.. Applications-numériques, 117 4.4.1 Optimisation fiabiliste : résolution d'unprobì eme, p.123

P. De and P. Piecg, Example 2 : configuration fiable de la section (comparaison des résultats, p.123

B. Abraham, J. Mackay, G. Box, R. N. Kacker, T. J. Lorenzen et al., Taguchi's parameter design : a panel discussion, Technometrics, vol.34, pp.127-161, 1992.

N. Acharjee-2007-]-swagato-acharjee and . Zabaras, A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes, ¡ce :title¿Computational Stochastic Mechanics¡/ce :title¿. (Cité en page 33, pp.244-254, 2007.
DOI : 10.1016/j.compstruc.2006.10.004

. Agarwal-2007-]-harish, C. Agarwal, J. Mozumder, L. Renaud, and . Watson, An inverse-measure-based unilevel architecture for reliability-based design optimization. Structural and Multidisciplinary Optimization, pp.217-227, 2007.

]. G. Anderson-2009, A. Anderson, O. Guionnet, and . Zeitouni, An introduction to random matrices, 2009.
DOI : 10.1017/CBO9780511801334

]. C. Andrieu-renaud-2004, B. Andrieu-renaud, M. Sudret, and . Lemaire, The PHI2 method : a way to compute time-variant reliability. Reliability Engineering &amp ; System Safety, pp.75-86, 2004.

. Ang, L. George, A. H. Ang, W. H. Ang, and . Tang, Optimal Importance???Sampling Density Estimator, Journal of Engineering Mechanics, vol.118, issue.6, pp.1146-1163, 1992.
DOI : 10.1061/(ASCE)0733-9399(1992)118:6(1146)

A. Aoues and . Chateauneuf, Benchmark study of numerical methods for reliability-based design optimization. Structural and Multidisciplinary Optimization, pp.277-294, 0118.

H. Arenbeck and S. Missoum, Anirban Basudhar et Parviz Nikravesh Reliability-Based Optimal Design and Tolerancing for Multibody Systems Using Explicit Design Space Decomposition, Journal of Mechanical Design, vol.132, issue.2, pp.2010-103, 2010.

]. 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

]. M. Arnst and R. Ghanem, A variational-inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity, International Journal for Numerical Methods in Engineering, vol.30, issue.13, p.29, 2011.
DOI : 10.1002/nme.3307

]. J. Arora, Optimization of structural and mechanical systems, 2007.
DOI : 10.1142/6214

]. S. Au-1999, J. L. Au, and . Beck, A new adaptive importance sampling scheme for reliability calculations, Structural Safety, vol.21, issue.2, pp.135-158, 1999.
DOI : 10.1016/S0167-4730(99)00014-4

]. S. Au-2001a, J. L. Au, and . Beck, First excursion probabilities for linear systems by very efficient importance sampling, Probabilistic Engineering Mechanics, vol.16, issue.3, pp.193-207, 2001.

. Au, . Siu-kui, J. L. Au, and . Beck, Estimation of small failure probabilities in high dimensions by subset simulation, Probabilistic Engineering Mechanics, vol.16, issue.4, pp.263-277, 2001.
DOI : 10.1016/S0266-8920(01)00019-4

]. S. Au-2005 and . Au, Reliability-based design sensitivity by efficient simulation Computers &amp ; Structures, Uncertainties in Structural Mechanics and Analysis?Computational Methods. (Cité en pages 104 et 105, pp.1048-1061, 2005.

]. Babuska, R. Tempone-et-georgios, and 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

]. 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

]. Barty, J. Roy, and C. Strugarek, Hilbert-Valued Perturbed Subgradient Algorithms, Mathematics of Operations Research, vol.32, issue.3, pp.551-562, 2007.
DOI : 10.1287/moor.1070.0253

]. A. Batou and C. Soize, Rigid multibody system dynamics with uncertain rigid bodies, Multibody System Dynamics, vol.198, issue.1, pp.1-35, 2011.
DOI : 10.1007/s11044-011-9279-2

URL : https://hal.archives-ouvertes.fr/hal-00701567

]. R. Bellman, The theory of dynamic programming, Bulletin of the American Mathematical Society, vol.60, issue.6, pp.503-515, 1954.
DOI : 10.1090/S0002-9904-1954-09848-8

A. Ben-taltal and . Nemirovski, Robust optimization ? methodology and applications, Mathematical Programming, pp.453-480, 2002.

P. S. Beran, C. L. Pettit, and D. R. Millman, Uncertainty quantification of limit-cycle oscillations, ¡ce :title¿Uncertainty Quantification in Simulation Science¡/ce :title¿. (Cité en page 30, pp.217-247, 2006.
DOI : 10.1016/j.jcp.2006.03.038

]. D. Bertsekas, Dynamic programming and stochastic control Mathematics in Science and Engineering 111 Fifth Avenue, 1976.

