T. V. Apanasovich, M. G. Genton, and Y. Sun, A Valid Mat??rn Class of Cross-Covariance Functions for Multivariate Random Fields With Any Number of Components, Journal of the American Statistical Association, vol.21, issue.497, pp.180-193, 2012.
DOI : 10.1080/01621459.2011.643197

M. Arnst, R. G. 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

P. Bocchini and G. Deodatis, Critical review and latest developments of a class of simulation algorithms for strongly non-Gaussian random fields, Probabilistic Engineering Mechanics, vol.23, issue.4, pp.393-407, 2008.
DOI : 10.1016/j.probengmech.2007.09.001

Z. I. Botev, J. F. Grotowski, and D. P. Kroese, Kernel density estimation via diffusion, The Annals of Statistics, vol.38, issue.5, pp.2916-2957, 2010.
DOI : 10.1214/10-AOS799

G. Chan and A. T. Wood, Simulation of stationary Gaussian vector fields, Statistics and Computing, vol.9, issue.4, pp.265-268, 1999.
DOI : 10.1023/A:1008903804954

C. Dietrich and G. Newsam, Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix, SIAM Journal on Scientific Computing, vol.18, issue.4, pp.1088-1107, 1997.
DOI : 10.1137/S1064827592240555

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 1992.

P. Kree and C. Soize, Mathematics of Random Phenomena, 1986.
DOI : 10.1007/978-94-009-4770-2
URL : https://hal.archives-ouvertes.fr/hal-00770408

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, A. Noshadravan, R. G. Ghanem, and C. Soize, 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, Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model, International Journal for Numerical Methods in Engineering, vol.94, issue.108, pp.613-635, 2012.
DOI : 10.1002/nme.3338
URL : https://hal.archives-ouvertes.fr/hal-00699345

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 & Simulation, vol.11, issue.3, pp.840-870, 2013.
DOI : 10.1137/120898346
URL : https://hal.archives-ouvertes.fr/hal-00854121

J. Guilleminot and C. Soize, On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties, Journal of Elasticity, vol.21, issue.5, pp.109-130, 2013.
DOI : 10.1007/s10659-012-9396-z
URL : https://hal.archives-ouvertes.fr/hal-00724048

E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration illustrated by the StrmerVerlet method, Acta Numerica, vol.12, pp.399-450, 2003.
DOI : 10.1017/S0962492902000144

E. Hairer and G. Sörderlind, Explicit, Time Reversible, Adaptive Step Size Control, SIAM Journal on Scientific Computing, vol.26, issue.6, pp.1838-1851, 2005.
DOI : 10.1137/040606995

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

R. and D. Vogelaere, Methods of integration which preserve the contact transformation property of the Hamiltonian equations, 1956.

R. V. Field and M. Grigoriu, A method for the efficient construction and sampling of vector-valued translation random fields, Probabilistic Engineering Mechanics, vol.29, pp.79-91, 2012.
DOI : 10.1016/j.probengmech.2011.09.003

M. Frigo and S. G. Johnson, The Design and Implementation of FFTW3, Proceedings of the IEEE, Special issue on " Program Generation, Optimization, and Platform Adaptation, pp.216-231, 2005.

R. Ghanem and P. D. Spanos, Stochastic finite elements: A spectral approach, 1991.
DOI : 10.1007/978-1-4612-3094-6

T. Gneiting, H. Seveikova, D. B. Percival, M. Schlather, and Y. Jiang, : Exploring the Limits, Journal of Computational and Graphical Statistics, vol.15, issue.3, pp.483-501, 2006.
DOI : 10.1198/106186006X128551

M. Grigoriu, Existence and construction of translation models for stationary non-Gaussian processes, Probabilistic Engineering Mechanics, pp.545-551, 2009.

E. T. Jaynes, Information Theory and Statistical Mechanics, Physical Review, vol.106, issue.4, pp.620-630, 1957.
DOI : 10.1103/PhysRev.106.620

E. T. Jaynes, Information Theory and statistical mechanics. II, Physical Review, pp.171-190, 1957.

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

N. Hofmann, T. Müller-gronbach, and K. Ritter, Optimal approximation of stochastic differential equations by adaptive step-size control, Mathematics of Computation, vol.69, issue.231, pp.1017-1034, 2000.
DOI : 10.1090/S0025-5718-99-01177-1

N. Hofmann, T. Müller-gronbach, and K. Ritter, The Optimal Discretization of Stochastic Differential Equations, Journal of Complexity, vol.17, issue.1, pp.117-153, 2001.
DOI : 10.1006/jcom.2000.0570

H. Lamba, J. C. Mattingly, and A. M. Stuart, An adaptive Euler-Maruyama scheme for SDEs: convergence and stability, IMA Journal of Numerical Analysis, vol.27, issue.3, pp.479-506, 2007.
DOI : 10.1093/imanum/drl032

D. Lamberton and G. Pagès, Computation of the Invariant Distribution of a Diffusion, Bernoulli, vol.8, issue.3, pp.367-405, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00104799

B. Leimkuhler and C. Matthews, Rational Construction of Stochastic Numerical Methods for Molecular Sampling, Applied Mathematics Research eXpress, vol.1, pp.34-56, 2013.
DOI : 10.1093/amrx/abs010

V. Lemaire, An adaptive scheme for the approximation of dissipative systems, Stochastic Processes and their Applications, pp.1491-1518, 2007.
DOI : 10.1016/j.spa.2007.02.004
URL : https://hal.archives-ouvertes.fr/hal-00004266

O. P. Le-ma??trema??tre and O. M. Knio, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, 2010.
DOI : 10.1007/978-90-481-3520-2

M. Matsumoto, T. Nishimura, and M. Twister, Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Transactions on Modeling and Computer Simulation, vol.8, issue.1, pp.3-30, 1998.
DOI : 10.1145/272991.272995

J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations : locally Lipschitz vector fields and degenerate noise, Stochastic Process, Appl, vol.101, issue.2, pp.185-232, 2002.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics, vol.21, issue.6, pp.1087-1092, 1953.
DOI : 10.1063/1.1699114

G. N. Milstein and M. V. Tretyakov, Quasi-symplectic methods for Langevin-type equations, IMA Journal of Numerical Analysis, vol.23, issue.4, pp.593-626, 2003.
DOI : 10.1093/imanum/23.4.593

T. Müller-gronbach, The optimal uniform approximation of systems of stochastic differential equations, The Annals of Applied Probability, pp.664-690, 2002.

G. O. Roberts, A. Gelman, and W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms, The Annals of Applied Probability, vol.7, issue.1, pp.110-120, 1997.
DOI : 10.1214/aoap/1034625254

G. Perrin, C. Soize, D. Duhamel, and C. Fünfschilling, 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

B. Puig, F. Poirion, and C. Soize, Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms, Probabilistic Engineering Mechanics, vol.17, issue.3, pp.277-294, 2000.
DOI : 10.1016/S0266-8920(02)00010-3
URL : https://hal.archives-ouvertes.fr/hal-00686282

M. D. Shields, G. Deodatis, and P. Bocchini, A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic process by a translation process, Probabilistic Engineering Mechanics, vol.26, issue.4, pp.511-519, 2011.
DOI : 10.1016/j.probengmech.2011.04.003

C. Soize, A nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probabilistic Engineering Mechanics, vol.15, issue.3, pp.277-294, 2000.
DOI : 10.1016/S0266-8920(99)00028-4
URL : https://hal.archives-ouvertes.fr/hal-00686293

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, Construction of probability distributions in high dimension usign the maximum entropy principle: Applications to stochastic processes, random fields and random matrices, International Journal for Numerical Methods in Engineering, pp.76-1583, 2008.

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, pp.395-410, 2004.
DOI : 10.1137/S1064827503424505
URL : https://hal.archives-ouvertes.fr/hal-00686211

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

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

M. L. Stein, Fast and exact simulation of fractional Brownian motion