M. Ainsworth and J. Oden, A posteriori error estimation in finite element analysis, Computer Methods in Applied Mechanics and Engineering, vol.142, issue.1-2, 2000.
DOI : 10.1016/S0045-7825(96)01107-3

A. Ammar, F. Chinesta, P. Diez, and A. Huerta, An error estimator for separated representations of highly multidimensional models, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.25-28, pp.1872-1880, 2010.
DOI : 10.1016/j.cma.2010.02.012

URL : https://hal.archives-ouvertes.fr/hal-01004991

P. Babin, G. Valle, R. Dendievel, N. Lassoued, and L. Salvo, Mechanical properties of bread crumbs from tomography based Finite Element simulations, Journal of Materials Science, vol.23, issue.22, pp.5867-5873, 2005.
DOI : 10.1007/s10853-005-5021-x

URL : https://hal.archives-ouvertes.fr/hal-00141295

B. Bader and T. Kolda, Matlab tensor toolbox version 2.4, 2011.

A. Bensoussan, J. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 1978.
DOI : 10.1090/chel/374

D. L. Camacho, R. H. Hopper, G. M. Lin, and B. S. Myers, An improved method for finite element mesh generation of geometrically complex structures with application to the skullbase, Journal of Biomechanics, vol.30, issue.10, pp.1067-1070, 1997.
DOI : 10.1016/S0021-9290(97)00073-0

E. Cances, V. Ehrlacher, and T. Lelievre, CONVERGENCE OF A GREEDY ALGORITHM FOR HIGH-DIMENSIONAL CONVEX NONLINEAR PROBLEMS, Mathematical Models and Methods in Applied Sciences, vol.21, issue.12, pp.2433-2467, 2011.
DOI : 10.1142/S0218202511005799

URL : https://hal.archives-ouvertes.fr/hal-00469622

J. D. Carroll and J. Chang, Analysis of individual differences in multidimensional scaling via an n-way generalization of ???Eckart-Young??? decomposition, Psychometrika, vol.12, issue.3, pp.283-319, 1970.
DOI : 10.1007/BF02310791

G. T. Charras and R. E. Guldberg, Improving the local solution accuracy of large-scale digital image-based finite element analyses, Journal of Biomechanics, vol.33, issue.2, pp.255-259, 2000.
DOI : 10.1016/S0021-9290(99)00141-4

F. Chinesta, A. Ammar, and E. Cueto, Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models, Archives of Computational Methods in Engineering, vol.190, issue.1, pp.327-350, 2010.
DOI : 10.1007/s11831-010-9049-y

URL : https://hal.archives-ouvertes.fr/hal-01007235

F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, and F. Boust, Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.5, pp.400-413, 2008.
DOI : 10.1016/j.cma.2007.07.022

URL : https://hal.archives-ouvertes.fr/hal-01004980

F. Chinesta, P. Ladeveze, and E. Cueto, A Short Review on Model Order Reduction Based on Proper Generalized Decomposition, Archives of Computational Methods in Engineering, vol.69, issue.9, pp.395-404, 2011.
DOI : 10.1007/s11831-011-9064-7

URL : https://hal.archives-ouvertes.fr/hal-01004940

A. Duster, J. Parvizian, Z. Yang, and E. Rank, The finite cell method for three-dimensional problems of solid mechanics, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.45-48, pp.3768-3782, 2008.
DOI : 10.1016/j.cma.2008.02.036

A. Falco and W. Hackbusch, On Minimal Subspaces in Tensor Representations, Foundations of Computational Mathematics, vol.91, issue.4
DOI : 10.1007/s10208-012-9136-6

A. Falco and A. Nouy, A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart???Young approach, Journal of Mathematical Analysis and Applications, vol.376, issue.2, pp.469-480, 2011.
DOI : 10.1016/j.jmaa.2010.12.003