]. M. Berveiller, B. Sudret, and M. Lemaire, Stochastic finite element: a non intrusive approach by regression, Revue europ??enne de m??canique num??rique, vol.15, issue.1-2-3, pp.81-92, 2006.
DOI : 10.3166/remn.15.81-92

]. H. Beyer and B. Sendhoff, Robust optimization ??? A comprehensive survey, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.33-34, pp.33-34, 2007.
DOI : 10.1016/j.cma.2007.03.003

]. G. Blatman, Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis, 2009.
URL : https://hal.archives-ouvertes.fr/tel-00440197

]. Bourinet, F. Deheeger, and M. Lemaire, Assessing small failure probabilities by combined subset simulation and Support Vector Machines. Structural Safety, pp.2011-98, 2011.

]. C. Broyden, A class of methods for solving nonlinear simultaneous equations, Mathematics of Computation, vol.19, issue.92, pp.577-593, 1965.
DOI : 10.1090/S0025-5718-1965-0198670-6

]. C. Broyden, The Convergence of a Class of Double-rank Minimization Algorithms, IMA Journal of Applied Mathematics, vol.6, issue.3, pp.76-90, 1970.
DOI : 10.1093/imamat/6.3.222

]. Bungartz and S. Dirnstorfer, Multivariate Quadrature on Adaptive Sparse Grids, Computing, vol.71, issue.1, pp.89-114, 1007.
DOI : 10.1007/s00607-003-0016-4

]. X. Chen, T. K. Hasselman, and D. J. Neill, Reliability based structural design optimization for practical applications, 38th Structures, Structural Dynamics, and Materials Conference, pp.2724-2732, 1997.
DOI : 10.2514/6.1997-1403

M. M. Wei-chen, J. Wiecek, . Zhang, . W. Dr, and . Chen, Quality Utility???A Compromise Programming Approach to Robust Design, Journal of Mechanical Design, vol.121, issue.2, pp.179-187, 1999.
DOI : 10.1115/1.2829440

]. Cheng, L. Xu, and L. Jiang, A sequential approximate programming strategy for reliability-based structural optimization, Computers & Structures, vol.84, issue.21, pp.1353-1367, 2006.
DOI : 10.1016/j.compstruc.2006.03.006

]. , C. , and W. Hsu, Converting reliability constraints by adaptive quantile estimation Structural Safety Probabilistic Methods for Modeling, Simulation and Optimization of Engineering Structures under Uncertainty in honor of Jim Beck's 60th Birthday, pp.316-325, 2010.

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

]. R. Croquet and E. Souza-de-cursi, Statistics of uncertain dynamical systems, 10th International Conference on Computational Structures Technology, 2010.

]. I. Das and J. E. Dennis, A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Structural and Multidisciplinary Optimization, pp.63-69, 1997.

K. Manas, . Deb, M. Ivo, J. Babu?ka, and . Tinsley-oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.48, pp.6359-6372, 2001.

]. A. Dempster, A Generalization of Bayesian Inference, Journal of the Royal Statistical Society. Series B (Methodological), vol.30, pp.205-247, 1968.
DOI : 10.1007/978-3-540-44792-4_4

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

]. C. Desceliers, C. Soize, and R. Ghanem, Identification of Chaos Representations of Elastic Properties of Random Media Using Experimental Vibration Tests, Computational Mechanics, vol.60, issue.5, pp.831-838, 2007.
DOI : 10.1007/s00466-006-0072-7

URL : https://hal.archives-ouvertes.fr/hal-00686150

]. O. Ditlevsen and H. O. Madsen, Structural Reliability Methods, pp.14-89, 2005.

A. Doostan, R. G. Ghanem, and 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

M. Dorigo and C. Blum, Ant colony optimization theory: A survey, Theoretical Computer Science, vol.344, issue.2-3, pp.243-278, 2005.
DOI : 10.1016/j.tcs.2005.05.020

]. Du and W. Chen, Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design, Journal of Mechanical Design, vol.126, issue.2, pp.225-233, 2004.
DOI : 10.1115/1.1649968

]. C. Du, D. Kurowicka, and R. M. Cooke, Techniques for generic probabilistic inversion, Computational Statistics & Data Analysis, vol.50, issue.5, pp.1164-1187, 2006.
DOI : 10.1016/j.csda.2005.01.002

S. Michael and . Eldred, Design Under Uncertainty Employing Stochastic Expension Methods, International Journal for Uncertainty Quantification, vol.1, issue.2, pp.119-146, 2011.

