February  2015, 8(1): 1-27. doi: 10.3934/dcdss.2015.8.1

Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem

1. 

Université Paris-Est, Institut Navier, LAMI, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2

2. 

Université Paris-Est, CERMICS, École des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France

Received  February 2013 Revised  June 2013 Published  July 2014

We consider a nonlinear convex stochastic homogenization problem, in a stationary setting. In practice, the deterministic homogenized energy density is approximated by a random apparent energy density, obtained by solving the corrector problem on a truncated domain.
    We show that the technique of antithetic variables can be used to reduce the variance of the computed quantities, and thereby decrease the computational cost at equal accuracy. This leads to an efficient approach for approximating expectations of the apparent homogenized energy density and of related quantities.
    The efficiency of the approach is numerically illustrated on several test cases. Some elements of analysis are also provided.
Citation: Frédéric Legoll, William Minvielle. Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 1-27. doi: 10.3934/dcdss.2015.8.1
References:
[1]

A. Abdulle and G. Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems,, Numer. Math., 121 (2012), 397. doi: 10.1007/s00211-011-0438-4. Google Scholar

[2]

A. Anantharaman, R. Costaouec, C. Le Bris, F. Legoll and F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges: Some recent developments,, W. Bao and Q. Du eds., 22 (2011), 197. doi: 10.1142/9789814360906_0004. Google Scholar

[3]

S. N. Armstrong, P. Cardaliaguet and P. E. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations,, J. Amer. Math. Soc., 27 (2014), 479. doi: 10.1090/S0894-0347-2014-00783-9. Google Scholar

[4]

J. W. Barrett and W. B. Liu, Finite Element approximation of the p-Laplacian,, Maths. of Comp., 61 (1993), 523. doi: 10.2307/2153239. Google Scholar

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, Studies in Mathematics and its Applications, (1978). Google Scholar

[6]

X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization: The technique of antithetic variables,, in Numerical Analysis and Multiscale Computations (eds. B. Engquist, (2012), 47. doi: 10.1007/978-3-642-21943-6_3. Google Scholar

[7]

X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization using antithetic variables,, Markov Processes and Related Fields, 18 (2012), 31. Google Scholar

[8]

A. Bourgeat and A. Piatnitski, Approximation of effective coefficients in stochastic homogenization,, Ann I. H. Poincaré - PR, 40 (2004), 153. doi: 10.1016/j.anihpb.2003.07.003. Google Scholar

[9]

S.-S. Chow, Finite Element error estimates for nonlinear elliptic equations of monotone type,, Numer. Math., 54 (1989), 373. doi: 10.1007/BF01396320. Google Scholar

[10]

D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999). Google Scholar

[11]

R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization: Proof of concept, using antithetic variables,, Boletin Soc. Esp. Mat. Apl., 50 (2010), 9. Google Scholar

[12]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization,, Annali di Matematica Pura ed Applicata, 144 (1986), 347. doi: 10.1007/BF01760826. Google Scholar

[13]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic-theory,, J. Reine Angewandte Mathematik, 368 (1986), 28. Google Scholar

[14]

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization,, Acta Numerica, 17 (2008), 147. doi: 10.1017/S0962492906360011. Google Scholar

[15]

A. Gloria and S. Neukamm, Commutability of homogenization and linearization at identity in finite elasticity and applications,, Ann. I. H. Poincaré- AN, 28 (2011), 941. doi: 10.1016/j.anihpc.2011.07.002. Google Scholar

[16]

A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations,, Ann. Appl. Probab., 22 (2012), 1. doi: 10.1214/10-AAP745. Google Scholar

[17]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la réesolution, par péenalisation-dualité d'une classe de problèmes de Dirichlet non linéaires,, RAIRO Anal. Numér., 9 (1975), 41. Google Scholar

[18]

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

[19]

