P. Absil, R. Mahony, and R. Sepulchre, Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation, Acta Applicandae Mathematicae, vol.80, issue.2, pp.199-220, 2004.
DOI : 10.1023/B:ACAP.0000013855.14971.91

M. Arnst, R. Ghanem, and C. Soize, Identification of Bayesian posteriors for coefficients of chaos expansions, Journal of Computational Physics, vol.229, issue.9, pp.3134-3154, 2010.
DOI : 10.1016/j.jcp.2009.12.033

URL : https://hal.archives-ouvertes.fr/hal-00684317

I. Babuska, R. Tempone, and G. E. Zouraris, Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations, SIAM Journal on Numerical Analysis, vol.42, issue.2, pp.800-825, 2004.
DOI : 10.1137/S0036142902418680

I. Babuska, R. Tempone, and G. E. Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.12-16, pp.12-16, 2005.
DOI : 10.1016/j.cma.2004.02.026

I. Babuska, F. Nobile, and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.45, issue.3, pp.1005-1034, 2007.
DOI : 10.1137/050645142

I. Babuska, F. Nobile, and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Review, vol.52, issue.2, pp.317-355, 2010.
DOI : 10.1137/100786356

S. Boyaval, C. Le-bris, Y. Maday, N. C. Nguyen, and A. T. Patera, A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.41-44, pp.41-44, 2009.
DOI : 10.1016/j.cma.2009.05.019

URL : https://hal.archives-ouvertes.fr/inria-00311463

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, pp.385-392, 1947.
DOI : 10.2307/1969178

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. Charrier, Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients, SIAM Journal on Numerical Analysis, vol.50, issue.1, pp.216-246, 2012.
DOI : 10.1137/100800531

URL : https://hal.archives-ouvertes.fr/inria-00490045

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, pp.615-646, 2010.
DOI : 10.1007/s10208-010-9072-2

A. Cohen, A. Chkifa, and C. , Schwab Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, in review

S. Das, R. Ghanem, and J. Spall, Asymptotic Sampling Distribution for Polynomial Chaos Representation from Data: A Maximum Entropy and Fisher Information Approach, SIAM Journal on Scientific Computing, vol.30, issue.5, pp.2207-2234, 2008.
DOI : 10.1137/060652105

S. Das, R. Ghanem, and S. Finette, Polynomial chaos representation of spatio-temporal random fields from experimental measurements, Journal of Computational Physics, vol.228, issue.23, pp.8726-8751, 2009.
DOI : 10.1016/j.jcp.2009.08.025

M. Deb, I. Babuska, and J. T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.48, pp.190-6359, 2001.
DOI : 10.1016/S0045-7825(01)00237-7

V. D. Oliveira, B. Kedem, and D. A. Short, Bayesian Prediction of Transformed Gaussian Random Fields, Journal of the American Statistical Association, vol.92, issue.440, pp.1422-1433, 1997.
DOI : 10.2307/2965412

C. Desceliers, R. Ghanem, and C. Soize, Maximum likelihood estimation of stochastic chaos representations from experimental data, International Journal for Numerical Methods in Engineering, vol.11, issue.6, pp.978-1001, 2006.
DOI : 10.1002/nme.1576

URL : https://hal.archives-ouvertes.fr/hal-00686154

A. Doostan, R. Ghanem, and J. , Red-Horse, Stochastic model reduction for chaos representations, Computer Methods in Applied Mechanics and Engineering, vol.196, pp.37-40, 2007.

A. Doostan and G. Iaccarino, A least-squares approximation of partial differential equations with high-dimensional random inputs, Journal of Computational Physics, vol.228, issue.12, pp.4332-4345, 2009.
DOI : 10.1016/j.jcp.2009.03.006

A. Edelman, T. A. Arias, and S. T. Smith, The Geometry of Algorithms with Orthogonality Constraints, SIAM Journal on Matrix Analysis and Applications, vol.20, issue.2, pp.303-353, 1998.
DOI : 10.1137/S0895479895290954

O. G. Ernst, A. Mugler, H. J. Starkloff, and E. Ullmann, On the convergence of generalized polynomial chaos expansions, ESAIM: Mathematical Modelling and Numerical Analysis, vol.46, issue.2, pp.317-339, 2012.
DOI : 10.1051/m2an/2011045

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. Falco and A. Nouy, Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numerische Mathematik, vol.115, issue.45???48, pp.503-530, 2012.
DOI : 10.1007/s00211-011-0437-5

URL : https://hal.archives-ouvertes.fr/hal-00609108

P. Frauenfelder, C. Schwab, and R. A. Todor, Finite elements for elliptic problems with stochastic coefficients, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.2-5, pp.205-228, 2005.
DOI : 10.1016/j.cma.2004.04.008

J. Galvis and M. Sarkis, Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity, SIAM Journal on Numerical Analysis, vol.47, issue.5, pp.3624-3651, 2009.
DOI : 10.1137/080717924

URL : http://digitalcommons.wpi.edu/cgi/viewcontent.cgi?article=1043&context=mathematicalsciences-pubs

R. Ghanem and P. D. Spanos, Stochastic Finite Elements: A spectral Approach, Spinger-verlag, 1991.

R. Ghanem and R. M. Kruger, Numerical solution of spectral stochastic finite element systems, Computer Methods in Applied Mechanics and Engineering, vol.129, issue.3, pp.289-303, 1996.
DOI : 10.1016/0045-7825(95)00909-4

R. Ghanem and R. Doostan, Characterization of stochastic system parameters from experimen- Random fields representations for statistical inverse boundary value problems 33

C. J. Gittelson, STOCHASTIC GALERKIN DISCRETIZATION OF THE LOG-NORMAL ISOTROPIC DIFFUSION PROBLEM, Mathematical Models and Methods in Applied Sciences, vol.20, issue.02, pp.237-263, 2010.
DOI : 10.1142/S0218202510004210

J. Guilleminot, A. Noshadravan, C. Soize, and R. Ghanem, A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.17-20, pp.17-20, 2011.
DOI : 10.1016/j.cma.2011.01.016

URL : https://hal.archives-ouvertes.fr/hal-00684305

J. Guilleminot and C. Soize, Non-Gaussian positive-definite matrix-valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries, International Journal for Numerical Methods in Engineering, vol.31, issue.3, pp.1128-1151, 2011.
DOI : 10.1002/nme.3212

URL : https://hal.archives-ouvertes.fr/hal-00684290

J. Guilleminot and C. Soize, Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media, Multiscale Modeling and Simulation, A SIAM Interdisciplinary Journal), vol.11, issue.3, pp.840-870, 2013.

W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Series in Computational Mathematics, vol.42, 2012.
DOI : 10.1007/978-3-642-28027-6

D. T. Hristopulos, Spartan Gibbs Random Field Models for Geostatistical Applications, SIAM Journal on Scientific Computing, vol.24, issue.6, pp.2125-2162, 2003.
DOI : 10.1137/S106482750240265X

B. N. Khoromskij and C. Schwab, Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs, SIAM Journal on Scientific Computing, vol.33, issue.1, pp.364-385, 2011.
DOI : 10.1137/100785715

T. Kühn, Eigenvalues of integral operators with smooth positive definite kernels, Archiv der Mathematik, vol.71, issue.6, pp.525-534, 1987.
DOI : 10.1007/BF01194301

O. P. Le-maitre, O. M. Knio, and H. N. Najm, Uncertainty propagation using Wiener???Haar expansions, Journal of Computational Physics, vol.197, issue.1, pp.28-57, 2004.
DOI : 10.1016/j.jcp.2003.11.033

O. P. Le-maitre and O. M. Knio, Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics, 2010.

D. Lucor, C. H. Su, and G. E. Karniadakis, Generalized polynomial chaos and random oscillators, International Journal for Numerical Methods in Engineering, vol.60, issue.3, pp.571-596, 2004.
DOI : 10.1002/nme.976

Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, Journal of Computational Physics, vol.228, issue.6, pp.1862-1902, 2009.
DOI : 10.1016/j.jcp.2008.11.024

H. G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.12-16, pp.12-16, 2005.
DOI : 10.1016/j.cma.2004.05.027

H. G. Matthies and E. Zander, Solving stochastic systems with low-rank tensor compression, Linear Algebra and its Applications, vol.436, issue.10, pp.3819-3838, 2012.
DOI : 10.1016/j.laa.2011.04.017

V. A. Menegatto and C. P. Oliveira, Eigenvalue and singular value estimates for integral operators: a unifying approach, Mathematische Nachrichten, vol.93, issue.17-18, pp.17-18
DOI : 10.1002/mana.201100257

A. Mugler and H. Starkloff, On elliptic partial differential equations with random coefficients, Studia Universitatis Babes-Bolyai -Series Mathematica, pp.473-487, 2011.

