Accelerated failure identification sampling for probability analysis of rare events



Critical engineering systems generally demand high reliability while considering uncertainties, which makes failure event identification and reliability analysis based on computer simulation codes computationally very expensive. Rare event can be defined as a failure event that has a small probability of failure value, normally less than 10−5. This paper presents a new approach, referred to as Accelerated Failure Identification Sampling (AFIS), for probability analysis of rare events, enabling savings of vast computational efforts. To efficiently identify rare failure sample points in probability analysis, the proposed AFIS technique will first construct a Gaussian process (GP) model for system performance of interest and then utilize the developed model to predict unknown responses of Monte Carlo sample points. Second, a new quantitative measure, namely “failure potential”, is developed to iteratively search sample points that have the best chance to be a failure sample point. Third, the identified sample points with highest failure potentials are evaluated for the true performance and then used to update the GP model. The failure identification process will be iteratively preceded and the Monte Carlo simulation will then be employed to estimate probabilities of rare events if the maximum failure potential of existing Monte Carlo samples falls below a given target value. Two case studies are used to demonstrate the effectiveness of the developed AFIS approach for rare events identification and probability analysis.


Reliability Probability of failure Rare events Adaptive sampling 



Probability of failure


Standard Gaussian cumulative distribution function


Probability density function


Failure indicator for sample point x


Confidence interval


Failure potential threshold


The limit state function


The number of failure sample points


The number of random sample points generated



This research is supported by National Science Foundation under Faculty Early Career Development (CAREER) Award CMMI-1351414 and the Award CMMI-1538508.