C. Howard, O. G. Elman, D. P. Ernst, M. Leary, and . Stewart, Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.911, pp.1037-1055, 2005.

]. I. Enevoldsen and J. D. Sorensen, Reliability???Based Optimization of Series Systems of Parallel Systems, Journal of Structural Engineering, vol.119, issue.4, pp.1069-1084, 1993.
DOI : 10.1061/(ASCE)0733-9445(1993)119:4(1069)

L. Faravelli, Response???Surface Approach for Reliability Analysis, Journal of Engineering Mechanics, vol.115, issue.12, pp.2763-2781, 1989.
DOI : 10.1061/(ASCE)0733-9399(1989)115:12(2763)

]. R. Field, J. Red-horse, and T. Paez, A nondeterministic shock and vibration application using polynomial chaos expansions, Proc. 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability, 2000.

]. R. Field, Numerical methods to estimate the coefficients of the polynomial chaos expansion, Proceedings of the 15 th ASCE Engineering Mechanics Conference. Columbia University, 2002.

]. R. Field and M. Grigoriu, On the accuracy of the polynomial chaos approximation, Fourth International Conference on Computational Stochastic Mechanics. (Cité en page 32, pp.65-80, 2004.
DOI : 10.1016/j.probengmech.2003.11.017

]. B. Fiessler, R. Rackwitz, and H. Neumann, Quadratic Limit States in Structural Reliability, J. Eng. Mech. Div, vol.105, pp.661-676, 1979.

]. R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients, The Computer Journal, vol.7, issue.2, pp.149-154, 1964.
DOI : 10.1093/comjnl/7.2.149

URL : http://comjnl.oxfordjournals.org/cgi/content/short/7/2/149

]. R. Forster and R. Kornhuber, A polynomial chaos approach to stochastic variational inequalities, Journal of Numerical Mathematics, vol.18, issue.4, pp.235-255, 2010.
DOI : 10.1515/jnum.2010.012

]. R. Foschi, H. Li, and J. Zhang, Reliability and performance-based design: a computational approach and applications, Structural Safety, vol.24, issue.2-4, pp.205-218, 2002.
DOI : 10.1016/S0167-4730(02)00025-5

M. Gasser, . Gerhart-iwo, and . Schuëller, Reliability-Based Optimization of structural systems, Mathematical Methods of Operations Research, vol.1, issue.3, pp.287-307, 1997.
DOI : 10.1007/BF01194858

]. N. Gayton, A. Mohamed, J. D. Sorensen, M. Pendola, and M. Lemaire, Calibration methods for reliability-based design codes, Structural Safety, vol.26, issue.1, pp.91-121, 2004.
DOI : 10.1016/S0167-4730(03)00024-9

C. Geuzaine, J. Geuzaine, and . Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, vol.69, issue.4, pp.1309-1331, 2009.
DOI : 10.1002/nme.2579

]. R. Ghanem and P. D. Spanos, Stochastic Finite Elements : A Spectral Approach, pp.24-31, 1991.
DOI : 10.1007/978-1-4612-3094-6

G. Roger, R. M. Ghanem, and . Kruger, Numerical solution of spectral stochastic finite element systems, Computer Methods in Applied Mechanics and Engineering, vol.129, issue.3, pp.289-303, 1996.

G. Roger, A. Ghanem, and . Doostan, On the construction and analysis of stochastic models : Characterization and propagation of the errors associated with limited data, Journal of Computational Physics Uncertainty Quantification in Simulation Science, vol.217, issue.1, pp.63-81, 2006.

]. Ghanem and D. Ghosh, Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition, International Journal for Numerical Methods in Engineering, vol.64, issue.4, pp.486-504, 2007.
DOI : 10.1002/nme.2025

M. Dan, R. G. Ghiocel, and . Ghanem, Stochastic Finite-Element Analysis of Seismic Soil?Structure Interaction, Journal of Engineering Mechanics, vol.128, issue.1, pp.66-77, 2002.

L. Pault-goethals, . Aragon-et-byung-rae, and . Cho, Experimental investigations of estimated response surface functions with different variability measures, International Journal of Experimental Design and Process Optimisation, vol.1, issue.2/3, pp.123-163, 2009.
DOI : 10.1504/IJEDPO.2009.030316

]. Goldberg, Genetic algorithms in search, optimization, and machine learning, 1989.

V. Ramana, L. Grandhi, and . Wang, Reliability-based structural optimization using improved two-point adaptive nonlinear approximations, Finite Elements in Analysis and Design, vol.29, issue.1, pp.35-48, 1998.

