Skip to main content
Log in

Analysis of adaptive directional stratification for the controlled estimation of rare event probabilities

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

In the context of structural reliability, a small probability to be assessed, a high computational time model and a relatively large input dimension are typical constraints which brought together lead to an interesting challenge. Indeed, in this framework many existing stochastic methods fail in estimating the failure probability with robustness.

Therefore, the aim of this article is to present and prove theoretical results about the validity of an original method we have introduced to overcome the specific constraints mentioned above. This new method turns out to be competitive compared with the existing techniques. It is a variant of accelerated Monte Carlo simulation method, named ADS-2—Adaptive Directional Stratification. It combines, in a two steps adaptive strategy, the stratification into quadrants and the directional simulation techniques. Two ADS-2 estimators are presented and their properties are studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Au, S., Beck, J.: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16, 263–277 (2001)

    Article  Google Scholar 

  • Blatman, G., Sudret, B.: An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab. Eng. Mech. 25, 183–197 (2010)

    Article  Google Scholar 

  • Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230, 2345–2367 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Bungartz, H., Dirnstorfer, S.: Multivariate quadrature on adaptive sparse grids. Computing 71, 89–114 (1985)

    Article  MathSciNet  Google Scholar 

  • Cannamela, C.: Apport des méthodes probabilistes dans la simulation du comportement sous irradiation du combustible à particules. Ph.D. thesis, University of Paris VII (2007)

  • Chan, J., Kroese, A.: Rare-event probability estimation with conditional Monte Carlo. Ann. Oper. Res. 189, 43–61 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Cochran, W.: Sampling Techniques, 3rd edn. Wiley, New York (1977)

    MATH  Google Scholar 

  • Crestaux, T., Le Maître, O., Martinez, J.M.: Plynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94, 1161–1172 (2009)

    Article  Google Scholar 

  • Dean, T., Dupuis, P.: Splitting for rare event simulation: a large deviations approach to design and analysis. Stoch. Process. Appl. 119(2), 562–587 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Del Moral, P., Garnier, J.: Genealogical particle analysis of rare events. Ann. Appl. Probab. 15(4), 2496–2534 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Fang, K.T., Li, R., Sudjianto, A.: Design and modeling for computer experiments. Chapman & Hall/CRC, London (2006)

    MATH  Google Scholar 

  • Fang, K.T., Kotz, S., Ng, K.: Symmetric multivariate and related distributions. In: Cox, D.R., Hinkley, D.V., Rubin, D., Silverman, B.W. (eds.) Monographs on Statistics and Applied Probability. Chapman and Hall, London/New York (1990)

    Google Scholar 

  • Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18, 209–232 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Gille-Genest, A.: Utilisation des méthodes numériques probabilistes dans les applications au domaine de fiabilite des structures. Ph.D. thesis, University of Paris VI (1999)

  • Helton, J.: Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliab. Eng. Syst. Saf. 42(2–3), 327–347 (1993)

    Article  MathSciNet  Google Scholar 

  • Helton, J., Davis, F., Johnson, J.: A comparison of uncertainty and sensitivity analysis results obtained with random and latin hypercube sampling. Reliab. Eng. Syst. Saf. 89(3), 305–330 (2005)

    Article  Google Scholar 

  • Homem-de-Mello, T., Rubinstein, R.: Estimation of rare event probabilities using cross-entropy. In: Proceedings of the 2002 Winter Simulation Conference (2002)

    Google Scholar 

  • Lagnoux-Renaudie, A.: A two-step branching splitting model under cost constraint for rare event analysis. J. Appl. Probab. 46, 429–452 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Lapeyre, B., Pardoux, E., Sentis, R.: Introduction aux Méthodes de Monte Carlo. Springer, Berlin (1997)

    Google Scholar 

  • L’Ecuyer, P., Demers, V., Tuffin, B.: Splitting for rare-event simulation. In: Proceedings of the 2006 Winter Simulation Conference (2006)

    Google Scholar 

  • L’Ecuyer, P., Demers, V., Tuffin, B.: Rare events, splitting, and quasi-Monte Carlo. ACM Trans. Model. Comput. Simul. 17(2) (2007)

  • Li, G., Wang, S.W., Georgopoulos, P., Schoendorf, J., Rabitz, H.: Random sampling-high dimensional model representation (rs-hdmr) and orthogonality of its different order component functions. J. Phys. Chem. 110(7), 2474–2485 (2006)

    Google Scholar 

  • Liu, P., Kiureghian, A.D.: Structural reliability under incomplete probability information 112, 85–104 (1986)

  • Madsen, H., Ditlevsen, O.: Strutural Reliability Methods. Wiley, New York (1996)

    Google Scholar 

  • Madsen, H., Krenk, S., Lind, N.: Methods of Structural Safety (2000). Odile Jacob

    Google Scholar 

  • Munoz Zuniga, M.: Méthodes stochastiques pour l’estimation contrôlée de faibles probabilités sur des modèles physiques complexes. application au domaine nucléaire. Ph.D. thesis, University of Paris VII (2011)

  • Munoz Zuniga, M., Garnier, J., Lefebvre, Y.: Controlled estimation of the probability of rare event for a complex physical model—examination of monotoneous variation models (2008)

  • Munoz Zuniga, M., Garnier, J., Remy, E., de Rocquigny, E.: Adaptative Directional Stratification: an adaptive directional simulation method in a stratified space (2010)

  • Rasmussen, C., Williams, C.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)

    MATH  Google Scholar 

  • Ripley, B.: Stochastic Simulation. Wiley Series in Probability and Statistics. Wiley, New York (1987)

    Book  MATH  Google Scholar 

  • Rubinstein, R., Kroese, D.: Simulation and the Monte Carlo Method, 2nd edn. Wiley, New York (2007)

    Book  Google Scholar 

  • Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, Berlin (1999)

    Google Scholar 

  • Siegmund, D.: Importance sampling in the Monte Carlo study of sequential tests. Anal. Stat. 4, 673–684 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • Soize, C., Ghanem, R.: Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26(2), 395–410 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93, 964–979 (2008)

    Article  Google Scholar 

  • Todor, R., Schwab, C.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27, 232–261 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, P.: Nonparametric importance sampling. J. Am. Stat. Assoc. 91(435), 1245–1253 (1996)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Miguel Munoz Zuniga.

Electronic Supplementary Material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Munoz Zuniga, M., Garnier, J., Remy, E. et al. Analysis of adaptive directional stratification for the controlled estimation of rare event probabilities. Stat Comput 22, 809–821 (2012). https://doi.org/10.1007/s11222-011-9277-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-011-9277-5

Keywords

Navigation