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Efficient Evaluation of Reliability-Oriented Sensitivity Indices

  • G. PerrinEmail author
  • G. Defaux
Article
  • 36 Downloads

Abstract

The role of simulation keeps increasing for the reliability analysis of complex systems. Most of the time, these analyses can be reduced to estimating the probability of occurrence of an undesirable event, also called failure probability, using a stochastic model of the system. If the considered event is rare, sophisticated sample-based procedures are generally introduced to get a relevant estimate of the failure probability. Based on the samples constructed for the evaluation of this estimate, this work considers two types of reliability-oriented sensitivity indices (ROSI). The first ones are introduced to identify the model inputs whose variability has to be reduced in priority to decrease this probability. The second ones are used to find the model inputs whose distribution has to be particularly well-characterized for the available estimate to be realistic. It is also shown how these ROSI can be derived when the true model is approximated by a surrogate model. In particular, an innovative procedure is proposed to take into account the surrogate model uncertainty in the estimation of these ROSI. The proposed approach is then applied to the reliability analysis of a series of numerical and industrial examples.

Keywords

Sobol indices Gaussian process Sensitivity analysis Risk analysis 

Mathematics Subject Classification

49Q12 62L05 62K20 

Notes

References

  1. 1.
    Rubinstein, R.T., Kroese, D.: Simulation and the Monte Carlo Method. Wiley, Hoboken (2008)zbMATHGoogle Scholar
  2. 2.
    Freudenthal, A.N.: Safety and the probability of structural failure. Trans. ASCE 121, 1337–1397 (1956)Google Scholar
  3. 3.
    Rackwitz, R., Fiessler, B.: Structural reliability under combined load sequences. J. Comput. Struct. 9, 479–484 (1978)CrossRefzbMATHGoogle Scholar
  4. 4.
    Lemaire, M.: Structural Reliability. Wiley, New York (2009)CrossRefGoogle Scholar
  5. 5.
    Castillo, E., Sarabia, J., Solares, C., Gomez, P.: Uncertainty analyses in fault trees and Bayesian networks using FORM/SORM methods. Reliab. Eng. Syst. Saf. 65, 29–40 (1999)CrossRefGoogle Scholar
  6. 6.
    Kahn, H., Harris, T.: Estimation of particle transmission by random sampling. Natl. Bur. Stand. Appl. Math. Ser. 12, 27–30 (1951)Google Scholar
  7. 7.
    Raguet, H., Marrel, A.: arXiv:1801.10047 (2018)
  8. 8.
    Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D.: Global Sensitivity Analysis: The Primer. Wiley, New York (2008)zbMATHGoogle Scholar
  9. 9.
    Cui, L., Lu, Z., Zhao, X.: Moment-independent importance measure of basic random variable and its probability density evolution solution. Sci. China Technol. Sci. 53(4), 1138–1145 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Li, L., Lu, Z., Feng, J., Wang, B.: Moment-independent importance measure of basic variable and its state dependent parameter solution. Struct. Saf. 38, 40–47 (2012)CrossRefGoogle Scholar
  11. 11.
    Sobol, I., Tarantola, S., Gatelli, D., Kucherenko, S., Mauntz, W.: Estimating the approximation error when fixing unessential factors in global sensitivity analysis. Reliab. Eng. Syst. Saf. 92, 957–960 (2007)CrossRefGoogle Scholar
  12. 12.
    Wei, P., Lu, Z., Hao, W., Feng, J., Wang, B.: Efficient sampling methods for global reliability sensitivity analysis. Comput. Phys. Commun. 183(8), 1728–1743 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yun, W., Lu, Z., Jiang, X., Liu, S.: An efficient method for estimating global sensitivity indices. Int. J. Numer. Methods Eng. 108, 1275–1289 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rubinstein, R.T.: The score function approach for sensitivity analysis of computer simulation models. Math. Comput. Simul. 28(5), 351–379 (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lemaitre, P., Sergienko, E., Arnaud, A., Bousquet, N., Gamboa, F., Iooss, B.: Density modification-based reliability sensitivity analysis. J. Stat. Comput. Simul. 85(6), 1200–1223 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments. Stat. Sci. 4, 409–435 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Santner, T.J., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23, 470–472 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nataf, A.: Determination des distributions dont les marges sont donnees. C. R. Acad. Sci. Paris 225, 42–43 (1962)zbMATHGoogle Scholar
  20. 20.
    Lebrun, R., Dutfoy, A.: An innovating analysis of the Nataf transformation from the copula viewpoint. Prob. Eng. Mech. 24, 312–320 (2009)CrossRefGoogle Scholar
  21. 21.
    Cérou, F., Guyader, A.: Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25(2), 417–443 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Walter, C.: Moving particles: a parallel optimal multilevel splitting method with application in quantiles estimation and meta-model based algorithms. Struct. Saf. 55, 10–25 (2015)CrossRefGoogle Scholar
  23. 23.
    Au, S., Beck, J.: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16(4), 263–277 (2001)CrossRefGoogle Scholar
  24. 24.
    Cérou, F., Moral, P.D., Furon, T., Guyader, A.: Sequential Monte Carlo for rare event simulation. Stat. Comput. 22(3), 795–808 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. 44, 335–341 (1949)CrossRefzbMATHGoogle Scholar
  26. 26.
    Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sobol, I.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55, 271–280 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sobol, I., Myshetskaya, E.: Monte Carlo estimators for small sensitivity indices. Monte Carlo Methods Appl. 13, 455–465 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Owen, A.: Better estimation of small Sobol sensitivity indices. ACM Trans. Model. Comput. Simul. 23, 11 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, C., Mahadevan, S.: An efficient modularized sample-based method to estimate the first-order Sobol’ index. Reliab. Eng. Syst. Saf. 153, 110–121 (2016)CrossRefGoogle Scholar
  31. 31.
    Kucherenko, S., Song, S.: Different numerical estimators for main effect global sensitivity indices. Reliab. Eng. Syst. Saf. 165, 222–238 (2017)CrossRefGoogle Scholar
  32. 32.
    Xiao, S., Lu, Z.: Structural reliability sensitivity analysis based on classification of model output. Aerosp. Sci. Technol. 57, 279–291 (2018)Google Scholar
  33. 33.
    Perrin, G., Soize, C., Ouhbi, N.: Data-driven kernel representations for sampling with an unknown block dependence structure under correlation constraints. J. Comput. Stat. Data Anal. 119, 139–154 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Perrin, G., Soize, C., Duhamel, D., Funfschilling, C.: Identification of polynomial chaos representations in high dimension from a set of realizations. SIAM J. Sci. Comput. 34(6), 2917–2945 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lemaitre, P.: Analyse de sensibilité en fiabilité des structures. Ph.D. thesis, University of Bordeaux, France (2014)Google Scholar
  36. 36.
    Gratiet, L.L., Cannamela, C., Iooss, B.: A Bayesian approach for global sensitivity analysis of (multifidelity) computer codes. SIAM/ASA J. Uncertain. Quantif. 2(1), 336–363 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Iooss, B., Gratiet, L.L.: Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes. Reliab. Eng. Syst. Saf. (2017).  https://doi.org/10.1016/j.ress.2017.11.022
  38. 38.
    McKay, M., Beckman, R., Conover, W.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239 (1979)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Park, J.S.: Optimal Latin-hypercube designs for computer experiments. J. Stat. Plan. Inference 39, 95–111 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Morris, M., Mitchell, T.: Exploratory designs for computationnal experiments. J. Stat. Plan. Inference 43, 381–402 (1995)CrossRefzbMATHGoogle Scholar
  41. 41.
    Fang, K., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments. Computer Science and Data Analysis Series. Chapman & Hall, London (2006)zbMATHGoogle Scholar
  42. 42.
    Perrin, G., Cannamela, C.: A repulsion-based method for the definition and the enrichment of opotimized space filling designs in constrained input spaces. J. Soc. Fr. Stat. 158(1), 37–67 (2017)zbMATHGoogle Scholar
  43. 43.
    Bect, J., Ginsbourger, D., Li, L., Picheny, V., Vasquez, E.: Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22, 773–797 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Chevalier, C., Bect, J., Ginsburger, D., Vasquez, E., Picheny, V., Richet, Y.: Fast kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. Technometrics 56(4), 455–465 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Bichon, B., Eldred, M., Swiler, L., Mahadevan, S., McFarland, J.: Efficient global reliability analysis for non linear implicit performance functions. AIAA J. 46(10), 2459–2468 (2008)CrossRefGoogle Scholar
  46. 46.
    Echard, B., Gayton, N., Lemaire, M.: AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Struct. Saf. 22, 145–154 (2011)CrossRefGoogle Scholar
  47. 47.
    Fauriat, W., Gayton, N.: AK-SYS: an adaptation of the AK-MCS method for system reliability. Reliab. Eng. Syst. Saf. 123, 137–144 (2014)CrossRefGoogle Scholar
  48. 48.
    Perrin, G.: Active learning surrogate models for the conception of systems with multiple failure modes. Reliab. Eng. Syst. Saf. 149, 130–136 (2016)CrossRefGoogle Scholar
  49. 49.
    Defaux, G., Walter, C., Iooss, B., Moutoussamy, V.: R package version 2.1-0 (2014)Google Scholar
  50. 50.
    Iooss, B., Janon, A., Pujol, G.: R package version 1.15.2 (2018)Google Scholar
  51. 51.
    Echard, B., Gayton, N., Lemaire, M., Relun, N.: A combined importane sampling and kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab. Eng. Syst. Saf. 111, 232–240 (2013)CrossRefGoogle Scholar
  52. 52.
    Bect, J., Li, L., Vasquez, E.: Bayesian subset simulation. SIAM/ASA J. Uncertain. Quantif. 5(1), 762–786 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Kucherenko, S., Tarantola, S., Annoni, P.: Estimation of global sensitivity indices for models with dependent variables. Comput. Phys. Commun. 183(4), 937–946 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Mara, T., Tarantola, S., Annoni, P.: Non-parametric methods for global sensitivity analysis of model output with dependent inputs. Environ. Model. Softw. 72, 173–183 (2015)CrossRefGoogle Scholar
  55. 55.
    Sueur, R., Iooss, B., Delage, T.: Sensitivity analysis using perturbed-law based indices for quantiles and application to an industrial case. In: 10th International Conference on Mathematical Methods in Reliability (2017)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CEA/DAM/DIFArpajonFrance

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