K. James, T. Guest, and . Igusa, Structural optimization under uncertain loads and nodal locations, Computer Methods in Applied Mechanics and Engineering Computational Methods in Optimization Considering Uncertainties, vol.198, issue.1, pp.116-124, 2008.

]. 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.2011-2041, 2011.
DOI : 10.1002/nme.3338

URL : https://hal.archives-ouvertes.fr/hal-00699345

]. A. Hasofer and N. Lind, An exact and invariant first order reliability format, Journal of the Engineering Mechanics Division, vol.100, pp.111-121, 1974.

]. S. Hernandez, Structural Optimization: 1960-2010 and Beyond, Computational Technology Reviews, vol.2, pp.177-222, 2010.
DOI : 10.4203/ctr.2.8

P. Hoai-2002-]-an-le-thi-hoai, V. Dinh-tao-et-nguyen, and . Thoai, Combination between global and local methods for solving an optimization problem over the efficient set, European Journal of Operational Research, vol.142, issue.2, pp.258-270, 2002.

H. Lopez-2010 and ]. Lopez, Optimisation en présence d'incertitudes, pp.30-34, 2010.

H. Lopez, E. Souza-de-cursi, and D. Lemosse, Approximating the probability density function of the optimal point of an optimization problem, Engineering Optimization, vol.1, issue.3, pp.281-303, 2011.
DOI : 10.1137/S1064827501387826

E. Jorge, D. A. Hurtado, and . Alvarez, Neural-network-based reliability analysis : a comparative study, Computer Methods in Applied Mechanics and Engineering, vol.191, issue.1-2, pp.113-132, 2001.

E. Jorge and . Hurtado, Filtered importance sampling with support vector margin : A powerful method for structural reliability analysis, Structural Safety, vol.29, issue.1, pp.2-15, 2007.

]. S. Isukapalli, Uncertainty analysis of transport-transformation models, pp.33-52, 1999.

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

A. Hector, M. A. Jensen, and . Catalan, On the effects of non-linear elements in the reliability-based optimal design of stochastic dynamical systems, Cité en page 103, pp.802-816, 2007.

]. H. Jensen, M. A. Valdebenito, G. I. Schuëller, and D. S. Kusanovic, Reliability-based optimization of stochastic systems using line search, Press, Corrected ProofCité en page 105, 2009.
DOI : 10.1016/j.cma.2009.08.016

K. Irfan and C. A. Mcmahon, A response surface method based on weighted regression for structural reliability analysis, Probabilistic Engineering Mechanics, vol.20, issue.1, pp.11-17, 2005.

]. R. Baker and . Kearfott, An interval branch and bound algorithm for bound constrained optimization problems, Journal of Global Optimization, vol.2, pp.259-280, 1992.

A. Keese, G. Hermann, and . Matthies, Hierarchical parallelisation for the solution of stochastic finite element equations Computers &amp ; Structures, ¡ce :title¿Uncertainties in Structural Mechanics and Analysis?Computational Methods¡/ce :title¿. (Cité en page 29, pp.1033-1047, 2005.

]. J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings of ICNN'95, International Conference on Neural Networks, pp.1942-1948, 1995.
DOI : 10.1109/ICNN.1995.488968

]. G. Kharmanda, A. Mohsine, A. Makloufi, and A. El-hami, Recent methodologies for reliability-based design optimization, International Journal for Simulation and Multidisciplinary Design Optimization, vol.2, issue.1, pp.11-23, 2008.
DOI : 10.1051/smdo:2008002

URL : http://doi.org/10.1051/smdo:2008002

]. J. Kiefer and J. Wolfowitz, Stochastic Estimation of the Maximum of a Regression Function, The Annals of Mathematical Statistics, vol.23, issue.3, pp.462-466, 1952.
DOI : 10.1214/aoms/1177729392

]. N. Kim, N. Queipo, and H. Wang, Efficient Shape Optimization Under Uncertainty Using Polynomial Chaos Expansions and Local Sensitivities, AIAA Journal, vol.44, issue.5, pp.1112-1116, 2006.
DOI : 10.2514/1.13011

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.462.5702

]. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by Simulated Annealing, Science, vol.220, issue.4598, pp.671-680, 1983.
DOI : 10.1126/science.220.4598.671

J. Michaa-l-kleiber, J. Law-knabel, and . Rojek, Response surface method for probabilistic assessment of metal forming failures, International Journal for Numerical Methods in Engineering, vol.60, issue.1, pp.51-67, 2004.
DOI : 10.1002/nme.954

]. Koehler and A. Owen, Computer Experiments, editeurs, Handbook of Statistics, 13 : Design and Analysis of Experiments, pp.261-308, 1996.

