American Institute of Mathematical Sciences

• Previous Article
A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items
• NACO Home
• This Issue
• Next Article
Effective approximation method for solving linear Fredholm-Volterra integral equations
March  2017, 7(1): 51-76. doi: 10.3934/naco.2017003

Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence

 University of the Bundeswehr Munich, Faculty of Informatics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

* Corresponding author: Erik Kropat

Received  August 2016 Revised  February 2016 Published  February 2017

The homogenization of optimal control problems on periodic networks is considered. Traditional approaches for a homogenization of uncontrolled problems on graphs often rely on an artificial extension of branches. The main result shows that such an extension to thin domains is not required. A two-scale transform for network functions leads to a representation of the microscopic optimal control problem on the graph in terms of a two-scale transformed minimization problem that allows for a further homogenization. Here, the concept of $S$-homogenization is applied in order to prove the existence of an absolutely $S$-homogenized optimal control problem with respect to the superior domain and the microscopic scale encoded in the reference graph of the network. In addition, results on the $Γ$-convergence of optimal control problems on periodic networks are discussed.

Citation: Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
References:
 [1] A. Braides, Γ-Convergence for Beginners Oxford University Press, Oxford Lecture Series in Mathematics and Its Applications, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. [2] G. Buttazzo, Gamma-convergence and its applications to some problem in the calculus of variations, Gamma-convergence and its applications to some problem in the calculus of variations, In School on homogenization, ICTP, Trieste, September 6-17,1993, 1993 (1994), 303-325. [3] G. Buttazzo and G. Dal Maso, Gamma-convergence and optimal control problems, Gamma-convergence and optimal control problems, Journal of Optimization Theory and Applications, 38 (1982), 385-407. doi: 10.1007/BF00935345. [4] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6. [5] G. Dal Maso, An Introduction to Γ-Convergence Birkhäuser, Progress in Nonlinear Differential Equations and Their Applications, Basel, 1993. doi: 10.1007/978-1-4612-0327-8. [6] S. Göktepe and C. Miehe, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, Journal of the Mechanics and Physics of Solids, 53 (2005), 2259-2283. doi: 10.1016/j.jmps.2005.04.010. [7] B. Hassani and E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software Springer, London, 2011. doi: 10.1007/978-1-4471-0891-7. [8] V. V. Jikov and S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. [9] P. Kogut and G. Leugering, S-Homogenization of Optimal Control Problems in Banach Spaces, S-Homogenization of Optimal Control Problems in Banach Spaces, Mathematische Nachrichten, 1 (2002), 141-169. doi: 10.1002/1522-2616(200201)233:1<141::AID-MANA141>3.0.CO;2-I. [10] P. Kogut and G. Leugering, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs in Optimal Control Problems for Partial Differential Equations on Reticulated Domains (eds. P. Kogut and G. Leugering), Birkhäuser Boston, (2011), 409-440. doi: 10.1007/978-0-8176-8149-4. [11] P. Kogut, Homogenization of Optimal Control Problems for Distributed Systems Cybernetics Institute of Ukrainian National Academic Science, Kyiv, Glushkov, 40,1998 (in Russian). [12] E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen Wissenschaftlicher Verlag Berlin, Berlin, 2007. [13] E. Kropat and S. Meyer-Nieberg, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, in Proceedings of the 15th IFAC Workshop on Control Applications of Optimization (CAO 2012), September 13-16,2012, Rimini, Italy, (2012), 83-88. [14] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 22 (2015), 293-324. [15] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Numerical Algebra, Control and Optimization, 9 (2016), 183-219. doi: 10.3934/naco.2016008. [16] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 23 (2016), 155-193. [17] M. Lenczner, Homogénéisation d'un circuit électrique, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Academie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542. [18] M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers, Homogenization of periodic electrical networks including voltage to current amplifiers, Multiscale Modeling and Simulation, 2 (2004), 359-397. doi: 10.1137/S1540345903423919. [19] M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers, Homogenization of electrical networks including voltage-to-voltage amplifiers, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899-932. doi: 10.1142/S0218202599000415. [20] C. Miehe and S. Göktepe, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, Journal of the Mechanics and Physics of Solids, 53 (2005), 2231-2258. doi: 10.1016/j.jmps.2005.04.006. [21] C. Miehe, S. Göktepe and F. Lulei, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, Journal of the Mechanics and Physics of Solids, 52 (2004), 2617-2660. doi: 10.1016/j.jmps.2004.03.011. [22] E. Nuhn, E. Kropat, W. Reinhardt and S. Pckl, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, in Proceedings of the 45th Annual Hawaii International Conference on System Sciences (HICSS-45), January 4-7,2012, Grand Wailea, Maui, Hawaii(eds. Ralph H. Sprague, Jr.), IEEE Computer Society, (2012), 1089-1096. [23] G. A. Pavliotis and Andrew Stuart, Multiscale Methods. Averaging and Homogenization Springer, Texts in Applied Mathematics, 53, New York, 2008. [24] L. Tartar, The General Theory of Homogenization: A Personalized Introduction Springer, Berlin, Heidelberg, 2010. [25] M. Vogelius, A homogenization result for planar, polygonal networks, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514.

show all references

References:
 [1] A. Braides, Γ-Convergence for Beginners Oxford University Press, Oxford Lecture Series in Mathematics and Its Applications, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. [2] G. Buttazzo, Gamma-convergence and its applications to some problem in the calculus of variations, Gamma-convergence and its applications to some problem in the calculus of variations, In School on homogenization, ICTP, Trieste, September 6-17,1993, 1993 (1994), 303-325. [3] G. Buttazzo and G. Dal Maso, Gamma-convergence and optimal control problems, Gamma-convergence and optimal control problems, Journal of Optimization Theory and Applications, 38 (1982), 385-407. doi: 10.1007/BF00935345. [4] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6. [5] G. Dal Maso, An Introduction to Γ-Convergence Birkhäuser, Progress in Nonlinear Differential Equations and Their Applications, Basel, 1993. doi: 10.1007/978-1-4612-0327-8. [6] S. Göktepe and C. Miehe, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, Journal of the Mechanics and Physics of Solids, 53 (2005), 2259-2283. doi: 10.1016/j.jmps.2005.04.010. [7] B. Hassani and E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software Springer, London, 2011. doi: 10.1007/978-1-4471-0891-7. [8] V. V. Jikov and S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. [9] P. Kogut and G. Leugering, S-Homogenization of Optimal Control Problems in Banach Spaces, S-Homogenization of Optimal Control Problems in Banach Spaces, Mathematische Nachrichten, 1 (2002), 141-169. doi: 10.1002/1522-2616(200201)233:1<141::AID-MANA141>3.0.CO;2-I. [10] P. Kogut and G. Leugering, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs in Optimal Control Problems for Partial Differential Equations on Reticulated Domains (eds. P. Kogut and G. Leugering), Birkhäuser Boston, (2011), 409-440. doi: 10.1007/978-0-8176-8149-4. [11] P. Kogut, Homogenization of Optimal Control Problems for Distributed Systems Cybernetics Institute of Ukrainian National Academic Science, Kyiv, Glushkov, 40,1998 (in Russian). [12] E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen Wissenschaftlicher Verlag Berlin, Berlin, 2007. [13] E. Kropat and S. Meyer-Nieberg, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, in Proceedings of the 15th IFAC Workshop on Control Applications of Optimization (CAO 2012), September 13-16,2012, Rimini, Italy, (2012), 83-88. [14] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 22 (2015), 293-324. [15] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Numerical Algebra, Control and Optimization, 9 (2016), 183-219. doi: 10.3934/naco.2016008. [16] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 23 (2016), 155-193. [17] M. Lenczner, Homogénéisation d'un circuit électrique, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Academie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542. [18] M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers, Homogenization of periodic electrical networks including voltage to current amplifiers, Multiscale Modeling and Simulation, 2 (2004), 359-397. doi: 10.1137/S1540345903423919. [19] M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers, Homogenization of electrical networks including voltage-to-voltage amplifiers, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899-932. doi: 10.1142/S0218202599000415. [20] C. Miehe and S. Göktepe, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, Journal of the Mechanics and Physics of Solids, 53 (2005), 2231-2258. doi: 10.1016/j.jmps.2005.04.006. [21] C. Miehe, S. Göktepe and F. Lulei, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, Journal of the Mechanics and Physics of Solids, 52 (2004), 2617-2660. doi: 10.1016/j.jmps.2004.03.011. [22] E. Nuhn, E. Kropat, W. Reinhardt and S. Pckl, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, in Proceedings of the 45th Annual Hawaii International Conference on System Sciences (HICSS-45), January 4-7,2012, Grand Wailea, Maui, Hawaii(eds. Ralph H. Sprague, Jr.), IEEE Computer Society, (2012), 1089-1096. [23] G. A. Pavliotis and Andrew Stuart, Multiscale Methods. Averaging and Homogenization Springer, Texts in Applied Mathematics, 53, New York, 2008. [24] L. Tartar, The General Theory of Homogenization: A Personalized Introduction Springer, Berlin, Heidelberg, 2010. [25] M. Vogelius, A homogenization result for planar, polygonal networks, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514.
Two-scale transform: Mapping from $\mathscr{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathscr{Y}$.
Function spaces and operators: Notation of abstract op-timal control problems.
 ${\cal U} = {L^2}(\Omega \times {\cal Y}),\;{\Phi _\varepsilon } = {{\hat \Phi }_\varepsilon }$ ${\cal V} = {L^2}(\Omega \times {\cal Y}),\;{{\cal C}_\varepsilon } = {\widehat {\cal C}_\varepsilon }$ ${\cal W} = {L^2}(\Omega \times {\cal Y}),\;{{\cal Z}^\varepsilon } = \widehat {{\zeta ^\varepsilon }} \in {L^2}(\Omega \times {\cal Y})$ ${{\cal G}^\varepsilon } = {\widehat {\cal G}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{L^2}(\Omega \times {\cal Y})),\;{{\cal G}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {g^\varepsilon }{\xi ^\varepsilon }$ ${{\cal P}^\varepsilon } = {\widehat {\cal P}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal P}^\varepsilon }\widehat {{\phi ^\varepsilon }}: = {p^\varepsilon }{\phi ^\varepsilon }$ ${{\cal Q}^\varepsilon } = {\widehat {\cal Q}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal Q}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {q^\varepsilon }{\xi ^\varepsilon }$
 ${\cal U} = {L^2}(\Omega \times {\cal Y}),\;{\Phi _\varepsilon } = {{\hat \Phi }_\varepsilon }$ ${\cal V} = {L^2}(\Omega \times {\cal Y}),\;{{\cal C}_\varepsilon } = {\widehat {\cal C}_\varepsilon }$ ${\cal W} = {L^2}(\Omega \times {\cal Y}),\;{{\cal Z}^\varepsilon } = \widehat {{\zeta ^\varepsilon }} \in {L^2}(\Omega \times {\cal Y})$ ${{\cal G}^\varepsilon } = {\widehat {\cal G}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{L^2}(\Omega \times {\cal Y})),\;{{\cal G}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {g^\varepsilon }{\xi ^\varepsilon }$ ${{\cal P}^\varepsilon } = {\widehat {\cal P}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal P}^\varepsilon }\widehat {{\phi ^\varepsilon }}: = {p^\varepsilon }{\phi ^\varepsilon }$ ${{\cal Q}^\varepsilon } = {\widehat {\cal Q}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal Q}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {q^\varepsilon }{\xi ^\varepsilon }$
 [1] Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223 [3] Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151 [4] Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353 [5] 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 & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016 [6] Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787 [7] Ryan Loxton, Qun Lin, Volker Rehbock, Kok Lay Teo. Control parameterization for optimal control problems with continuous inequality constraints: New convergence results. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 571-599. doi: 10.3934/naco.2012.2.571 [8] Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial & Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311 [9] Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477 [10] 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 [11] Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks & Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503 [12] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [13] Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757 [14] Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058 [15] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [16] X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial & Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287 [17] Jean Louis Woukeng. $\sum$-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753 [18] Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503 [19] Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707 [20] Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143

Impact Factor: