M. Arnst, R. Ghanem, and A. 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

M. Arnst, R. Ghanem, E. Phipps, and A. J. Red-horse, Dimension reduction in stochastic modeling of coupled problems, International Journal for Numerical Methods in Engineering, vol.125, issue.012075, pp.940-968, 2012.
DOI : 10.1002/nme.4364

M. Arnst, C. Soize, and A. R. Ghanem, Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems, SIAM/ASA Journal on Uncertainty Quantification, vol.1, issue.1, p.218243, 2013.
DOI : 10.1137/120894403
URL : https://hal.archives-ouvertes.fr/hal-00829060

M. Arnst, R. Ghanem, E. Phipps, and A. J. Red-horse, Reduced chaos expansions with random coefficients in reduced-dimensional stochastic modeling of coupled problems, International Journal for Numerical Methods in Engineering, vol.29, issue.5, pp.352-376, 2014.

A. W. Bowman and A. Azzalini, Applied Smoothing Techniques for Data Analysis, 1997.

A. Batou and C. Soize, Calculation of Lagrange Multipliers in the Construction of Maximum Entropy Distributions in High Stochastic Dimension, SIAM/ASA Journal on Uncertainty Quantification, vol.1, issue.1, pp.431-451, 2013.
DOI : 10.1137/120901386
URL : https://hal.archives-ouvertes.fr/hal-00851201

K. Burrage, I. Lenane, and A. G. Lythe, Numerical Methods for Second???Order Stochastic Differential Equations, SIAM Journal on Scientific Computing, vol.29, issue.1, pp.29-245264, 2007.
DOI : 10.1137/050646032

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

S. Das, R. Ghanem, and A. 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

B. J. Debusschere, H. N. Najm, P. P. Pebay, O. M. Knio, R. Ghanem et al., Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes, SIAM Journal on Scientific Computing, vol.26, issue.2, pp.698-719, 2004.
DOI : 10.1137/S1064827503427741

C. Desceliers, R. Ghanem, and A. 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 A. J. Red-horse, Stochastic model reduction for chaos representations, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.37-40, pp.37-40, 2007.
DOI : 10.1016/j.cma.2006.10.047

O. G. Ernst, A. Mugler, H. J. Starkloff, and A. 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

S. Geman-and-d and . Geman, Stochastic relaxation, Gibbs distribution and the Bayesian distribution of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.6, pp.721-741, 1984.

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

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, R. Doostan, and A. J. Red-horse, A probability construction of model validation, Computer Methods in Applied Mechanics and Engineering, vol.197, pp.29-32, 2008.

D. Ghosh-and-r and . Ghanem, Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions, International Journal for Numerical Methods in Engineering, vol.28, issue.2, pp.162-184, 2008.
DOI : 10.1002/nme.2066

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.

E. Hairer, C. Lubich, and A. G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01403326

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, vol.57, issue.1, pp.57-97, 1970.
DOI : 10.1093/biomet/57.1.97

V. Keshavarzzadeh, R. Ghanem, S. F. Masri, and A. O. Aldraihem, Convergence acceleration of polynomial chaos solutions via sequence transformation, Computer Methods in Applied Mechanics and Engineering, vol.271, pp.271-167, 2014.
DOI : 10.1016/j.cma.2013.12.003

R. Khasminskii, Stochastic Stability of Differential Equations, Series: Stochastic Modelling and Applied Probability Originally published in Russian, First English edition published in 1980 under R.Z. Has'minski in the series Mechanics: Analysis by Sijthoff & Noordhoff, 1969.

P. E. Kloeden-and-e and . Platen, Numerical Solution of Stochastic Differentials Equations, 1992.

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

D. Lucor, C. H. Su, and A. G. 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

N. S. Metropolis and . Ulam, The Monte Carlo Method, Journal of the American Statistical Association, vol.44, issue.247, p.335341, 1949.
DOI : 10.1080/01621459.1949.10483310

H. N. Najm, Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics, Annual Review of Fluid Mechanics, vol.41, issue.1, pp.35-52, 2009.
DOI : 10.1146/annurev.fluid.010908.165248

R. M. Neal, Slice sampling, The Annals of Statistics, vol.31, issue.3, pp.31-705767, 2003.
DOI : 10.1214/aos/1056562461

A. Nouy, Identification of multi-modal random variables through mixtures of polynomial chaos expansions, Comptes Rendus M??canique, vol.338, issue.12, pp.698-703, 2010.
DOI : 10.1016/j.crme.2010.09.003
URL : https://hal.archives-ouvertes.fr/hal-00521932

A. Nouy and C. Soize, Random field representations for stochastic elliptic boundary value problems and statistical inverse problems, European Journal of Applied Mathematics, vol.19, issue.03, pp.339-373, 2014.
DOI : 10.1023/B:ACAP.0000013855.14971.91

G. Perrin, C. Soize, D. Duhamel, and A. C. Funfschilling, Identification of Polynomial Chaos Representations in High Dimension from a Set of Realizations, SIAM Journal on Scientific Computing, vol.34, issue.6, pp.2917-2945, 2012.
DOI : 10.1137/11084950X
URL : https://hal.archives-ouvertes.fr/hal-00770006

C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, World Scientific, vol.17, 1994.
DOI : 10.1142/2347
URL : https://hal.archives-ouvertes.fr/hal-00770411

C. Soize-and-r and . 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

C. Soize, Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices, International Journal for Numerical Methods in Engineering, vol.195, issue.4, pp.1583-1611, 2008.
DOI : 10.1002/nme.2385
URL : https://hal.archives-ouvertes.fr/hal-00684517

C. Soize-and-r and . Ghanem, Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields, Computer Methods in Applied Mechanics and Engineering, vol.198, pp.21-26, 2009.

C. Soize and C. Desceliers, Computational Aspects for Constructing Realizations of Polynomial Chaos in High Dimension, SIAM Journal on Scientific Computing, vol.32, issue.5, pp.2820-2831, 2010.
DOI : 10.1137/100787830
URL : https://hal.archives-ouvertes.fr/hal-00684323

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

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

B. Sudret, Global sensitivity analysis using polynomial chaos expansions, Reliability Engineering & System Safety, pp.964-979, 2008.
DOI : 10.1016/j.ress.2007.04.002

D. Talay-and-l and . Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equation, Stochastic Analysis and Applications, pp.94-120, 1990.

D. Talay, Simulation and numerical analysis of stochastic differential systems, Probabilistic Methods in Applied Physics, p.5496, 1995.
URL : https://hal.archives-ouvertes.fr/inria-00075246

D. Talay, Stochastic Hamiltonian system: exponential convergence to the invariant measure and discretization by the implicit Euler scheme, Markov Processes and Related Fields, pp.163-198, 2002.

R. Tipireddy-and-r and . Ghanem, Basis adaptation in homogeneous chaos spaces, Journal of Computational Physics, vol.259, pp.304-317, 2014.
DOI : 10.1016/j.jcp.2013.12.009

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

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