T. Kanit, S. Forest, I. Galliet, V. Mounoury and D. Jeulin, Determination of the size of the representative volume element for random composites: Statistical and numerical approach,, International Journal of Solids and Structures, 40 (2003), 3647. doi: 10.1016/S0020-7683(03)00143-4. Google Scholar

[20]

U. Krengel, Ergodic Theorems,, de Gruyter Studies in Mathematics, (1985). doi: 10.1515/9783110844641. Google Scholar

[21]

C. Le Bris, Some numerical approaches for "weakly'' random homogenization,, in Numerical Mathematics and Advanced Applications (eds. G. Kreiss, (2009), 29. Google Scholar

[22]

F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem,, preprint, (). Google Scholar

[23]

P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity,, in Handbook of Numerical Analysis, (1994), 465. doi: 10.1016/S1570-8659(05)80018-3. Google Scholar

[24]

J. S. Liu, Monte-Carlo Strategies in Scientific Computing,, Springer Series in Statistics, (2001). Google Scholar

[25]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials,, Arch. Rational Mech. Anal., 99 (1987), 189. doi: 10.1007/BF00284506. Google Scholar

[26]

A. N. Shiryaev, Probability,, Graduate Texts in Mathematics, (1984). doi: 10.1007/978-1-4899-0018-0. Google Scholar

[27]

L. Tartar, Estimations of homogenized coefficients,, in Topics in the Mathematical Modelling of Composite Materials (eds. A. Cherkaev and R. Kohn), 31 (1987), 9. doi: 10.1007/978-1-4612-2032-9_2. Google Scholar

[28]

A. A. Tempel'man, Ergodic theorems for general dynamical systems,, Trudy Moskov. Mat. Obsc., 26 (1972), 95. Google Scholar

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems,, Springer Series in Computational Mathematics, (2006). Google Scholar

show all references

References:
[1]

A. Abdulle and G. Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems,, Numer. Math., 121 (2012), 397. doi: 10.1007/s00211-011-0438-4. Google Scholar

[2]

A. Anantharaman, R. Costaouec, C. Le Bris, F. Legoll and F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges: Some recent developments,, W. Bao and Q. Du eds., 22 (2011), 197. doi: 10.1142/9789814360906_0004. Google Scholar

[3]

S. N. Armstrong, P. Cardaliaguet and P. E. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations,, J. Amer. Math. Soc., 27 (2014), 479. doi: 10.1090/S0894-0347-2014-00783-9. Google Scholar

[4]

J. W. Barrett and W. B. Liu, Finite Element approximation of the p-Laplacian,, Maths. of Comp., 61 (1993), 523. doi: 10.2307/2153239. Google Scholar

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, Studies in Mathematics and its Applications, (1978). Google Scholar

[6]

X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization: The technique of antithetic variables,, in Numerical Analysis and Multiscale Computations (eds. B. Engquist, (2012), 47. doi: 10.1007/978-3-642-21943-6_3. Google Scholar

[7]

X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization using antithetic variables,, Markov Processes and Related Fields, 18 (2012), 31. Google Scholar

[8]

A. Bourgeat and A. Piatnitski, Approximation of effective coefficients in stochastic homogenization,, Ann I. H. Poincaré - PR, 40 (2004), 153. doi: 10.1016/j.anihpb.2003.07.003. Google Scholar

[9]

S.-S. Chow, Finite Element error estimates for nonlinear elliptic equations of monotone type,, Numer. Math., 54 (1989), 373. doi: 10.1007/BF01396320. Google Scholar

[10]

D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999). Google Scholar

[11]

R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization: Proof of concept, using antithetic variables,, Boletin Soc. Esp. Mat. Apl., 50 (2010), 9. Google Scholar

[12]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization,, Annali di Matematica Pura ed Applicata, 144 (1986), 347. doi: 10.1007/BF01760826. Google Scholar

[13]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic-theory,, J. Reine Angewandte Mathematik, 368 (1986), 28. Google Scholar

[14]

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization,, Acta Numerica, 17 (2008), 147. doi: 10.1017/S0962492906360011. Google Scholar

[15]

A. Gloria and S. Neukamm, Commutability of homogenization and linearization at identity in finite elasticity and applications,, Ann. I. H. Poincaré- AN, 28 (2011), 941. doi: 10.1016/j.anihpc.2011.07.002. Google Scholar

[16]

A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations,, Ann. Appl. Probab., 22 (2012), 1. doi: 10.1214/10-AAP745. Google Scholar

[17]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la réesolution, par péenalisation-dualité d'une classe de problèmes de Dirichlet non linéaires,, RAIRO Anal. Numér., 9 (1975), 41. Google Scholar

[18]

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

[19]

T. Kanit, S. Forest, I. Galliet, V. Mounoury and D. Jeulin, Determination of the size of the representative volume element for random composites: Statistical and numerical approach,, International Journal of Solids and Structures, 40 (2003), 3647. doi: 10.1016/S0020-7683(03)00143-4. Google Scholar

[20]

U. Krengel, Ergodic Theorems,, de Gruyter Studies in Mathematics, (1985). doi: 10.1515/9783110844641. Google Scholar

[21]

C. Le Bris, Some numerical approaches for "weakly'' random homogenization,, in Numerical Mathematics and Advanced Applications (eds. G. Kreiss, (2009), 29. Google Scholar

[22]

F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem,, preprint, (). Google Scholar

[23]

P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity,, in Handbook of Numerical Analysis, (1994), 465. doi: 10.1016/S1570-8659(05)80018-3. Google Scholar

[24]

J. S. Liu, Monte-Carlo Strategies in Scientific Computing,, Springer Series in Statistics, (2001). Google Scholar

[25]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials,, Arch. Rational Mech. Anal., 99 (1987), 189. doi: 10.1007/BF00284506. Google Scholar

[26]

A. N. Shiryaev, Probability,, Graduate Texts in Mathematics, (1984). doi: 10.1007/978-1-4899-0018-0. Google Scholar

[27]

L. Tartar, Estimations of homogenized coefficients,, in Topics in the Mathematical Modelling of Composite Materials (eds. A. Cherkaev and R. Kohn), 31 (1987), 9. doi: 10.1007/978-1-4612-2032-9_2. Google Scholar

[28]

A. A. Tempel'man, Ergodic theorems for general dynamical systems,, Trudy Moskov. Mat. Obsc., 26 (1972), 95. Google Scholar

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems,, Springer Series in Computational Mathematics, (2006). Google Scholar

[1]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

[2]

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

[3]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645

[4]

Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354

[5]

Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295

[6]

Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49

[7]

Liselott Flodén, Jens Persson. Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales. Networks & Heterogeneous Media, 2016, 11 (4) : 627-653. doi: 10.3934/nhm.2016012

[8]

Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181

[9]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189

[10]

Tasnim Fatima, Ekeoma Ijioma, Toshiyuki Ogawa, Adrian Muntean. Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers. Networks & Heterogeneous Media, 2014, 9 (4) : 709-737. doi: 10.3934/nhm.2014.9.709

[11]

Yan Zeng, Zhongfei Li, Jingjun Liu. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. Journal of Industrial & Management Optimization, 2010, 6 (3) : 483-496. doi: 10.3934/jimo.2010.6.483

[12]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977

[13]

Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks & Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523

[14]

Ming-Zheng Wang, M. Montaz Ali. Penalty-based SAA method of stochastic nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 241-257. doi: 10.3934/jimo.2010.6.241

[15]

Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks & Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002

[16]

Hakima Bessaih, Yalchin Efendiev, Razvan Florian Maris. Stochastic homogenization for a diffusion-reaction model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5403-5429. doi: 10.3934/dcds.2019221

[17]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[18]

Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049

[19]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[20]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]