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Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem
Deterministic homogenization for media with barriers
| 1. | Saratovskaya 9, 160, Moscow, 109518, Russian Federation |
In this article we consider coefficients with barriers. We show how the averaged coefficient may be inadequate near the barriers and propose a generalization which can detect the potential problems and improve the accuracy of the averaged solution.
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G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
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A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978. |
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Y. Capdeville and J. J. Marigo, Second order homogenization of the elastic wave equation for non-periodic layered media, Geophysical Journal International, 170 (2007), 823-838.
doi: 10.1111/j.1365-246X.2007.03462.x. |
| [4] |
Y. Chen, L. J. Durlofsky, M. Gerritsen and X. H. Wen, A coupled local-global upscaling ap- proach for simulating flow in highly heterogeneous formations, Advances in Water Resources, 26 (2003), 1041-1060.
doi: 10.1016/S0309-1708(03)00101-5. |
| [5] |
C. C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comput., 79 (2010), 1915-1955.
doi: 10.1090/S0025-5718-2010-02372-5. |
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L. J. Durlofsky, Numerical calculation of equivalent gridblock permeability tensors for heterogeneous porous media, Water Resources Research, 27 (1991), 699-708.
doi: 10.1029/91WR00107. |
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L. J. Durlofsky, Upscaling and gridding of fine scale geological models for flow simulation, Proceedings of the 8th International Forum on Reservoir Simulation in Stresa, Italy, 2005, 59 pp. |
| [8] |
Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys., 251 (2013), 116-135.
doi: 10.1016/j.jcp.2013.04.045. |
| [9] |
C. L. Farmer, Upscaling: A review, Numerical Methods in Fluids, 40 (2002), 63-78.
doi: 10.1002/fld.267. |
| [10] |
H. Hajibeygi, G. Bonfigli, M. A. Hesse and P. Jenny, Iterative multiscale finite-volume method, Journal of Computational Physics, 227 (2008), 8604-8621.
doi: 10.1016/j.jcp.2008.06.013. |
| [11] |
T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134 (1997), 169-189.
doi: 10.1006/jcph.1997.5682. |
| [12] |
V. Laptev and S. Belouettar, On averaging of the non-periodic conductivity coefficient using two-scale extension, PAMM, 5 (2005), 681-682.
doi: 10.1002/pamm.200510316. |
| [13] |
V. Laptev, Two-scale extensions for non-periodic coefficients,, preprint, ().
|
| [14] |
V. Laptev, On numerical averaging of the conductivity coefficient using two-scale extensions,, preprint, ().
|
| [15] |
V. D. Laptev, Construction and practical use of two-scaled extensions for rapidly oscillating functions, Journal of Mathematical Sciences, 158 (2009), 211-218.
doi: 10.1007/s10958-009-9384-4. |
| [16] |
V. Laptev, work, in progress., ().
|
| [17] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.
doi: 10.1137/0520043. |
| [18] |
H. Owhadi and L. Zhang, Metric based up-scaling,, preprint, ().
|
| [19] |
H. Owhadi and L. Zhang, Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast, Multiscale Model. Simul., 9 (2011), 1373-1398.
doi: 10.1137/100813968. |
| [20] |
X. H. Wen, L. J. Durlofsky and M. G. Edwards, Use of border regions for improved permeability upscaling, Mathematical Geology, 35 (2003), 521-547.
doi: 10.1023/A:1026230617943. |
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] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978. |
| [3] |
Y. Capdeville and J. J. Marigo, Second order homogenization of the elastic wave equation for non-periodic layered media, Geophysical Journal International, 170 (2007), 823-838.
doi: 10.1111/j.1365-246X.2007.03462.x. |
| [4] |
Y. Chen, L. J. Durlofsky, M. Gerritsen and X. H. Wen, A coupled local-global upscaling ap- proach for simulating flow in highly heterogeneous formations, Advances in Water Resources, 26 (2003), 1041-1060.
doi: 10.1016/S0309-1708(03)00101-5. |
| [5] |
C. C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comput., 79 (2010), 1915-1955.
doi: 10.1090/S0025-5718-2010-02372-5. |
| [6] |
L. J. Durlofsky, Numerical calculation of equivalent gridblock permeability tensors for heterogeneous porous media, Water Resources Research, 27 (1991), 699-708.
doi: 10.1029/91WR00107. |
| [7] |
L. J. Durlofsky, Upscaling and gridding of fine scale geological models for flow simulation, Proceedings of the 8th International Forum on Reservoir Simulation in Stresa, Italy, 2005, 59 pp. |
| [8] |
Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys., 251 (2013), 116-135.
doi: 10.1016/j.jcp.2013.04.045. |
| [9] |
C. L. Farmer, Upscaling: A review, Numerical Methods in Fluids, 40 (2002), 63-78.
doi: 10.1002/fld.267. |
| [10] |
H. Hajibeygi, G. Bonfigli, M. A. Hesse and P. Jenny, Iterative multiscale finite-volume method, Journal of Computational Physics, 227 (2008), 8604-8621.
doi: 10.1016/j.jcp.2008.06.013. |
| [11] |
T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134 (1997), 169-189.
doi: 10.1006/jcph.1997.5682. |
| [12] |
V. Laptev and S. Belouettar, On averaging of the non-periodic conductivity coefficient using two-scale extension, PAMM, 5 (2005), 681-682.
doi: 10.1002/pamm.200510316. |
| [13] |
V. Laptev, Two-scale extensions for non-periodic coefficients,, preprint, ().
|
| [14] |
V. Laptev, On numerical averaging of the conductivity coefficient using two-scale extensions,, preprint, ().
|
| [15] |
V. D. Laptev, Construction and practical use of two-scaled extensions for rapidly oscillating functions, Journal of Mathematical Sciences, 158 (2009), 211-218.
doi: 10.1007/s10958-009-9384-4. |
| [16] |
V. Laptev, work, in progress., ().
|
| [17] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.
doi: 10.1137/0520043. |
| [18] |
H. Owhadi and L. Zhang, Metric based up-scaling,, preprint, ().
|
| [19] |
H. Owhadi and L. Zhang, Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast, Multiscale Model. Simul., 9 (2011), 1373-1398.
doi: 10.1137/100813968. |
| [20] |
X. H. Wen, L. J. Durlofsky and M. G. Edwards, Use of border regions for improved permeability upscaling, Mathematical Geology, 35 (2003), 521-547.
doi: 10.1023/A:1026230617943. |
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