URL : https://hal.archives-ouvertes.fr/hal-00461094

A. Falcó and A. Nouy, Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numerische Mathematik, vol.115, issue.45???48, 2012.
DOI : 10.1007/s00211-011-0437-5

D. Golanski, K. Terada, and N. Kikuchi, Macro and micro scale modeling of thermal residual stresses in metal matrix composite surface layers by the homogenization method, Computational Mechanics, vol.19, issue.3, pp.188-202, 1997.
DOI : 10.1007/s004660050168

W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, 2012.
DOI : 10.1007/978-3-642-28027-6

W. Hackbusch and S. Kühn, A New Scheme for the Tensor Representation, Journal of Fourier Analysis and Applications, vol.5, issue.3, pp.706-722, 2009.
DOI : 10.1007/s00041-009-9094-9

R. A. Harshman, Foundations of the PARAFAC procedure: Model and conditions for an " explanatory " multi-mode factor analysis, UCLA Working Papers in Phonetics, vol.16, pp.1-84, 1970.

S. Hollister and N. Kikuchi, Homogenization theory and digital imaging: A basis for studying the mechanics and design principles of bone tissue, Biotechnology and Bioengineering, vol.36, issue.7, pp.586-596, 1994.
DOI : 10.1002/bit.260430708

M. Jiang, I. Jasiuk, and M. Ostoja-starzewski, Apparent thermal conductivity of periodic two-dimensional composites, Computational Materials Science, vol.25, issue.3, pp.329-338, 2002.
DOI : 10.1016/S0927-0256(02)00234-3

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, vol.40, issue.13-14, pp.3647-3679, 2003.
DOI : 10.1016/S0020-7683(03)00143-4

T. Kanit, F. N-'guyen, S. Forest, D. Jeulin, M. Reed et al., Apparent and effective physical properties of heterogeneous materials: Representativity of samples of two materials from food industry, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.33-36, pp.3960-3982, 2006.
DOI : 10.1016/j.cma.2005.07.022

URL : https://hal.archives-ouvertes.fr/hal-00139164

J. Keyak, J. Meagher, H. Skinner, C. Mote, and . Jr, Automated three-dimensional finite element modelling of bone: a new method, Journal of Biomedical Engineering, vol.12, issue.5, pp.389-397, 1990.
DOI : 10.1016/0141-5425(90)90022-F

B. Khoromskij, Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ??? d, Constructive Approximation, vol.27, issue.4, pp.599-620, 2009.
DOI : 10.1007/s00365-009-9068-9

T. G. Kolda and B. W. Bader, Tensor Decompositions and Applications, SIAM Review, vol.51, issue.3, p.455, 2009.
DOI : 10.1137/07070111X

P. Ladevèze, J. Passieux, and D. Néron, The LATIN multiscale computational method and the Proper Generalized Decomposition, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.21-22, pp.1287-1296, 2010.
DOI : 10.1016/j.cma.2009.06.023

P. Ladevèze and L. Chamoin, On the verification of model reduction methods based on the proper generalized decomposition, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.23-24, pp.2032-2047, 2011.
DOI : 10.1016/j.cma.2011.02.019

H. Lamari, A. Ammar, P. Cartraud, F. Chinesta, F. Jacquemin et al., Recent advances in material homogenization, International Journal of Material Forming, vol.144, issue.S1, pp.899-902, 2010.
DOI : 10.1007/s12289-010-0913-y

URL : https://hal.archives-ouvertes.fr/hal-01008739

H. Lamari, A. Ammar, P. Cartraud, G. Legrain, F. Chinesta et al., Routes for Efficient Computational Homogenization of??Nonlinear Materials Using the??Proper Generalized Decompositions, Archives of Computational Methods in Engineering, vol.198, issue.33???36, pp.373-391, 2010.
DOI : 10.1007/s11831-010-9051-4

A. N. Langville and W. J. Stewart, A Kronecker product approximate preconditioner for SANs, Numerical Linear Algebra with Applications, vol.11, issue.89, pp.723-752, 2004.
DOI : 10.1002/nla.344

G. Laschet, Homogenization of the thermal properties of transpiration cooled multi-layer plates, Computer Methods in Applied Mechanics and Engineering, vol.191, issue.41-42, pp.4535-4554, 2002.
DOI : 10.1016/S0045-7825(02)00319-5

G. Legrain, R. Allais, and P. Cartraud, On the use of the extended finite element method with quadtree/octree meshes, International Journal for Numerical Methods in Engineering, vol.77, issue.4, pp.717-743, 2011.
DOI : 10.1002/nme.3070

G. Legrain, P. Cartraud, I. Perreard, and N. Moës, An X-FEM and level set computational approach for image-based modelling: Application to homogenization, International Journal for Numerical Methods in Engineering, vol.76, issue.4, pp.915-934, 2011.
DOI : 10.1002/nme.3085

G. Legrain, N. Chevaugeon, and K. Dréau, Computational homogenization using high order x-fem and levelsets: Uncoupling geometry and approximation, Computer Methods in Applied Mechanics and Engineering, 2012.
DOI : 10.1016/j.cma.2012.06.001

URL : https://hal.archives-ouvertes.fr/hal-00703239/file/legrainChevaugeonDreauPreprint.pdf

A. Lewis and A. Geltmacher, Image-based modeling of the response of experimental 3D microstructures to mechanical loading, Scripta Materialia, vol.55, issue.1, pp.81-85, 2006.
DOI : 10.1016/j.scriptamat.2006.01.043

W. Lian, ContributionàContribution`Contributionà l'Homogénéisation Numérique du Comportement Elastique de MatériauxMatériaux`Matériauxà Microstructure Complexe Caractérisés par Imagerie, 2011.

W. D. Lian, G. Legrain, and P. Cartraud, Image-based computational homogenization and localization: comparison between X-FEM/levelset and voxel-based approaches, Computational Mechanics, vol.42, issue.13, 2012.
DOI : 10.1007/s00466-012-0723-9

URL : https://hal.archives-ouvertes.fr/hal-00703243

K. Madi, S. Forest, M. Boussuge, S. Gailliégue, E. Lataste et al., Finite element simulations of the deformation of fused-cast refractories based on X-ray computed tomography, Computational Materials Science, vol.39, issue.1, pp.224-229, 2007.
DOI : 10.1016/j.commatsci.2006.01.033

URL : https://hal.archives-ouvertes.fr/hal-00140558

E. Maire, A. Fazekas, L. Salvo, R. Dendievel, S. Youssef et al., X-ray tomography applied to the characterization of cellular materials. Related finite element modeling problems, Composites Science and Technology, vol.63, issue.16, pp.2431-2443, 2003.
DOI : 10.1016/S0266-3538(03)00276-8

URL : https://hal.archives-ouvertes.fr/hal-00475150

M. Jr and L. L. , Automatic voxel-based generation of 3d microstructural fe models and its application to the damage analysis of composites, Materials Science and Engineering: A, vol.407, p.11, 2005.

H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, vol.157, issue.1-2, pp.69-94, 1998.
DOI : 10.1016/S0045-7825(97)00218-1

URL : https://hal.archives-ouvertes.fr/hal-01282728

A. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.23-24, pp.1603-1626, 2010.
DOI : 10.1016/j.cma.2010.01.009

URL : https://hal.archives-ouvertes.fr/hal-00455635

A. Nouy, Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems, Archives of Computational Methods in Engineering, vol.225, issue.1, pp.403-434, 2010.
DOI : 10.1007/s11831-010-9054-1

URL : https://hal.archives-ouvertes.fr/hal-00461099

I. V. Oseledets, Tensor-Train Decomposition, SIAM Journal on Scientific Computing, vol.33, issue.5, p.2295, 2011.
DOI : 10.1137/090752286

