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 and 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), 397-431. doi: 10.1007/s00211-011-0438-4.

[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., Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 22 (2011), 197-272. doi: 10.1142/9789814360906_0004.

[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), 479-540. doi: 10.1090/S0894-0347-2014-00783-9.

[4]

J. W. Barrett and W. B. Liu, Finite Element approximation of the p-Laplacian, Maths. of Comp., 61 (1993), 523-537. doi: 10.2307/2153239.

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, Vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978.

[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, O. Runborg and R. Tsai), Lect. Notes Comput. Sci. Eng., 82, Springer, 2012, 47-70. doi: 10.1007/978-3-642-21943-6_3.

[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), 31-66. Preliminary version available from: http://cermics.enpc.fr/~legoll/hdr/FL24.pdf.

[8]

A. Bourgeat and A. Piatnitski, Approximation of effective coefficients in stochastic homogenization, Ann I. H. Poincaré - PR, 40 (2004), 153-165. doi: 10.1016/j.anihpb.2003.07.003.

[9]

S.-S. Chow, Finite Element error estimates for nonlinear elliptic equations of monotone type, Numer. Math., 54 (1989), 373-393. doi: 10.1007/BF01396320.

[10]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, Vol. 17, Oxford University Press, New York, 1999.

[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), 9-26.

[12]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Annali di Matematica Pura ed Applicata, 144 (1986), 347-389. doi: 10.1007/BF01760826.

[13]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic-theory, J. Reine Angewandte Mathematik, 368 (1986), 28-42.

[14]

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numerica, 17 (2008), 147-190. doi: 10.1017/S0962492906360011.

[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), 941-964. doi: 10.1016/j.anihpc.2011.07.002.

[16]

A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., 22 (2012), 1-28. doi: 10.1214/10-AAP745.

[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), 41-76.

[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.

[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), 3647-3679. doi: 10.1016/S0020-7683(03)00143-4.

[20]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, Vol. 6, de Gruyter, 1985. doi: 10.1515/9783110844641.

[21]

C. Le Bris, Some numerical approaches for "weakly'' random homogenization, in Numerical Mathematics and Advanced Applications (eds. G. Kreiss, P. Lötstedt, A. Malqvist and M. Neytcheva), Proceedings of ENUMATH 2009, Lect. Notes Comput. Sci. Eng., Springer, 2010, 29-45.

[22]

F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem, preprint, arXiv:1302.0038.

[23]

P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis, Vol. 3 (eds. P. Ciarlet and J.-L. Lions), North Holland, Amsterdam, 1994, 465-624. doi: 10.1016/S1570-8659(05)80018-3.

[24]

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

[25]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.

[26]

A. N. Shiryaev, Probability, Graduate Texts in Mathematics, Vol. 95, Springer, 1984. doi: 10.1007/978-1-4899-0018-0.

[27]

L. Tartar, Estimations of homogenized coefficients, in Topics in the Mathematical Modelling of Composite Materials (eds. A. Cherkaev and R. Kohn), Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, 31 (1987), 9-20. doi: 10.1007/978-1-4612-2032-9_2.

[28]

A. A. Tempel'man, Ergodic theorems for general dynamical systems, Trudy Moskov. Mat. Obsc., 26 (1972), 95-132.

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, 2006.

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), 397-431. doi: 10.1007/s00211-011-0438-4.

[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., Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 22 (2011), 197-272. doi: 10.1142/9789814360906_0004.

[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), 479-540. doi: 10.1090/S0894-0347-2014-00783-9.

[4]

J. W. Barrett and W. B. Liu, Finite Element approximation of the p-Laplacian, Maths. of Comp., 61 (1993), 523-537. doi: 10.2307/2153239.

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, Vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978.

[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, O. Runborg and R. Tsai), Lect. Notes Comput. Sci. Eng., 82, Springer, 2012, 47-70. doi: 10.1007/978-3-642-21943-6_3.

[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), 31-66. Preliminary version available from: http://cermics.enpc.fr/~legoll/hdr/FL24.pdf.

[8]

A. Bourgeat and A. Piatnitski, Approximation of effective coefficients in stochastic homogenization, Ann I. H. Poincaré - PR, 40 (2004), 153-165. doi: 10.1016/j.anihpb.2003.07.003.

[9]

S.-S. Chow, Finite Element error estimates for nonlinear elliptic equations of monotone type, Numer. Math., 54 (1989), 373-393. doi: 10.1007/BF01396320.

[10]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, Vol. 17, Oxford University Press, New York, 1999.

[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), 9-26.

[12]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Annali di Matematica Pura ed Applicata, 144 (1986), 347-389. doi: 10.1007/BF01760826.

[13]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic-theory, J. Reine Angewandte Mathematik, 368 (1986), 28-42.

[14]

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numerica, 17 (2008), 147-190. doi: 10.1017/S0962492906360011.

[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), 941-964. doi: 10.1016/j.anihpc.2011.07.002.

[16]

A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., 22 (2012), 1-28. doi: 10.1214/10-AAP745.

[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), 41-76.

[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.

[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), 3647-3679. doi: 10.1016/S0020-7683(03)00143-4.

[20]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, Vol. 6, de Gruyter, 1985. doi: 10.1515/9783110844641.

[21]

C. Le Bris, Some numerical approaches for "weakly'' random homogenization, in Numerical Mathematics and Advanced Applications (eds. G. Kreiss, P. Lötstedt, A. Malqvist and M. Neytcheva), Proceedings of ENUMATH 2009, Lect. Notes Comput. Sci. Eng., Springer, 2010, 29-45.

[22]

F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem, preprint, arXiv:1302.0038.

[23]

P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis, Vol. 3 (eds. P. Ciarlet and J.-L. Lions), North Holland, Amsterdam, 1994, 465-624. doi: 10.1016/S1570-8659(05)80018-3.

[24]

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

[25]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.

[26]

A. N. Shiryaev, Probability, Graduate Texts in Mathematics, Vol. 95, Springer, 1984. doi: 10.1007/978-1-4899-0018-0.

[27]

L. Tartar, Estimations of homogenized coefficients, in Topics in the Mathematical Modelling of Composite Materials (eds. A. Cherkaev and R. Kohn), Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, 31 (1987), 9-20. doi: 10.1007/978-1-4612-2032-9_2.

[28]

A. A. Tempel'man, Ergodic theorems for general dynamical systems, Trudy Moskov. Mat. Obsc., 26 (1972), 95-132.

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, 2006.

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