglas, Jr and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.

[4]

E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), 570-598.  doi: 10.1137/0322035.

[5]

A. Bourgeat, S. Luckhaus and A. Mikelić, A rigorous result for a double porosity model of immiscible two-phase flow, Comptes Rendusa l'Académie des Sciences, 320 (1994), 1289–1294.

[6]

A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. 

[7]

D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math., 339 (2004), 77-82.  doi: 10.1016/j.crma.2004.03.028.

[8]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.

[9]

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math., 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[10]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.

[11]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization., Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.

[12]

D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics. Springer-Verlag, New York, 1988.

[13]

J. Fischer and S. Neukamm, Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems, Arch. Ration. Mech. Anal., 242 (2021), 343-452.  doi: 10.1007/s00205-021-01686-9.

[14]

G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286. 

[15]

H. Hanke, Homogenization in gradient plasticity, Math. Models Methods Appl. Sci., 21 (2011), 1651-1684.  doi: 10.1142/S0218202511005520.

[16]

M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.  doi: 10.3233/ASY-2010-1022.

[17]

M. Heida, Stochastic homogenization of rate-independent systems and applications, Contin. Mech. Thermodyn., 29 (2017), 853-894.  doi: 10.1007/s00161-017-0564-z.

[18]

M. Heida and S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptot. Anal., 112 (2019), 185-212.  doi: 10.3233/ASY-181502.

[19]

M. Heida, S. Neukamm and M. Varga, Stochastic homogenization of $\Lambda$-convex gradient flows, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 427-453.  doi: 10.3934/dcdss.2020328.

[20]

H. Hoppe, Homogenization of Rapidly Oscillating Riemannian Manifolds, Dissertation, TU Dresden, 2020, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-743766.

[21]

H. Hoppe, S. Neukamm and M. Schäffner, Stochastic homogenization of non-convex integral functionals with degenerate growth, (in preparation), 2021.

[22]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[23]

S. M. Kozlov, Averaging of random operators, Mat. Sb., 109 (1979), 188-202. 

[24]

M. Liero and S. Reichelt, Homogenization of Cahn–Hilliard-type equations via evolutionary $\Gamma$-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.

[25]

D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. 

[26]

A. Mielke, S. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.  doi: 10.3934/nhm.2014.9.353.

[27]

A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.  doi: 10.1137/060672790.

[28]

S. Neukamm, Homogenization, linearization and dimension reduction in elasticity with variational methods, Technische Universität München, 2010.

[29]

S. Neukamm, M. Schäffner and A. Schlömerkemper, Stochastic homogenization of nonconvex discrete energies with degenerate growth, SIAM J. Math. Anal., 49 (2017), 1761-1809.  doi: 10.1137/16M1097705.

[30]

S. Neukamm and M. Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.  doi: 10.1137/17M1141230.

[31]

S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, Asymptot. Anal., 125 (2021), 247-287. 

[32]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.

[33]

G. C. Papanicolaou and S. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, 1 (1979), 835-873. 

[34]

M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342.

[35]

A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.  doi: 10.1051/cocv:2006012.

[36]

C. Vogt, A homogenization theorem leading to a Volterra-integrodifferential equation for permeation chromotography, Preprint No 155, SFB 123, Heidelberg, 1982.

[37]

V. V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sb. Math., 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

[38]

V. V. Zhikov and A. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.  doi: 10.1070/IM2006v070n01ABEH002302.

[1]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[2]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[3]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks and Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[4]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[5]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure and Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[6]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
[7]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[8]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[9]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
[10]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[11]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic and Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[12]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic and Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[13]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143
[14]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic and Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65
[15]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[16]

Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108
[17]

Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016
[18]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
[19]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[20]

Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks and Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503

2021 Impact Factor: 1.41

Tools

Metrics

  • PDF downloads (128)
  • HTML views (89)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy