A. Nouy, Recent Developments in Spectral Stochastic Methods for??the??Numerical Solution of Stochastic Partial Differential Equations, Archives of Computational Methods in Engineering, vol.24, issue.2, pp.251-285, 2009.
DOI : 10.1007/978-94-017-2838-6
URL : https://hal.archives-ouvertes.fr/hal-00366636

D. Xiu, Fast numerical methods for stochastic computations: a review, Communications in computational physics, vol.5, issue.2-4, pp.242-272, 2009.

O. P. Le-ma??trema??tre and O. M. Knio, Spectral Methods for Uncertainty Quantification With Applications to Computational Fluid Dynamics, 2010.

Y. Thomas, X. Hou, and . Wu, A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media, Journal of Computational Physics, vol.134, issue.1, pp.169-189, 1997.

J. R. Thomas, G. R. Hughes, L. Feijóo, J. Mazzei, and . Quincy, The variational multiscale method?a paradigm for computational mechanics, Advances in Stabilized Methods in Computational Mechanics, pp.3-24, 1998.

E. Weinan and B. Engquist, The heterogeneous multiscale methods, Communications in Mathematical Sciences, vol.1, issue.1, pp.87-132, 2003.

X. and F. Xu, A multiscale stochastic finite element method on elliptic problems involving uncertainties, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.25-28, pp.25-282723, 2007.
DOI : 10.1016/j.cma.2007.02.002

B. Velamur-asokan, N. Narayanan, and . Zabaras, Variational multiscale stabilized FEM formulations for transport equations: stochastic advection?diffusion and incompressible stochastic Navier?Stokes equations, Journal of Computational Physics, vol.202, issue.1, pp.94-133, 2005.

B. Velamur-asokan and N. Zabaras, A stochastic variational multiscale method for diffusion in heterogeneous random media, Journal of Computational Physics, vol.218, issue.2, pp.654-676, 2006.
DOI : 10.1016/j.jcp.2006.02.026

B. Ganapathysubramanian and N. Zabaras, Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method, Journal of Computational Physics, vol.226, issue.1, pp.326-353, 2007.
DOI : 10.1016/j.jcp.2007.04.009

P. Dostert, Y. Efendiev, and T. Y. Hou, Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.43-44, pp.43-443445, 2008.
DOI : 10.1016/j.cma.2008.02.030

X. F. Xu, X. Chen, and L. Shen, A Green-function-based multiscale method for uncertainty quantification of finite body random heterogeneous materials, Computers & Structures, vol.87, issue.21-22, pp.21-221416, 2009.
DOI : 10.1016/j.compstruc.2009.05.009

B. Ganapathysubramanian and N. Zabaras, A stochastic multiscale framework for modeling flow through random heterogeneous porous media, Journal of Computational Physics, vol.228, issue.2, pp.591-618, 2009.
DOI : 10.1016/j.jcp.2008.10.006

V. Ginting, A. Målqvist, and M. Presho, A Novel Method for Solving Multiscale Elliptic Problems with Randomly Perturbed Data, Multiscale Modeling & Simulation, vol.8, issue.3, pp.977-996, 2010.
DOI : 10.1137/090771302

L. Bris, . Claude, . Legoll, . Frédéric, and F. Thomines, Multiscale Finite Element approach for ???weakly??? random problems and related issues, ESAIM: M2AN, pp.815-858, 2014.
DOI : 10.1007/978-3-642-84659-5
URL : https://hal.archives-ouvertes.fr/hal-00639349

C. Jin, X. Cai, and C. Li, Parallel Domain Decomposition Methods for Stochastic Elliptic Equations, SIAM Journal on Scientific Computing, vol.29, issue.5, pp.2096-2114, 2007.
DOI : 10.1137/060662381
URL : http://www.cs.colorado.edu/~cai/papers/jcl07.pdf

K. Zhang, R. Zhang, Y. Yin, and S. Yu, Domain decomposition methods for linear and semilinear elliptic stochastic partial differential equations, Applied Mathematics and Computation, vol.195, issue.2, pp.630-640, 2008.
DOI : 10.1016/j.amc.2007.05.009

A. Sarkar, N. Benabbou, and R. Ghanem, Domain decomposition of stochastic PDEs: Theoretical formulations, International Journal for Numerical Methods in Engineering, vol.31, issue.4, pp.689-701, 2009.
DOI : 10.1007/978-1-4612-3094-6

B. Ganis, I. Yotov, and M. Zhong, A Stochastic Mortar Mixed Finite Element Method for Flow in Porous Media with Multiple Rock Types, SIAM Journal on Scientific Computing, vol.33, issue.3, pp.1439-1474, 2011.
DOI : 10.1137/100790689

M. F. Wheeler, T. Wildey, and I. Yotov, A multiscale preconditioner for stochastic mortar mixed finite elements, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.9-12, pp.9-121251, 2011.
DOI : 10.1016/j.cma.2010.10.015
URL : http://www.math.pitt.edu/%7Eyotov/research/publications/smfe-msbasis-CMAME.pdf

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-refinement Techniques, 1996.

E. Stein and S. Ohnimus, Coupled model- and solution-adaptivity in the finite-element method, Computer Methods in Applied Mechanics and Engineering, vol.150, issue.1-4, pp.327-350, 1997.
DOI : 10.1016/S0045-7825(97)00082-0

T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, vol.55, issue.5, pp.601-620, 1999.
DOI : 10.1115/1.3173676

N. Moës, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, vol.8, issue.1, pp.131-150, 1999.
DOI : 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

T. Strouboulis, I. Babu?ka, and K. Copps, The design and analysis of the Generalized Finite Element Method, Computer Methods in Applied Mechanics and Engineering, vol.181, issue.1-3, pp.43-69, 2000.
DOI : 10.1016/S0045-7825(99)00072-9

J. Lions and O. Pironneau, Domain decomposition methods for CAD. Comptes Rendus de l'Académie des Sciences -Series I -Mathematics, pp.73-80, 1999.
DOI : 10.1016/s0764-4442(99)80015-9

R. Glowinski, J. He, A. Lozinski, J. Rappaz, and J. Wagner, Finite element approximation of multi-scale elliptic problems using patches of elements, Numerische Mathematik, vol.49, issue.4, pp.663-687, 2005.
DOI : 10.1016/S0764-4442(97)89801-1
URL : https://hal.archives-ouvertes.fr/hal-00113130

J. He, A. Lozinski, and J. Rappaz, Accelerating the method of finite element patches using approximately harmonic functions, Comptes Rendus Mathematique, vol.345, issue.2, pp.107-112, 2007.
DOI : 10.1016/j.crma.2007.06.006

L. Joseph, C. Steger, . Dougherty, A. John, and . Benek, A Chimera grid scheme, Advances in Grid Generation, pp.59-69, 1983.

F. Brezzi, J. Lions, and O. Pironneau, Analysis of a Chimera method, Comptes Rendus de l'Académie des Sciences -Series I -Mathematics, pp.655-660, 2001.
DOI : 10.1016/S0764-4442(01)01904-8

O. Pironneau, A. Lozinski, A. Perronnet, and F. Hecht, Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems -A, pp.265-280, 2009.

L. Gendre, O. Allix, P. Gosselet, and F. Comte, Non-intrusive and exact global/local techniques for structural problems with local plasticity, Computational Mechanics, vol.36, issue.1, pp.233-245, 2009.
DOI : 10.1007/s00466-009-0372-9
URL : https://hal.archives-ouvertes.fr/hal-00437023

A. Lozinski, Méthodes numériques et modélisation pour certainsprobì emes multi-´ echelles. HabilitationàHabilitationà diriger des recherches, 2010.

L. Gendre, O. Allix, and P. Gosselet, A two-scale approximation of the Schur complement and its use for non-intrusive coupling, International Journal for Numerical Methods in Engineering, vol.64, issue.1-4, pp.889-905, 2011.
DOI : 10.1016/S0045-7949(96)00165-4
URL : https://hal.archives-ouvertes.fr/hal-01224373

C. Hager, P. Hauret, P. L. Tallec, and B. I. Wohlmuth, Solving dynamic contact problems with local refinement in space and time, Computer Methods in Applied Mechanics and Engineering, vol.201, issue.204, pp.201-20425, 2012.
DOI : 10.1016/j.cma.2011.09.006
URL : https://hal.archives-ouvertes.fr/hal-01393141

A. Gravouil, J. Rannou, and M. Ba¨?ettoba¨?etto, A Local Multi-grid X-FEM approach for 3D fatigue crack growth, International Journal of Material Forming, vol.1, issue.S1, pp.1103-1106, 2008.
DOI : 10.1007/s12289-008-0212-z

J. Rannou, A. Gravouil, and M. C. Ba¨?ettoba¨?etto-dubourg, A local multigrid X-FEM strategy for 3-D crack propagation, International Journal for Numerical Methods in Engineering, vol.41, issue.11-12, pp.581-600, 2009.
DOI : 10.1002/nme.2427
URL : https://hal.archives-ouvertes.fr/hal-00938285

J. C. Passieux, A. Gravouil, J. Réthoré, and M. C. Ba¨?ettoba¨?etto, Direct estimation of generalized stress intensity factors using a three-scale concurrent multigrid X-FEM, International Journal for Numerical Methods in Engineering, vol.63, issue.5, pp.1648-1666, 2011.
DOI : 10.1016/j.crme.2010.03.001
URL : https://hal.archives-ouvertes.fr/hal-00708392

J. Passieux, J. Réthoré, A. Gravouil, and M. Ba¨?ettoba¨?etto, Local/global non-intrusive crack propagation simulation using a multigrid X-FEM solver, Computational Mechanics, vol.89, issue.2, pp.1381-1393, 2013.
DOI : 10.1002/nme.3234
URL : https://hal.archives-ouvertes.fr/hal-00824125

D. Hachmi-ben, Probì emes mécaniques multi-´ echelles: la méthode Arlequin -Multiscale mechanical problems: the Arlequin method. Comptes Rendus de l'Académie des Sciences -Series IIB -Mechanics-Physics-Astronomy, pp.899-904, 1998.

J. T. Ludovic-chamoin, S. Oden, and . Prudhomme, A stochastic coupling method for atomic-to-continuum Monte-Carlo simulations, Computer Methods in Applied Mechanics and Engineering, vol.197, pp.43-443530, 2008.

R. Cottereau, D. Clouteau, H. B. Dhia, and C. Zaccardi, A stochastic-deterministic coupling method for continuum mechanics, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.47-48, pp.47-483280, 2011.
DOI : 10.1016/j.cma.2011.07.010
URL : https://hal.archives-ouvertes.fr/hal-00709540

M. Chevreuil, A. Nouy, and E. Safatly, A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties, Computer Methods in Applied Mechanics and Engineering, vol.255, pp.255-274, 2013.
DOI : 10.1016/j.cma.2012.12.003
URL : https://hal.archives-ouvertes.fr/hal-00733739

P. Henning, . Målqvist, . Axel, and D. Peterseim, A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM: M2AN, pp.1331-1349, 2014.
DOI : 10.1137/10081839X

Y. Efendiev, T. Y. Hou, and V. Ginting, Multiscale Finite Element Methods for Nonlinear Problems and Their Applications, Communications in Mathematical Sciences, vol.2, issue.4, pp.553-589, 2004.
DOI : 10.4310/CMS.2004.v2.n4.a2
URL : http://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2004/0002/0004/CMS-2004-0002-0004-a002.pdf

Y. Efendiev, Y. Thomas, and . Hou, Multiscale Finite Element Methods: Theory and Applications, of Surveys and Tutorials in the Applied Mathematical Sciences, 2009.

J. Martin-nordbotten, Variational and Heterogeneous Multiscale Methods, pp.713-720, 2010.
DOI : 10.1007/978-3-642-11795-4_76

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Discrete and Continuous Dynamical Systems -Series S, pp.119-150, 2015.

O. Allix, L. Gendre, P. Gosselet, and G. Guguin, Non-intrusive Coupling: An Attempt to Merge Industrial and Research Software Capabilities, Recent Developments and Innovative Applications in Computational Mechanics, chapter 15, pp.125-133, 2011.
DOI : 10.1007/978-3-642-17484-1_15
URL : https://hal.archives-ouvertes.fr/hal-00591345

A. Chkifa, A. Cohen, R. Devore, and C. Schwab, Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs, ESAIM: M2AN, pp.253-280, 2013.
DOI : 10.1051/m2an/2012027

A. D. Bazykin, Hypothetical Mechanism of Speciaton, Evolution, vol.23, issue.4, pp.685-687, 1969.
DOI : 10.2307/2406862

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, pp.5-49, 1975.
DOI : 10.1109/TCT.1965.1082476

B. Bradshaw-hajek and P. Broadbridge, A robust cubic reaction-diffusion system for gene propagation, Mathematical and Computer Modelling, vol.39, issue.9-10, pp.1151-1163, 2004.
DOI : 10.1016/S0895-7177(04)90537-7
URL : https://doi.org/10.1016/s0895-7177(04)90537-7

A. Hanna, A. Saul, and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, Journal of the American Chemical Society, vol.104, issue.14, pp.3838-3844, 1982.
DOI : 10.1021/ja00378a011

J. Huang and B. F. Edwards, Pattern formation and evolution near autocatalytic reaction fronts in a narrow vertical slab, Physical Review E, vol.53, issue.3, pp.2620-2627, 1996.
DOI : 10.1103/PhysRevE.53.6012

F. Boyd and . Edwards, Poiseuille Advection of Chemical Reaction Fronts, Phys. Rev. Lett, vol.89, p.104501, 2002.

M. Kaern and M. Menzinger, Propagation of Excitation Pulses and Autocatalytic Fronts in Packed-Bed Reactors, The Journal of Physical Chemistry B, vol.106, issue.14, pp.3751-3758, 2002.

S. Robert, B. F. Spangler, and . Edwards, Poiseuille advection of chemical reaction fronts: Eikonal approximation, The Journal of Chemical Physics, vol.118, issue.13, pp.5911-5915, 2003.

M. Leconte, J. Martin, N. Rakotomalala, and D. Salin, Pattern of Reaction Diffusion Fronts in Laminar Flows, Physical Review Letters, vol.219, issue.12, p.128302, 2003.
DOI : 10.1017/S0022112097004928

I. V. Koptyug, V. V. Zhivonitko, and R. Z. Sagdeev, Advection of Chemical Reaction Fronts in a Porous Medium, The Journal of Physical Chemistry B, vol.112, issue.4, pp.1170-1176, 2008.
DOI : 10.1021/jp077612r

S. Saha, S. Atis, D. Salin, and L. Talon, Phase diagram of sustained wave fronts opposing the flow in disordered porous media, EPL (Europhysics Letters), vol.101, issue.3, p.38003, 2013.
DOI : 10.1209/0295-5075/101/38003

K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, 2009.

D. Xiu and D. M. Tartakovsky, Numerical Methods for Differential Equations in Random Domains, SIAM Journal on Scientific Computing, vol.28, issue.3, pp.1167-1185, 2006.
DOI : 10.1137/040613160

M. Daniel, D. Tartakovsky, and . Xiu, Stochastic analysis of transport in tubes with rough walls, Journal of Computational Physics Uncertainty Quantification in Simulation Science, vol.217, issue.1, pp.248-259, 2006.

C. Canuto and T. Kozubek, A fictitious domain approach to the numerical solution of PDEs in stochastic domains, Numerische Mathematik, vol.28, issue.2, p.257, 2007.
DOI : 10.1007/978-1-4684-9464-8

A. Nouy, A. Clément, F. Schoefs, and N. Moës, An extended stochastic finite element method for solving stochastic partial differential equations on random domains, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.51-52, pp.4663-4682, 2008.
DOI : 10.1016/j.cma.2008.06.010
URL : https://hal.archives-ouvertes.fr/hal-00366617

A. Nouy, M. Chevreuil, and E. Safatly, Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.45-46, pp.45-463066, 2011.
DOI : 10.1016/j.cma.2011.07.002
URL : https://hal.archives-ouvertes.fr/hal-00662564

. Faker-ben and . Belgacem, The Mortar finite element method with Lagrange multipliers, Numerische Mathematik, vol.84, issue.2, pp.173-197, 1999.

I. Barbara and . Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lecture Notes in Computational Science and Engineering, vol.17, 2001.

C. Kim, R. Lazarov, J. Pasciak, and P. Vassilevski, Multiplier Spaces for the Mortar Finite Element Method in Three Dimensions, SIAM Journal on Numerical Analysis, vol.39, issue.2, pp.519-538, 2001.
DOI : 10.1137/S0036142900367065

M. Duval, J. Passieux, M. Salaün, and S. Guinard, Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition, Archives of Computational Methods in Engineering, vol.138, issue.1???4, pp.17-38, 2016.
DOI : 10.1016/S0045-7825(96)01106-1
URL : https://hal.archives-ouvertes.fr/hal-01065538

G. Hermann, A. Matthies, and . Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.194, pp.12-161295, 2005.

L. Giraldi, A. Litvinenko, D. Liu, H. Matthies, and A. Nouy, To Be or Not to Be Intrusive? The Solution of Parametric and Stochastic Equations---the ???Plain Vanilla??? Galerkin Case, SIAM Journal on Scientific Computing, vol.36, issue.6, pp.2720-2744, 2014.
DOI : 10.1137/130942802
URL : https://hal.archives-ouvertes.fr/hal-00859536

L. Giraldi, D. Liu, H. G. Matthies, and A. Nouy, To Be or Not to be Intrusive? The Solution of Parametric and Stochastic Equations---Proper Generalized Decomposition, SIAM Journal on Scientific Computing, vol.37, issue.1, pp.347-368, 2015.
DOI : 10.1137/140969063
URL : https://hal.archives-ouvertes.fr/hal-00987530

A. Chkifa, A. Cohen, G. Migliorati, F. Nobile, and R. Tempone, Discrete least squares polynomial approximation with random evaluations ??? application to parametric and stochastic elliptic PDEs, ESAIM: M2AN, pp.815-837, 2015.
DOI : 10.1137/070680540
URL : https://hal.archives-ouvertes.fr/hal-01352276

M. S. Bartlett, An Inverse Matrix Adjustment Arising in Discriminant Analysis, The Annals of Mathematical Statistics, vol.22, issue.1, pp.107-111, 1951.
DOI : 10.1214/aoms/1177729698
URL : http://doi.org/10.1214/aoms/1177729698

C. Gavin, N. L. Cawley, and . Talbot, Fast exact leave-one-out cross-validation of sparse least-squares support vector machines, Neural Networks, vol.17, issue.10, pp.1467-1475, 2004.

M. Bruce, . Irons, C. Robert, and . Tuck, A version of the Aitken accelerator for computer iteration, International Journal for Numerical Methods in Engineering, vol.1, issue.3, pp.275-277, 1969.

A. J. Macleod, Acceleration of vector sequences by multi-dimensional ??2 methods, Communications in Applied Numerical Methods, vol.49, issue.4, pp.385-392, 1986.
DOI : 10.1002/cnm.1630020409

U. Küttler and W. A. Wall, Fixed-point fluid???structure interaction solvers with dynamic relaxation, Computational Mechanics, vol.35, issue.6???8, pp.61-72, 2008.
DOI : 10.1007/s00466-008-0255-5

Y. J. Liu, Q. Sun, and X. L. Fan, A non-intrusive global/local algorithm with non-matching interface: Derivation and numerical validation, Computer Methods in Applied Mechanics and Engineering, vol.277, issue.0, pp.81-103, 2014.
DOI : 10.1016/j.cma.2014.04.012

B. Sudret, Global sensitivity analysis using polynomial chaos expansions, Reliability Engineering & System Safety, vol.93, issue.7, pp.964-979, 2008.
DOI : 10.1016/j.ress.2007.04.002
URL : https://hal.archives-ouvertes.fr/hal-01432217

O. Chapelle, V. Vapnik, and Y. Bengio, Model Selection for Small Sample Regression, Machine Learning, pp.9-23, 2002.

G. Blatman and B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, Journal of Computational Physics, vol.230, issue.6, pp.2345-2367, 2011.
DOI : 10.1016/j.jcp.2010.12.021

T. Roubí?ek, Nonlinear Partial Differential Equations with Applications, 2005.
DOI : 10.1007/978-3-0348-0513-1