F. Nobile, R. Tempone, and C. G. Webster, A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.46, issue.5, pp.2309-2345, 2008.
DOI : 10.1137/060663660

A. Nouy, A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.45-48, pp.45-48, 2007.
DOI : 10.1016/j.cma.2007.05.016

URL : https://hal.archives-ouvertes.fr/hal-00366619

A. Nouy, Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.51-52, pp.51-52, 2008.
DOI : 10.1016/j.cma.2008.06.012

URL : https://hal.archives-ouvertes.fr/hal-00366613

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/s11831-009-9034-5

URL : https://hal.archives-ouvertes.fr/hal-00366636

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

R. Schneider and A. Uschmajew, Approximation rates for the hierarchical tensor format in periodic Sobolev spaces, Journal of Complexity, vol.30, issue.2, 2013.
DOI : 10.1016/j.jco.2013.10.001

C. Schwab and C. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numerica, vol.15, pp.291-467, 2011.
DOI : 10.1016/0022-247X(77)90186-X

C. Schwab and A. M. Stuart, Sparse deterministic approximation of Bayesian inverse problems, Inverse Problems, vol.28, issue.4, 2012.
DOI : 10.1088/0266-5611/28/4/045003

R. J. Serfling, Approximation Theorems of Mathematical Statistics, 1980.

C. Soize and R. Ghanem, Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure, SIAM Journal on Scientific Computing, vol.26, issue.2, pp.395-410, 2004.
DOI : 10.1137/S1064827503424505

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

C. Soize, Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.1-3, pp.26-64, 2006.
DOI : 10.1016/j.cma.2004.12.014

URL : https://hal.archives-ouvertes.fr/hal-00686157

C. Soize and R. Ghanem, Reduced chaos decomposition with random coefficients of vectorvalued random variables and random fields, Computer Methods in Applied Mechanics and Engineering, vol.198, pp.21-26, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00684487

C. Soize, Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data, Computer Methods in Applied Mechanics and Engineering, vol.199, pp.33-36, 2010.
DOI : 10.1061/9780784412237.ch08

URL : https://hal.archives-ouvertes.fr/hal-00684324

C. Soize, A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.45-46, pp.45-46, 2011.
DOI : 10.1016/j.cma.2011.07.005

URL : https://hal.archives-ouvertes.fr/hal-00684294

J. C. Spall, Introduction to Stochastic Search and Optimization, 2003.
DOI : 10.1002/0471722138

G. Stefanou, A. Nouy, and A. Clément, Identification of random shapes from images through polynomial chaos expansion of random level set functions, International Journal for Numerical Methods in Engineering, vol.24, issue.2, pp.127-155, 2009.
DOI : 10.1002/nme.2546

URL : https://hal.archives-ouvertes.fr/hal-00366640

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, vol.19, pp.451-559, 2010.
DOI : 10.1017/S0962492910000061

Q. A. Ta, D. Clouteau, and R. Cottereau, Modeling of random anisotropic elastic media and impact on wave propagation, Revue europ??enne de m??canique num??rique, vol.19, issue.1-3, pp.241-253, 2010.
DOI : 10.3166/ejcm.19.241-253

URL : https://hal.archives-ouvertes.fr/hal-00709537

R. A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA Journal of Numerical Analysis, vol.27, issue.2, pp.232-261, 2007.
DOI : 10.1093/imanum/drl025

E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data, 1997.

X. L. Wan and G. E. Karniadakis, Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures, SIAM Journal on Scientific Computing, vol.28, issue.3, pp.901-928, 2006.
DOI : 10.1137/050627630

X. L. Wan and G. E. Karniadakis, Solving elliptic problems with non-Gaussian spatiallydependent random coefficients, Computer Methods in Applied Mechanics and Engineering, vol.198, pp.21-26, 2009.

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 G. E. Karniadakis, The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations, SIAM Journal on Scientific Computing, vol.24, issue.2, pp.619-644, 2002.
DOI : 10.1137/S1064827501387826

N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, Journal of Computational Physics, vol.227, issue.9, pp.4697-4735, 2008.
DOI : 10.1016/j.jcp.2008.01.019

P. P. Zabreyko, A. I. Koshelev, M. A. Krasnosel-'skii, S. G. Mikhlin, L. S. Rakovshchik et al., Stet'senko, Integral Equations -A Reference Text, 1975.