K. ,

R. Beck, J. , and S. Au, Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation, Journal of Engineering Mechanics, vol.128, issue.4, pp.380-391, 2002.

J. L. Beck and L. S. Katafygiotis, Updating models and their uncertainties. I: Bayesian statistical framework, Journal of Engineering Mechanics, vol.124, issue.4, pp.455-461, 1998.

A. Birolleau, G. Poëtte, and D. Lucor, Adaptive Bayesian inference for discontinuous inverse problems, application to hyperbolic conservation laws, Communications in Computational Physics, vol.16, issue.1, pp.1-34, 2014.

C. M. Bishop, Pattern recognition and machine learning. Information Science and Statistics, 2006.

G. Blatman and B. Sudret, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, vol.25, pp.183-197, 2010.

G. Blatman and B. Sudret, Adaptive sparse polynomial chaos expansion based on Least Angle Regression, Journal of Computational Physics, vol.230, pp.2345-2367, 2011.

S. P. Brooks and A. Gelman, General methods for monitoring convergence of iterative simulations, Journal of Computational and Graphical Statistics, vol.7, issue.4, pp.434-455, 1998.

O. Chapelle, V. Vapnik, and Y. Bengio, Model selection for small sample regression, Journal of Machine Learning Research, vol.48, issue.1, pp.9-23, 2002.

J. Ching and Y. Chen, Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging, Journal of Engineering Mechanics, vol.133, issue.7, pp.816-832, 2007.

J. Dick, R. N. Gantner, Q. T. Le-gia, and C. Schwab, Multilevel higher-order quasi-Monte Carlo Bayesian estimation, Mathematical Models and Methods in Applied Sciences, vol.27, issue.5, pp.953-995, 2017.

T. A. El-moselhy and Y. M. Marzouk, Bayesian inference with optimal maps, Journal of Computational Physics, vol.231, issue.23, pp.7815-7850, 2012.

N. Fajraoui, S. Marelli, and B. Sudret, Sequential design of experiment for sparse polynomial chaos expansions, SIAM/ASA Journal on Uncertainty Quantification, vol.5, issue.1, pp.1061-1085, 2017.

R. N. Gantner and M. D. Peters, Higher-order quasi-Monte Carlo for Bayesian shape inversion, SIAM/ASA Journal on Uncertainty Quantification, vol.6, issue.2, pp.707-736, 2018.

W. Gautschi, Orthogonal polynomials: Computation and approximation. Numerical Mathematics and Scientific Computation, 2004.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari et al., , 2014.

, Texts in Statistical Science, Bayesian Data Analysis

A. Gelman and D. B. Rubin, Inference from iterative simulation using multiple sequences, Statistical Science, vol.7, issue.4, pp.457-472, 1992.

R. G. Ghanem and P. Spanos, Stochastic finite elements -A spectral approach, 1991.

J. Goodman and J. Weare, Ensemble samplers with affine invariance, Communications in Applied Mathematics and Computational Science, vol.5, issue.1, pp.65-80, 2010.

H. Haario, E. Saksman, and J. Tamminen, An adaptive Metropolis algorithm, Bernoulli, vol.7, issue.2, pp.223-242, 2001.

H. L. Harney, Bayesian Inference: Data Evaluation and Decisions, 2016.

J. Jakeman, M. S. Eldred, and K. Sargsyan, Enhancing 1 -minimization estimates of polynomial chaos expansions using basis selection, Journal of Computational Physics, vol.289, pp.18-34, 2015.

E. T. Jaynes, Probability theory: The logic of science, 2003.

H. Jeffreys, An invariant form for the prior probability in estimation problems, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.186, pp.453-461, 1007.

M. C. Kennedy and A. O'hagan, Bayesian calibration of computer models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol.63, issue.3, pp.425-464, 2001.

J. Li and Y. M. Marzouk, Adaptive construction of surrogates for the Bayesian solution of inverse problems, SIAM Journal on Scientific Computing, vol.36, issue.3, pp.1163-1186, 2014.

J. Lin, Divergence measures based on the Shannon entropy, Transactions on Information Theory, vol.37, issue.1, pp.145-151, 1991.

N. Lüthen, S. Marelli, and B. Sudret, Sparse polynomial chaos expansions: Review and benchmark, SIAM/ASA Journal on Uncertainty Quantification. submitted, 2020.

S. Marelli and B. Sudret, UQLab: A framework for uncertainty quantification in Matlab, Vulnerability, Uncertainty, and Risk (Proceedings of the 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM2014), pp.2554-2563, 2014.

S. Marelli and B. Sudret, UQLab user manual -Polynomial chaos expansions, Chair of Risk, Safety and Uncertainty Quantification, 2019.

S. Marelli, P. Wagner, C. Lataniotis, and B. Sudret, Stochastic spectral embedding, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02539515

J. Marin, P. Pudlo, C. P. Robert, and R. J. Ryder, Approximate Bayesian computational methods, Statistics and Computing, vol.22, issue.6, pp.1167-1180, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00567240

Y. Marzouk, T. Moselhy, M. Parno, and A. Spantini, Sampling via measure transport: An introduction, 2016.

Y. M. Marzouk, H. N. Najm, and L. A. Rahn, Stochastic spectral methods for efficient Bayesian solution of inverse problems, Journal of Computational Physics, vol.224, pp.560-586, 2007.

Y. M. Marzouk and D. Xiu, A stochastic collocation approach to Bayesian inference in inverse problems, Communications in Computational Physics, vol.6, issue.4, pp.826-847, 2009.

J. Nagel and B. Sudret, Spectral likelihood expansions for Bayesian inference, Journal of Computational Physics, vol.309, pp.267-294, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01432170

R. M. Neal, MCMC using Hamiltonian dynamics, 2011.

, Handbook of Markov Chain Monte Carlo, Handbooks of Modern Statistical Methods, pp.113-162

M. D. Parno, Transport maps for accelerated Bayesian computation, 2015.

C. P. Robert and G. Casella, , 2004.

, Monte Carlo statistical methods (2 nd Ed, Springer Series in Statistics

M. Rosenblatt, Remarks on a multivariate transformation, Annals of Mathematical Statistics, vol.23, issue.3, pp.470-472, 1952.

Y. Shin and D. Xiu, Nonadaptive quasi-optimal points selection for least squares linear regression, SIAM Journal on Scientific Computing, vol.38, issue.1, pp.385-411, 2016.

S. A. Sisson and Y. , Handbook of approximate Bayesian computation, 2018.

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, issue.SIAM, 2005.

L. Tierney and J. B. Kadane, Accurate approximations for posterior moments and marginal densities, Journal of the American Statistical Association, vol.81, issue.393, pp.82-86, 1986.

L. Tierney, R. E. Kass, and J. B. Kadane, Approximate marginal densities of nonlinear functions, Biometrika, vol.76, issue.3, pp.425-433, 1989.

L. Tierney, R. E. Kass, and J. B. Kadane, Fully exponential Laplace approximations to expectations and variances of nonpositive functions, Journal of the American Statistical Association, vol.84, issue.407, pp.710-716, 1989.

P. Wagner, R. Fahrni, M. Klippel, A. Frangi, and B. Sudret, Bayesian calibration and sensitivity analysis of heat transfer models for fire insulation panels, Engineering Structures, vol.205, issue.15, p.110063, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02290027

M. Wand and M. C. Jones, Kernel smoothing, 1995.

D. Xiu, Numerical methods for stochastic computations -A spectral method approach, 2010.

D. 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.