P. Abrahamsen, A review of Gaussian random fields and correlation functions, second edition, 1997.

R. J. Adler and J. E. Taylor, Random Fields and Geometry, 2007.
DOI : 10.1137/1.9780898718980

S. Amari and H. Nagaoka, Transactions of mathematical monographs: methods of information geometry, 2000.

E. B. Anderes, Estimating deformations of isotropic Gaussian random fields on the plane, The Annals of Statistics, vol.36, issue.2, 2005.
DOI : 10.1214/009053607000000893

E. B. Anderes and S. Chatterjee, Consistent estimates of deformed isotropic Gaussian random fields on the plane. The Annals of Statistics, pp.2324-2350, 2009.

E. B. Anderes and M. L. Stein, Estimating deformations of isotropic Gaussian random fields on the plane, The Annals of Statistics, vol.36, issue.2, pp.719-741, 2008.
DOI : 10.1214/009053607000000893

J. Bect, D. Ginsbourger, L. Li, V. Picheny, and E. Vazquez, Sequential design of computer experiments for the estimation of a probability of failure, Statistics and Computing, vol.34, issue.4, 2011.
DOI : 10.1007/s11222-011-9241-4
URL : https://hal.archives-ouvertes.fr/hal-00689580

B. Bekka, P. De-la-harpe, and A. Valette, Kazhdan's property (T), 2008.
DOI : 10.1017/CBO9780511542749

C. Berg, J. P. Christensen, and P. , Harmonic Analysis on Semigroups, 1984.
DOI : 10.1007/978-1-4612-1128-0

A. Berlinet and C. Thomas-agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, 2003.
DOI : 10.1007/978-1-4419-9096-9

H. Cartan and R. Godement, Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts. Annales Scientifiques de l'Ecole Normale Supérieure, pp.79-99, 1947.

H. Cramér and M. R. Leadbetter, Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications, 1967.

N. A. Cressie, Statistics for spatial data Wiley series in probability and mathematical statistics, 1993.

F. C. Curriero, On the use of non-euclidean isotropy in geostatistics, 2005.

B. Echard, N. Gayton, and M. Lemaire, Kriging-based monte carlo simulation to compute the probability of failure efficiently: AK-MCS method, 6` emes Journées Nationales de Fiabilité, pp.24-26, 2010.

C. Gaetan and X. Guyon, Modélisation et Statistique Spatiales, 2008.

M. G. Genton, Classes of kernels for machine learning: A statistics perspective, Journal of Machine Learning Research, vol.2, pp.299-312, 2001.

R. B. Gramacy and H. K. Lee, Bayesian Treed Gaussian Process Models With an Application to Computer Modeling, Journal of the American Statistical Association, vol.103, issue.483, pp.1119-1130, 2008.
DOI : 10.1198/016214508000000689

P. Guttorp and P. D. Sampson, Nonparametric estimation of nonstationary spatial covariance structure, Journal of the American Statistical Association, vol.87, issue.417, pp.108-119, 1992.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, 2001.

D. T. Hristopoulos, Spatial random field models inspired from statistical physics with applications in the geosciences, Physica A: Statistical Mechanics and its Applications, vol.365, issue.1, pp.211-216, 2006.
DOI : 10.1016/j.physa.2006.01.037

J. Istas, Manifold indexed fractional fields, ESAIM: Probability and Statistics, vol.16, 2011.
DOI : 10.1051/ps/2011106
URL : https://hal.archives-ouvertes.fr/hal-00851597

D. R. Jones, M. Schonlau, and W. J. Welch, Efficient global optimization of expensive black-box functions, Journal of Global Optimization, vol.13, issue.4, pp.455-492, 1998.
DOI : 10.1023/A:1008306431147

D. G. Krige, A statistical approach to some basic mine valuation problems on the witwatersrand, J. of the Chem., Metal. and Mining Soc. of South Africa, vol.52, issue.6, p.119139, 1951.

H. Lee, D. Higdon, K. Calder, and C. Holloman, Spatial models via convolutions of intrinsic processes, Statistical Modelling, vol.5, pp.1-21, 2005.

P. Lévy, Processus stochastiques et mouvement Brownien, 1965.

G. Matheron, La théorie des variables régionalisées et ses applications, 1970.

A. O. Hagan, Bayesian analysis of computer code outputs: a tutorial, Reliability Engineering and System Safety, vol.91, issue.91, pp.1290-1300, 2006.

C. J. Paciorek and M. J. Schervish, Nonstationary covariance functions for Gaussian process regression, Advances in Neural Information Processing Systems 16 (NIPS), 2004.

C. J. Paciorek, Nonstationary Gaussian Processes for Regression and Spatial Modelling, 2003.

K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory, Lecture Notes in Mathematics, vol.272, 1972.
DOI : 10.1007/BFb0058340

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 1991.

T. J. Santner, B. J. Williams, and W. J. Notz, The Design and Analysis of Computer Experiments, 2003.
DOI : 10.1007/978-1-4757-3799-8

A. M. Schmidt and A. O. Hagan, Bayesian inference for non-stationary spatial covariance structure via spatial deformations, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol.87, issue.3, pp.745-758, 2003.
DOI : 10.1002/(SICI)1099-095X(199809/10)9:5<565::AID-ENV324>3.0.CO;2-S

I. J. Schoenberg, Metric spaces and positive definite functions. Transactions of the, pp.522-536, 1938.

B. Schölkopf, The kernel trick for distances, Neural Information Processing Systems, 2000.

L. Schueremans and D. Van-gemert, Benefit of splines and neural networks in simulation based structural reliability analysis. Structural safety, pp.246-261, 2005.

M. L. Stein, Interpolation of Spatial Data, Some Theory for Kriging, 1999.

V. N. Vapnik, Statistical Learning Theory, 1998.

E. Vazquez, Modélisation Comportementale de Systèmes Non-linéaires Multivariables par MéthodesMéthodesà Noyaux et Applications, 2005.

P. H. Waarts, Structural reliability using finite element methods: an appraisal of DARS, 2000.

G. Wahba, Spline Models for Observational Data. Siam, 1990.

Y. Xiong, W. Chen, D. Apley, and X. Ding, A non-stationary covariance-based Kriging method for metamodelling in engineering design, International Journal for Numerical Methods in Engineering, vol.128, issue.6, pp.733-756, 2007.
DOI : 10.1002/nme.1969

D. Ginsbourger, CH-3012 Bern, Switzerland, david.ginsbourger@stat.unibe.ch Xavier Bay and Olivier Roustant, Ecole Nationale Supérieure des Mines, Fayol-EMSE, LSTI, 158 cours Fauriel, F-42023 Saint-Etienne, France, roustant@emse .fr; bay@emse.fr Laurent Carraro, Télécom Saint-Etienne