]. N. Kuschel and R. Rackwitz, A new approach for structural optimization of series systems, Applications of Statistics and Probability. Mechers and Stewart, 2000.

]. Laporte, V. François, and . Louveaux, The integer L-shaped method for stochastic integer programs with complete recourse, Operations Research Letters, vol.13, issue.3, pp.133-142, 1993.
DOI : 10.1016/0167-6377(93)90002-X

L. P. Ma??trema??tre-2001-]-o, O. M. Le-ma??trema??tre, . Knio, N. Habib, R. G. Najm et al., A Stochastic Projection Method for Fluid Flow : I. Basic Formulation, Journal of Computational Physics, vol.173, issue.2, pp.481-511, 2001.

L. P. Ma??trema??tre-2002-]-o, M. T. Le-ma??trema??tre, . Reagan, N. Habib, R. G. Najm et al., A Stochastic Projection Method for Fluid Flow : II. Random Process, Journal of Computational Physics, vol.181, issue.1, pp.9-44, 2002.

L. P. Ma??trema??tre-2004-]-o, O. M. Le-ma??trema??tre, H. N. Knio, R. G. Najm, and . Ghanem, Uncertainty propagation using Wiener?Haar expansions, Journal of Computational Physics, vol.197, issue.1, pp.28-57, 2004.

L. L. Ma??trema??tre-2006-]-o and . Le-ma??trema??tre, Développement en polynômes de chaos d'un modèle lagrangien d'´ ecoulement autour d'un profil, Comptes Rendus Mécanique, vol.334, issue.11, pp.693-699, 2006.

L. L. Ma??trema??tre-2010a-]-o, L. Le-ma??trema??tre, O. M. Mathelin, . Knio, and Y. Hussaini, Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics, Discrete and Continuous Dynamical Systems -Series A (DCDS-A), pp.199-226, 2010.

L. P. Ma??trema??tre-2010b-]-o, L. Le-ma??trema??tre, and . Mathelin, Equation-free model reduction for complex dynamical systems, International Journal for Numerical Methods in Fluids, vol.63, issue.2, pp.163-184

]. M. Lemaire-2005 and . Lemaire, Fiabilité des structures -couplage mécano-fiabiliste statique, Hermes, 2005.

]. Liang and P. Zissimos, A Single-Loop Approach for System Reliability-Based Design Optimization, Journal of Mechanical Design, vol.129, issue.12, pp.1215-1224, 2007.
DOI : 10.1115/1.2779884

]. Liu and A. Der-kiureghian, Optimization Algorithms for Structural Reliability Analysis, 1986.

]. Liu-et-armen-der-kiureghian, Optimization algorithms for structural reliability Structural Safety, pp.167-177, 1991.

]. Liu and K. K. Lai, Focus on computational neurobiology, pp.13-27, 2004.

L. and S. Luke, Essentials of metaheuristics, 2009.

]. Luo, Z. Kang, and A. Li, Structural reliability assessment based on probability and convex set mixed model, Computers & Structures, vol.87, issue.21-22, pp.1408-1415, 2009.
DOI : 10.1016/j.compstruc.2009.06.001

]. H. Madsen, Omission sensitivity factors Structural Safety, pp.35-45, 1988.
DOI : 10.1016/0167-4730(88)90004-5

]. H. Madsen and F. Hensen, A Comparison of Some Algorithms for Reliability Based Structural Optimization and Sensitivity Analysis, Proceedings of the 4th IFIP WG 7, 1992.
DOI : 10.1007/978-3-642-84753-0_34

M. and K. Marti, Solving stochastic structural optimization problems by RSM-based stochastic approximation methods ? gradient estimation in case of intermediate variables, Mathematical Methods of Operations Research, vol.6, issue.5, pp.409-434, 1997.
DOI : 10.1007/BF01194863

M. Y. Mathelin and . Hussaini, A Stochastic Collocation Algorithm for Uncertainty Analysis, pp.2003-212153, 2003.

L. Madan and . Mehta, Random matrices Academic Press, 3 ´ edition, 2004.

]. A. Messac and A. Ismail-yahaya, Multiobjective robust design using physical programming. Structural and Multidisciplinary Optimization, pp.357-371, 2002.
DOI : 10.1007/s00158-002-0196-0

M. Rosenbluth, A. Rosenbluth, E. Teller, and . Teller, Equation of state calculations by fast computing machines, Journal of Medical Physics, vol.21, issue.6, pp.1087-1092, 1953.

M. Ramon and E. Moore, Interval analysis (automatic computation s.), 1967.

]. F. Moses and D. E. Kinser, Optimum structural design with failure probability constraints, AIAA Journal, vol.5, pp.1152-1158, 1967.

]. Nechak, S. Berger, and E. Aubry, A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems, European Journal of Mechanics - A/Solids, vol.30, issue.4, pp.594-607, 2011.
DOI : 10.1016/j.euromechsol.2011.03.002

URL : https://hal.archives-ouvertes.fr/hal-00753962

]. J. Nelder and R. Mead, A Simplex Method for Function Minimization, The Computer Journal, vol.7, issue.4, pp.308-313, 1965.
DOI : 10.1093/comjnl/7.4.308

]. M. Ngah and A. Young, Application of the spectral stochastic finite element method for performance prediction of composite structures, Composite Structures, vol.78, issue.3, pp.447-456, 2007.
DOI : 10.1016/j.compstruct.2005.11.009

]. X. Nguyen, Algorithmes probabilistes appliquésappliqués`appliquésà la durabilité etàet`età la mécanique des ouvrages de génie civil, 2007.

]. F. Nobile, R. Tempone, and C. G. Webster, An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.46, issue.5, pp.2411-2442, 2008.
DOI : 10.1137/070680540

]. 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

]. J. Nocedal and S. J. Wright, Numerical optimization, pp.15-117, 2000.
DOI : 10.1007/b98874

]. 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.4718-4736, 2008.
DOI : 10.1016/j.cma.2008.06.012

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

N. Kevin, E. K. Otto, and . Antonsson, Design Parameter Selection in the Presence of Noise, Research in Engineering Design, pp.234-246, 1993.

]. A. Papageorgiou and J. F. Traub, Faster Evaluation of Multidimensional Integrals, Computers in Physics, vol.11, issue.6, pp.574-578, 1997.
DOI : 10.1063/1.168616

Y. Sung, . Park, K. Anil, and . Bera, Maximum entropy autoregressive conditional heteroskedasticity model, Journal of Econometrics, vol.150, issue.2, pp.219-230, 2009.

]. M. Pellissetti and R. Ghanem, Iterative solution of systems of linear equations arising in the context of stochastic finite elements, Advances in Engineering Software, pp.607-616, 2000.
DOI : 10.1016/S0965-9978(00)00034-X

]. M. Pereira and L. M. Pinto, Multi-stage stochastic optimization applied to energy planning, Mathematical Programming, pp.359-375, 1991.
DOI : 10.1007/BF01582895

]. M. Powell, A hybrid method for nonlinear equations, Philip Rabinowitz, editeur, Numerical Methods for Nonlinear Algebraic Equations, pp.87-114, 1970.

]. H. Pradlwarter, G. I. Schueller, and G. S. Szekely, Random eigenvalue problems for large systems, Computers & Structures, vol.80, issue.27-30, pp.2415-2424, 2002.
DOI : 10.1016/S0045-7949(02)00237-7

]. H. Pradlwarter, G. I. Schuëller, P. S. Koutsourelakis, and D. C. Charmpis, Application of line sampling simulation method to reliability benchmark problems, Benchmark Study on Reliability in High Dimensions. (Cité en page 97, pp.208-221, 2007.
DOI : 10.1016/j.strusafe.2006.07.009

S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition : The Art of Scientific Computing, 2007.

]. S. Rahman and D. Wei, A univariate approximation at most probable point for higher-order reliability analysis, International Journal of Solids and Structures, vol.43, issue.9, pp.2820-2839, 2006.
DOI : 10.1016/j.ijsolstr.2005.05.053

]. Rahman, A solution of the random eigenvalue problem by a dimensional decomposition method, International Journal for Numerical Methods in Engineering, vol.23, issue.9, pp.1318-1340, 2006.
DOI : 10.1002/nme.1665

]. T. Ritto, C. Soize, and R. Sampaio, Probabilistic model identification of the bit???rock-interaction-model uncertainties in nonlinear dynamics of a drill-string, Mechanics Research Communications, vol.37, issue.6, pp.584-589, 2010.
DOI : 10.1016/j.mechrescom.2010.07.004

URL : https://hal.archives-ouvertes.fr/hal-00684312

R. Herbert and S. Monro, A Stochastic Approximation Method, The Annals of Mathematical Statistics, vol.22, issue.3, pp.400-407, 1951.

]. J. Rojas, F. A. Chegury, and D. Rade, Structural reliability analysis combining ant colony optimization and finite element modeling, 16th posmec. FEMEC/UFU, 2006.

]. M. Rosenblatt, Remarks on a Multivariate Transformation, The Annals of Mathematical Statistics, vol.23, issue.3, pp.470-472, 1952.
DOI : 10.1214/aoms/1177729394

]. J. Royset, A. Der-kiureghian, E. Polak, J. O. Royset, and E. Polak, Reliability-based optimal design using sample average approximations, Probabilistic Engineering Mechanics, vol.19, issue.4, pp.331-343, 2001.
DOI : 10.1016/j.probengmech.2004.03.001

]. R. Rubinstein, Simulation and the Monte Carlo method, 1981.

A. Saltelli, S. Tarantola, F. Campolongo, and M. Ratto, Sensitivity analysis in practice : A guide to assessing scientific models, 2004.
DOI : 10.1002/0470870958

. Sampaio and . Cataldo, Comparing Two Strategies to Model Uncertainties in Structural Dynamics, Shock and Vibration, vol.17, issue.2, pp.171-186, 2010.
DOI : 10.1155/2010/837362

]. D. Sarsri, L. Azrar, A. Jebbouri, and A. Hami, Component mode synthesis and polynomial chaos expansions for stochastic frequency functions of large linear FE models, Computers & Structures, vol.89, issue.3-4, pp.346-356, 2011.
DOI : 10.1016/j.compstruc.2010.11.009

]. G. Schuëller and R. Stix, A critical appraisal of methods to determine failure probabilities Structural Safety, pp.293-309, 1987.

G. I. Schuëller, H. J. Pradlwarter, and P. S. Koutsourelakis, A critical appraisal of reliability estimation procedures for high dimensions, Probabilistic Engineering Mechanics, pp.463-474, 2004.
DOI : 10.1016/j.probengmech.2004.05.004

G. I. Schuëller and H. J. Pradlwarter, Benchmark study on reliability estimation in higher dimensions of structural systems ? An overview Structural Safety, A Benchmark Study on Reliability in High Dimensions, vol.29, issue.3, pp.167-182, 2007.

G. I. Schuëller and H. A. Jensen, Computational methods in optimization considering uncertainties ??? An overview, Computational Methods in Optimization Considering Uncertainties. (Cité en page 22, pp.2-13, 2008.
DOI : 10.1016/j.cma.2008.05.004

]. C. Shannon, A mathematical theory of communication, Bell system technical journal, vol.27, 1948.

]. Shapiro, Stochastic programming approach to optimization under uncertainty, Mathematical Programming, pp.183-220, 2008.
DOI : 10.1007/s10107-006-0090-4

]. J. Shen and J. Chen, An efficient polynomial chaos method for uncertainty quantification in electromagnetic simulations, Antennas and Propagation Society International Symposium (APSURSI), pp.1-4, 2010.

]. I. Sloan and H. Wozniakowski, When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?, Journal of Complexity, vol.14, issue.1, pp.1-33, 1998.
DOI : 10.1006/jcom.1997.0463

]. I. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, The Second IMACS Seminar on Monte Carlo Methods. (Cité en, pp.271-280, 2001.
DOI : 10.1016/S0378-4754(00)00270-6

]. C. Soize, Random matrix theory and non-parametric model of random uncertainties in vibration analysis, Journal of Sound and Vibration, vol.263, issue.4, pp.893-916, 2003.
DOI : 10.1016/S0022-460X(02)01170-7

URL : https://hal.archives-ouvertes.fr/hal-00686213

]. C. Soize, A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics, Uncertainty in structural dynamics. (Cité en pages 10 et 29, pp.623-652, 2005.
DOI : 10.1016/j.jsv.2005.07.009

URL : https://hal.archives-ouvertes.fr/hal-00686182

]. C. Soize, Random matrix theory for modeling uncertainties in computational mechanics, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.12-16, pp.12-16, 2005.
DOI : 10.1016/j.cma.2004.06.038

URL : https://hal.archives-ouvertes.fr/hal-00686187

]. C. Soize, E. Capiez-lernout, J. Durand, C. Fernandez, and L. Gagliardini, Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation, Computational Methods in Optimization Considering Uncertainties. (Cité en page 24, pp.150-163, 2008.
DOI : 10.1016/j.cma.2008.04.007

URL : https://hal.archives-ouvertes.fr/hal-00686138

C. Soize, R. G. Soize, and . Ghanem, Reduced Chaos decomposition with random coefficients of vector-valued random variables and random fields Advances in Simulation-Based Engineering Sciences -Honoring, Computer Methods in Applied Mechanics and Engineering, vol.198, pp.21-26, 2009.

]. 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

]. C. Soize-2011, A. Soize, and . Batou, Stochastic Reduced-Order Model in Low-Frequency Dynamics in Presence of Numerous Local Elastic Modes, Journal of Applied Mechanics, vol.78, issue.6, pp.61003-2011
DOI : 10.1115/1.4002593

E. Souza-de-cursi and M. B. Cortes, Developments in neural networks and evolutionary computing for civil and structural engineering, chapitre General Genetic Algorithms and Simulated Annealing Perturbation of the Gradient Method with a Fixed Parameter, pp.189-198, 1995.