M. Ostoja-starzewski, Material spatial randomness: From statistical to representative volume element, Probabilistic Engineering Mechanics, vol.21, issue.2, pp.112-132, 2006.
DOI : 10.1016/j.probengmech.2005.07.007

I. Ozdemir, W. Brekelmans, and M. Geers, Computational homogenization for heat conduction in heterogeneous solids, International Journal for Numerical Methods in Engineering, vol.47, issue.2, pp.185-204, 2008.
DOI : 10.1002/nme.2068

E. Sanchez-palencia, Non homogeneous media and vibration theory, Lecture Notes in Physics, vol.127, 1980.

J. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge Monographs on Applied and Computational Mathematics, issue.3, 1999.

V. De-silva and L. Lim, Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem, SIAM Journal on Matrix Analysis and Applications, vol.30, issue.3, 1084.
DOI : 10.1137/06066518X

N. Takano, M. Zako, F. Kubo, and K. Kimura, Microstructure-based stress analysis and evaluation for porous ceramics by homogenization method with digital image-based modeling, International Journal of Solids and Structures, vol.40, issue.5, pp.1225-1242, 2003.
DOI : 10.1016/S0020-7683(02)00642-X

K. Terada, M. Asai, and M. Yamagishi, Finite cover method for linear and non-linear analyses of heterogeneous solids, International Journal for Numerical Methods in Engineering, vol.41, issue.9, pp.1321-1346, 2003.
DOI : 10.1002/nme.820

A. Touzene, A Tensor Sum Preconditioner for Stochastic Automata Networks, INFORMS Journal on Computing, vol.20, issue.2, pp.234-242, 2008.
DOI : 10.1287/ijoc.1070.0236

D. Ulrich, B. Van-rietbergen, H. Weinans, and P. Rüegsegger, Finite element analysis of trabecular bone structure: a comparison of image-based meshing techniques, Journal of Biomechanics, vol.31, issue.12, pp.1187-1192, 1998.
DOI : 10.1016/S0021-9290(98)00118-3

P. Young, T. Beresford-west, S. Coward, B. Notarberardino, B. Walker et al., An efficient approach to converting three-dimensional image data into highly accurate computational models, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.36, issue.2, pp.3155-3173, 2008.
DOI : 10.1016/S0021-9290(02)00309-3

A. Chapitre, Propriété d'espace produit tensoriel de l'espace de Sobolev anisotrope cassé L 2 (? 3 ) a ? . . . a ? L 2 (? 6 ) est dense dans H(?) Ainsi on a la relation

]. B. Bibliographie1, P. Almroth, F. Stern, I. Brogan, R. Babu²ka et al., Automatic choice of global shape functions in structural analysis Ammar et al. A new family of solvers for some classes of multidimensional partial dierential equations encountered in kinetic theory modeling of complex uids Journal of Non- Newtonian Fluid Mechanics 139 Ammar et al. A new family of solvers for some classes of multidimensional partial dierential equations encountered in kinetic theory modelling of complex uids -Part II : Transient simulation using space-time separated representations [4] I. Babu²ka, F. Nobile et R. Tempone. A Stochastic Collocation Method for Elliptic Partial Dierential Equations with Random Input Data Solving elliptic boundary value problems with uncertain coecients by the nite element method : the stochastic formulation, AIAA Journal of Non-Newtonian Fluid Mechanics SIAM Journal on Numerical Analysis Computer Methods in Applied Mechanics and Engineering, vol.1632336, issue.194, pp.525-153176, 1978.

F. Bach, J. Ballani, and L. Grasedyck, Optimization with Sparsity-Inducing Penalties A projection method to solve linear systems in tensor format, Foundations and Trends R in Machine Learning Numerical Linear Algebra with Applications, vol.418, pp.1106-1113, 2011.

R. Barrett, Templates for the Solution of Linear Systems : Building Blocks for Iterative Methods. 2 e éd Algorithm 432 : Solution of the matrix equation AX + XB = C Stochastic nite element : a non intrusive approach by regression, Society for Industrial et Applied Mathematics Commun. ACM, vol.15, issue.9, pp.1-2, 1972.

]. G. Beylkin and M. J. Mohlenkamp, Algorithms for Numerical Analysis in High Dimensions, SIAM Journal on Scientific Computing, vol.26, issue.6
DOI : 10.1137/040604959

]. B. Bognet, Advanced simulation of models dened in plate geometries : 3D solutions with 2D computational complexity, Computer Methods in Applied Mechanics and Engineering, vol.201204, p.112, 2012.

H. Bungartz and M. Griebel, Sparse grids, Acta Numerica, vol.13, p.147269, 2004.

R. H. Cameron and W. T. Martin, The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals, The Annals of Mathematics, vol.48, issue.2, p.385392, 1947.
DOI : 10.2307/1969178

S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic Decomposition by Basis Pursuit
DOI : 10.1137/s003614450037906x

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.7694

]. F. Chinesta, Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.5, p.400413, 2008.
DOI : 10.1016/j.cma.2007.07.022

URL : https://hal.archives-ouvertes.fr/hal-01004980

S. Choi, R. V. Grandhi, and R. A. Canfield, Structural reliability under non-Gaussian stochastic behavior, Computers & Structures, vol.82, issue.13-14, pp.1314-11131121, 2004.
DOI : 10.1016/j.compstruc.2004.03.015

]. A. Cohen, R. Devore, and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Analysis and Applications 09, pp.1-1147, 2011.

]. A. Cohen, R. Devore, and C. Schwab, Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs, Foundations of Computational Mathematics, vol.60, issue.6, p.615646, 2010.
DOI : 10.1007/s10208-010-9072-2

]. T. Crestaux, Méthode adaptative d'intégration multi-dimensionnelle et sélection d'une base de polynômes de chaos Thèse de doct, 21] L. De Lathauwer, B. De Moor et J. Vandewalle. A Multilinear Singular Value Decomposition . SIAM Journal on Matrix Analysis and Applications 21, p.12531278, 2000.

L. De-lathauwer, B. De-moor, J. Vandewaller1, and R. , ) Approximation of Higher-Order Tensors, SIAM Journal on Matrix Analysis and Applications, vol.21, issue.4, p.13241342, 2000.
DOI : 10.1137/S0895479898346995

A. Defant and K. Floret, Tensor Norms and Operator Ideals. 1 re éd. North-Holland, 1992. [24] A. Doostan et H. Owhadi. A non-adapted sparse approximation of PDEs with stochastic inputs, Journal of Computational Physics, vol.2308, p.30153034, 2011.

C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika, vol.1, issue.3, p.211218, 1936.
DOI : 10.1007/BF02288367

O. G. Ernst, On the convergence of generalized polynomial chaos expansions, ESAIM: Mathematical Modelling and Numerical Analysis, vol.46, issue.2, p.317339, 2011.
DOI : 10.1051/m2an/2011045

M. Espig and W. Hackbusch, A regularized Newton method for the ecient approximation of tensors represented in the canonical tensor format, Numerische Mathematik (, p.137

A. Falcó and W. Hackbusch, On Minimal Subspaces in Tensor Representations, Foundations of Computational Mathematics, vol.91, issue.4, 2010.
DOI : 10.1007/s10208-012-9136-6

A. Falcó and A. Nouy, Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numerische Mathematik, vol.115, issue.45???48, p.503530, 2012.
DOI : 10.1007/s00211-011-0437-5

R. W. Freund and N. M. , QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numerische Mathematik, vol.10, issue.1, p.315339, 1991.
DOI : 10.1007/BF01385726

]. R. Ghanem, Ingredients for a general purpose stochastic nite elements implementation, Computer Methods in Applied Mechanics and Engineering, vol.16814, p.1934, 1999.

R. G. Ghanem and R. M. Kruger, Numerical solution of spectral stochastic nite element systems, Computer Methods in Applied Mechanics and Engineering, vol.1293, p.289303, 1996.

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements : A Spectral Approach, 1991.
DOI : 10.1007/978-1-4612-3094-6

L. Grasedyck, Hierarchical Singular Value Decomposition of Tensors, SIAM Journal on Matrix Analysis and Applications, vol.31, issue.4, p.20292054, 2010.
DOI : 10.1137/090764189

W. Hackbusch and S. Kuehn, A New Scheme for the Tensor Representation, Journal of Fourier Analysis and Applications, vol.5, issue.3, p.706722, 2009.
DOI : 10.1007/s00041-009-9094-9

W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Series in Computational Mathematics 42 Hestenes et E. Stiefel. Methods of Conjugate Gradients for Solving Linear Systems, 2012.
DOI : 10.1007/978-3-642-28027-6

F. L. Hitchcock, The Expression of a Tensor or a Polyadic as a Sum of Products, Journal of Mathematics and Physics, vol.6, issue.1-4, p.164189, 1927.
DOI : 10.1002/sapm192761164

H. Hotelling, Analysis of a complex of statistical variables into principal components, Journal of Educational Psychology, vol.246, p.417441, 1933.

K. Karhunen, Zur Spektraltheorie stochastischer Prozesse, Annales Academiae Scientiarum Fennicae, vol.37, 1946.

]. B. Khoromskij, Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in R^d, Constructive Approximation, vol.303, p.599620, 2009.

T. G. Kolda and B. W. Bader, Tensor Decompositions and Applications, SIAM Review, vol.51, issue.3, p.455500, 2009.
DOI : 10.1137/07070111X

D. Kressner and C. Tobler, htucker -A MATLAB toolbox for tensors in hierarchical Tucker format, 2012.

D. Kressner and C. Tobler, Krylov Subspace Methods for Linear Systems with Tensor Product Structure, SIAM Journal on Matrix Analysis and Applications, vol.31, issue.4, pp.1688-1714, 2010.
DOI : 10.1137/090756843

D. Kressner and C. Tobler, Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems, SIAM Journal on Matrix Analysis and Applications, vol.32, issue.4, p.12881316, 2011.
DOI : 10.1137/100799010

D. Kressner and C. Tobler, Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems, Computational Methods in Applied Mathematics, vol.11, issue.3, p.363381, 2011.
DOI : 10.2478/cmam-2011-0020

P. Ladevèze, Nonlinear Computational Structural Mechanics, 1999.
DOI : 10.1007/978-1-4612-1432-8

C. Lanczos, Solution of systems of linear equations by minimized iterations, Journal of Research of the National Bureau of Standards, vol.49, issue.1, p.3353, 1952.
DOI : 10.6028/jres.049.006

A. N. Langville and W. J. Stewart, A Kronecker product approximate preconditioner for SANs. Numerical Linear Algebra with Applications 11, p.723752, 2004.

O. P. Le-maître and O. M. Knio, Spectral Methods for Uncertainty Quantication : With Applications to Computational Fluid Dynamics, 2010.

M. Loève, Probability Theory II. 4th, 1978.

J. Lumley, The Structure of Inhomogeneous Turbulent Flows Atmospheric turbulence and radio propagation, p.166178, 1967.

H. G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial dierential equations, Computer Methods in Applied Mechanics and Engineering, vol.194, pp.1216-12951331, 2005.

A. K. Noor and J. M. Peters, Reduced basis technique for nonlinear analysis of structures, 20th Structures, Structural Dynamics, and Materials Conference
DOI : 10.2514/6.1979-747

]. A. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial dierential equations, Computer Methods in Applied Mechanics and Engineering, vol.199, pp.2324-16031626, 2010.

]. A. Nouy, Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems, Archives of Computational Methods in Engineering, vol.225, issue.1, p.403434, 2010.
DOI : 10.1007/s11831-010-9054-1

URL : https://hal.archives-ouvertes.fr/hal-00461099

]. A. Nouy, Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Dierential Equations, Archives of Computational Methods in Engineering, vol.163, p.251285, 2009.

I. Oseledets, DMRG Approach to Fast Linear Algebra in the TT-Format, Computational Methods in Applied Mathematics, vol.11, issue.3, pp.382-393, 2011.
DOI : 10.2478/cmam-2011-0021

]. N. Parés, P. Díez, and A. Huerta, Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous Galerkin time discretization, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.19-20, p.16411660, 1920.
DOI : 10.1016/j.cma.2007.08.025

]. Y. Pati, R. Rezaiifar, and P. Krishnaprasad, Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, p.44, 1993.
DOI : 10.1109/ACSSC.1993.342465

K. Pearson, On lines and planes of closest t to systems of points in space, Philosophical Magazine, vol.2, issue.6, p.559572, 1901.

G. Rozza, D. B. Huynh, and A. T. Patera, Reduced basis approximation and a posteriori error estimation for anely parametrized elliptic coercive partial dierential equations, Archives of Computational Methods in Engineering, vol.153, p.229275, 2008.
DOI : 10.1007/bf03024948

Y. Saad and M. H. Schultz, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM Journal on Scientific and Statistical Computing, vol.7, issue.3, p.856, 1986.
DOI : 10.1137/0907058

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.476.951

Y. , S. Silva, and L. Lim, Iterative Methods for Sparse Linear Systems, Second Edition. 2 e éd Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem, Society for Industrial et Applied Mathematics Siam Journal on Matrix Analysis and Applications, vol.303, pp.1084-1127, 2003.

C. Soize and R. G. Ghanem, Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure, SIAM Journal on Scientific Computing, vol.26, issue.2, p.395410, 2004.
DOI : 10.1137/S1064827503424505

URL : https://hal.archives-ouvertes.fr/hal-00686211

P. Sonneveld, CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems, SIAM Journal on Scientific and Statistical Computing, vol.10, issue.1, p.3652, 1989.
DOI : 10.1137/0910004

B. Sudret, Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety 93, p.964979, 2008.
DOI : 10.1016/j.ress.2007.04.002

URL : https://hal.archives-ouvertes.fr/hal-01432217

V. N. Temlyakov, Greedy Approximation, Acta Numerica, vol.17, p.235409, 2008.
DOI : 10.1017/cbo9780511762291

R. Tibshirani, Regression Shrinkage and Selection Via the Lasso, Journal of the Royal Statistical Society, Series B, pp.58-267288, 1994.

A. Touzene, A Tensor Sum Preconditioner for Stochastic Automata Networks, INFORMS Journal on Computing, vol.202, p.234242, 2008.

L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, vol.64, issue.3, p.279311, 1966.
DOI : 10.1007/BF02289464

]. E. Ullmann, A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations, SIAM Journal on Scientific Computing, vol.32, issue.2, p.923946, 2010.
DOI : 10.1137/080742853

K. Veroy and A. T. Patera, Certied real-time solution of the parametrized steady incompressible Navier-Stokes equations : rigorous reduced-basis a posteriori error bounds, International Journal for Numerical Methods in Fluids, vol.47, pp.8-9, 2005.

H. A. Van and . Vorst, Bi-CGSTAB : A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems, SIAM Journal on Scientic and Statistical Computing, vol.13, issue.2, p.631644, 1992.

H. A. Van and . Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge Monographs on Applied and Computational Mathematics, 2003.

N. Wiener, The Homogeneous Chaos, American Journal of Mathematics, vol.60, issue.4, pp.897-936, 1938.
DOI : 10.2307/2371268

D. B. Xiu and J. S. Hesthaven, High-order collocation methods for dierential equations with random inputs, Siam Journal on Scientic Computing, vol.273, p.11181139, 2005.

]. E. Zander, Tensor Approximation Methods for Stochastic Problems, Thèse de doct. TU Braunschweig, 2012.