  1. Au SK, Ching J, Beck JL (2007) Application of subset simulation methods to reliability benchmark problems. Struct Saf 29(3):183–193CrossRefGoogle Scholar
  2. Berg BA (2005) Introduction to Markov chain Monte Carlo simulations and their statistical analysis. Markov Chain Monte Carlo Lect Notes Ser Inst Math Sci Natl Univ Singap 7:1–52MathSciNetMATHGoogle Scholar
  3. Blanchet J, Shi Y (2013) Efficient rare event simulation for heavy-tailed systems via cross entropy. Oper Res Lett 41(3):271–276MathSciNetCrossRefMATHGoogle Scholar
  4. Cérou F et al (2012) Sequential Monte Carlo for rare event estimation. Stat Comput 22(3):795–808MathSciNetCrossRefMATHGoogle Scholar
  5. Cornuet J-M, Marin J-M, Mira A, Robert CP (2012) Adaptive multiple importance sampling. Scand J Stat 39(4):798–812MathSciNetCrossRefMATHGoogle Scholar
  6. Crestaux T, Olivier L, Martinez JM (2009) Polynomial chaos expansion for sensitivity analysis. Reliab Eng Syst Saf 94(7):1161–1172CrossRefGoogle Scholar
  7. Ding J, Chen X (2013) Assessing small failure probability by importance splitting method and its application to wind turbine extreme response prediction. Eng Struct 54:180–191CrossRefGoogle Scholar
  8. Dubourg V, Sudret B (2014) Meta-model-based importance sampling for reliability sensitivity analysis. Struct Saf 49:27–36CrossRefGoogle Scholar
  9. Dubourg V, Bruno S, Bourinet JM (2011) Reliability-based design optimization using kriging surrogates and subset simulation. Struct Multidiscip Optim 44(5):673–690CrossRefGoogle Scholar
  10. Echard B, Gayton N, Lemairem M (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Struct Saf 33(2):145–154CrossRefGoogle Scholar
  11. Gamerman D, Lopes HF (2006) Markov chain Monte Carlo: stochastic simulation for Bayesian inference. CRC PressGoogle Scholar
  12. Hu Z, Du X (2014) First order reliability method for time-variant problems using series expansions. Struct Multidiscip Optim 1–21Google Scholar
  13. Hu C, Youn BD (2011) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidiscip Optim 43(3):419–442MathSciNetCrossRefMATHGoogle Scholar
  14. Janssen H (2013) Monte-Carlo based uncertainty analysis: sampling efficiency and sampling convergence. Reliab Eng Syst Saf 109:123–132CrossRefGoogle Scholar
  15. Jiang Z et al (2013) Reliability-based design optimization with model bias and data uncertainty. No. 2013-01-1384. SAE Technical PaperGoogle Scholar
  16. Lagnoux A (2006) Rare event simulation. Probab Eng Inf Sci 20(01):45–66MathSciNetCrossRefMATHGoogle Scholar
  17. Lee I, Choi KK, Gorsich D (2010) System reliability-based design optimization using the MPP-based dimension reduction method. Struct Multidiscip Optim 41(6):823–839CrossRefGoogle Scholar
  18. Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19:393–408CrossRefGoogle Scholar
  19. Rubino G, Bruno T (eds) (2009) Rare event simulation using Monte Carlo methods. Wiley, New YorkMATHGoogle Scholar
  20. Rubinstein RY, Kroese DP (2011) Simulation and the Monte Carlo method. Vol. 707. WileyGoogle Scholar
  21. Schunk D (2008) A Markov chain Monte Carlo algorithm for multiple imputations in large surveys. AStA Adv Stat Anal 92(1):101–114MathSciNetCrossRefMATHGoogle Scholar
  22. Shortle JF et al (2012) Optimal splitting for rare-event simulation. IIE Trans 44(5):352–367CrossRefGoogle Scholar
  23. Song S, Lu Z, Qiao H (2009) Subset simulation for structural reliability sensitivity analysis. Reliab Eng Syst Saf 94(2):658–665CrossRefGoogle Scholar
  24. Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979CrossRefGoogle Scholar
  25. Tang Y, Chen J, Wei J (2012) A sequential algorithm for reliability-based robust design optimization under epistemic uncertainty. J Mech Des 134(1):014502CrossRefGoogle Scholar
  26. Wang L, Grandhi R (1996) Safety index calculation using intervening variables for structural reliability. Comput Struct 59(6):1139–1148CrossRefMATHGoogle Scholar
  27. Wang Z, Wang P (2013) A new approach for reliability analysis with time-variant performance characteristics. Reliab Eng Syst Saf 115:70–81CrossRefGoogle Scholar
  28. Wang Z, Wang P (2014) A maximum confidence enhancement based sequential sampling scheme for simulation-based design. J Mech Des 136(2):021006CrossRefGoogle Scholar
  29. Wang P, Hu C, Youn BD (2011) A generalized complementary intersection method (GCIM) for system reliability analysis. J Mech Des 133(7):071003CrossRefGoogle Scholar
  30. Wei D, Rahman S (2007) Structural reliability analysis by univariate decomposition and numerical integration. Probab Eng Mech 22(1):27–38CrossRefGoogle Scholar
  31. Wei DL, Cui ZS, Chen J (2008) Uncertainty quantification using polynomial chaos expansion with points of monomial cubature rules. Comput Struct 86(23):2102–2108CrossRefGoogle Scholar
  32. Xu H, Rahman S (2004) A generalized dimension reduction method for multidimensional integration in stochastic mechanics. Int J Numer Methods Eng 61(12):1992–2019CrossRefMATHGoogle Scholar
  33. Xu H, Rahman S (2005) Decomposition methods for structural reliability analysis. Probab Eng Mech 20(3):239–250MathSciNetCrossRefGoogle Scholar
  34. Yadav V, Rahman S (2014) A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems. Probab Eng Mech 38:22–34CrossRefGoogle Scholar
  35. Youn BD, Wang P (2008) Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method. Struct Multidiscip Optim 36(2):107–123MathSciNetCrossRefGoogle Scholar
  36. Youn BD, Chio KK, Gu L, Yang R-J (2004) Reliability-based design optimization for crashworthiness of side impact. J Struct Multidiscip Optim 26(3–4):272–283CrossRefGoogle Scholar
  37. Youn BD, Choi KK, Du L (2005a) Enriched performance measure approach for reliability-based design optimization. AIAA J 43(4):874–884CrossRefGoogle Scholar
  38. Youn BD, Choi KK, Du L (2005b) Adaptive probability analysis using an enhanced hybrid mean value (HMV+) method. J Struct Multidiscip Optim 29(2):134–148CrossRefGoogle Scholar
  39. Youn BD, Xi Z, Wang P (2008) Eigenvector dimension-reduction (EDR) method for sensitivity-free uncertainty quantification. Struct Multidiscip Optim 37(1):13–28MathSciNetCrossRefMATHGoogle Scholar
  40. Zhang H, Dai H, Beer M, Wang W (2013) Structural reliability analysis on the basis of small samples: an interval quasi-Monte Carlo method. Mech Syst Signal Process 37(1):137–151CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Industrial and Manufacturing EngineeringWichita State UniversityWichitaUSA

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