]. J. Souza-de-cursi and . Souza-de-cursi, A general framework for the stochastic/deterministic hybridisation with convergence results. Rapport technique, 1999.

E. Souza-de-cursi, On the numerical determination of the probability distribution of uncertain natural frequencies and modes, 18th International Congress on Sound and Vibration, 2011.

]. G. Stefanou, A. Nouy, and A. Clement, 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

]. Sudret, M. Berveiller, and M. Lemaire, El??ments finis stochastiques en ??lasticit?? lin??aire, Comptes Rendus M??canique, vol.332, issue.7, pp.531-537, 2004.
DOI : 10.1016/j.crme.2004.02.024

]. A. Taflanidis and J. L. Beck, Stochastic Subset Optimization for optimal reliability problems, 5th International Conference on Computational Stochastic Mechanics. (Cité en page 104, pp.324-338, 2008.
DOI : 10.1016/j.probengmech.2007.12.011

]. Taguchi, System of experimental design : engineering methods to optimize quality and minimize costs, White Plains, N.Y, vol.2, 1987.

]. P. Thoft-christensen and Y. Murotsu, Application of structural systems reliability theory, 1986.
DOI : 10.1007/978-3-642-82764-8

]. J. Tu, K. K. Choi, and Y. H. Park, A New Study on Reliability-Based Design Optimization, Journal of Mechanical Design, vol.121, issue.4, pp.557-564, 1999.
DOI : 10.1115/1.2829499

M. Valdebenito and G. Schuëller, A survey on approaches for reliability-based optimization. Structural and Multidisciplinary Optimization, pp.645-663, 2010.

M. A. Valdebenito and G. I. Schuëller, Efficient strategies for reliability-based optimization involving non-linear, dynamical structures, Cité en page 103, pp.1797-1811, 2011.
DOI : 10.1016/j.compstruc.2010.10.014

]. M. Webster, M. A. Tatang, and G. J. Mcrae, Application of the probabilistic collocation method for an uncertainty analysis of a simple ocean model. Rapport technique, MIT Joint Program on the Science and Policy of Global Change, 1996.

A. Jeroen, A. Witteveen, S. Loeven, H. Sarkar, and . Bijl, Probabilistic collocation for period-1 limit cycle oscillations, Journal of Sound and Vibration, vol.311, issue.1-2, pp.421-439, 2008.

P. Wolfe and . Wolfe, Convergence Conditions for Ascent Methods, SIAM Review, vol.11, issue.2, pp.226-235, 1969.
DOI : 10.1137/1011036

]. D. Wolpert and W. G. Macready, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation, vol.1, issue.1, pp.67-82, 1997.
DOI : 10.1109/4235.585893

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.138.6606

]. Y. Wu and W. Wang, Efficient probabilistic design by converting reliability constraints to approximately equivalent deterministic constraints, Journal of Integrated Design and Process Sciences, vol.2, pp.13-21, 1998.

]. 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

]. Xiu and J. S. Hesthaven, High-Order Collocation Methods for Differential Equations with Random Inputs, SIAM Journal on Scientific Computing, vol.27, issue.3, pp.1118-1139, 2005.
DOI : 10.1137/040615201

]. H. Xu and S. Rahman, Decomposition methods for structural reliability analysis, Probabilistic Engineering Mechanics, vol.20, issue.3, pp.239-250, 2005.
DOI : 10.1016/j.probengmech.2005.05.005

]. R. Yang and L. Gu, Experience with approximate reliability-based optimization methods. Structural and Multidisciplinary Optimization, pp.152-159, 2004.
DOI : 10.2514/6.2003-1781

]. , Y. , and G. Cheng, Further study on efficiency of sequential approximate programming for probabilistic structural design optimization. Structural and Multidisciplinary Optimization, pp.509-522, 2008.

]. Youn, K. Choi, and Y. Park, Hybrid Analysis Method for Reliability-Based Design Optimization, Journal of Mechanical Design, vol.125, issue.2, pp.221-232, 2003.
DOI : 10.1115/1.1561042

]. Youn and K. Choi, An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches, Journal of Mechanical Design, vol.126, issue.3, pp.403-411, 2004.
DOI : 10.1115/1.1701880

]. Yu and X. Wang, Stochastic Programming Models in Financial Optimization : A Survey, 2003.

]. L. Zadeh, Fuzzy sets, Information and Control, vol.8, issue.3, pp.338-353, 1965.
DOI : 10.1016/S0019-9958(65)90241-X