Applications of Mathematics

, Volume 63, Issue 6, pp 713–737 | Cite as

Polynomial chaos in evaluating failure probability: A comparative study

  • Eliška JanouchováEmail author
  • Jan Sýkora
  • Anna Kučerová


Recent developments in the field of stochastic mechanics and particularly regarding the stochastic finite element method allow to model uncertain behaviours for more complex engineering structures. In reliability analysis, polynomial chaos expansion is a useful tool because it helps to avoid thousands of time-consuming finite element model simulations for structures with uncertain parameters. The aim of this paper is to review and compare available techniques for both the construction of polynomial chaos and its use in computing failure probability. In particular, we compare results for the stochastic Galerkin method, stochastic collocation, and the regression method based on Latin hypercube sampling with predictions obtained by crude Monte Carlo sampling. As an illustrative engineering example, we consider a simple frame structure with uncertain parameters in loading and geometry with prescribed distributions defined by realistic histograms.


uncertainty quantification reliability analysis probability of failure safety margin polynomial chaos expansion regression method stochastic collocation method stochastic Galerkin method Monte Carlo method 

MSC 2010

41A10 62P30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. Augustin, A. Gilg, M. Paffrath, P. Rentrop, M. Villegas, U. Wever: An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties. J. Math. Ind. 3 (2013), 24 pages.MathSciNetzbMATHGoogle Scholar
  2. [2]
    I. Babuška, F. Nobile, R. Tempone: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007), 1005–1034.MathSciNetCrossRefGoogle Scholar
  3. [3]
    I. Babuška, R. Tempone, G. E. Zouraris: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004), 800–825.MathSciNetCrossRefGoogle Scholar
  4. [4]
    G. Blatman, B. Sudret: An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilistic Engineering Mechanics 25 (2010), 183–197.CrossRefGoogle Scholar
  5. [5]
    G. Blatman, B. Sudret: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230 (2011), 2345–2367.MathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Cheng, A. Sandu: Efficient uncertainty quantification with the polynomial chaos method for stiff systems. Math. Comput. Simul. 79 (2009), 3278–3295.MathSciNetCrossRefGoogle Scholar
  7. [7]
    S.–K. Choi, R. V. Grandhi, R. A. Canfield, C. L. Pettit: Polynomial chaos expansion with latin hypercube sampling for estimating response variability. AIAA J. 42 (2004), 1191–1198.CrossRefGoogle Scholar
  8. [8]
    O. Ditlevsen, H. O. Madsen: Structural Reliability Methods. John Wiley & Sons, Chichester, 1996.Google Scholar
  9. [9]
    M. Eigel, C. J. Gittelson, C. Schwab, E. Zander: Adaptive stochastic Galerkin FEM. Comput. Methods Appl. Mech. Eng. 270 (2014), 247–269.CrossRefGoogle Scholar
  10. [10]
    M. S. Eldred, J. Burkardt: Comparison of non–intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. The 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando. AIAA 2009–976, 2009, pp. 20.CrossRefGoogle Scholar
  11. [11]
    H. C. Elman, C. W. Miller, E. T. Phipps, R. S. Tuminaro: Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int. J. Uncertain. Quantif. 1 (2011), 19–33.MathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Fülöp, M. Iványi: Safety of a column in a frame. Probabilistic Assessment of Structures Using Monte Carlo Simulation: Background, Exercises and Software (P. Marek et al., eds.). Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Praha, CD, Chapt. 8. 10, 2003.Google Scholar
  13. [13]
    R. G. Ghanem, P. D. Spanos: Stochastic Finite Elements: A Spectral Approach, Revised Edition. Dover Civil and Mechanical Engineering, Dover Publications, 2012.Google Scholar
  14. [14]
    M. Gutiérrez, S. Krenk: Stochastic finite element methods. Encyclopedia of Computational Mechanics (E. Stein et al., eds.). John Wiley & Sons, Chichester, 2004.Google Scholar
  15. [15]
    F. Heiss, V. Winschel: Likelihood approximation by numerical integration on sparse grids. J. Econom. 144 (2008), 62–80.MathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Hosder, R. W. Walters, M. Balch: Efficient sampling for non–intrusive polynomial chaos applications with multiple uncertain input variables. The 48th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu. AIAA 2007–1939, 2007, pp. 16.Google Scholar
  17. [17]
    C. Hu, B. D. Youn: Adaptive–sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct. Multidiscip. Optim. 43 (2011), 419–442.MathSciNetCrossRefGoogle Scholar
  18. [18]
    E. Janouchová, A. Kučerová: Competitive comparison of optimal designs of experiments for sampling–based sensitivity analysis. Comput. Struct. 124 (2013), 47–60.CrossRefGoogle Scholar
  19. [19]
    E. Janouchová, A. Kučerová, J. Sýkora: Polynomial chaos construction for structural reliability analysis. Proceedings of the Fourth International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering (Y. Tsompanakis et al., eds.). Civil–Comp Press, Stirlingshire, 2015, Paper 9.Google Scholar
  20. [20]
    J. Li, J. Li, D. Xiu: An efficient surrogate–based method for computing rare failure probability. J. Comput. Phys. 230 (2011), 8683–8697.MathSciNetCrossRefGoogle Scholar
  21. [21]
    J. Li, D. Xiu: Evaluation of failure probability via surrogate models. J. Comput. Phys. 229 (2010), 8966–8980.MathSciNetCrossRefGoogle Scholar
  22. [22]
    X. Ma, N. Zabaras: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228 (2009), 3084–3113.MathSciNetCrossRefGoogle Scholar
  23. [23]
    H. G. Matthies: Uncertainty quantification with stochastic finite elements. Encyclopedia of Computational Mechanics (E. Stein et al., eds.). John Wiley & Sons, Chichester, 2007.Google Scholar
  24. [24]
    H. G. Matthies, A. Keese: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194 (2005), 1295–1331.MathSciNetCrossRefGoogle Scholar
  25. [25]
    H. N. Najm: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annual Review of Fluid Mechanics 41 (S. H. Davis et al., eds.). Annual Reviews, Palo Alto, 2009, pp. 35–52.zbMATHGoogle Scholar
  26. [26]
    F. Nobile, R. Tempone, C. G. Webster: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008), 2309–2345.MathSciNetCrossRefGoogle Scholar
  27. [27]
    M. Paffrath, U. Wever: Adapted polynomial chaos expansion for failure detection. J. Comput. Phys. 226 (2007), 263–281.MathSciNetCrossRefGoogle Scholar
  28. [28]
    M. P. Pettersson, G. Iaccarino, J. Nordström: Polynomial chaos methods. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Mathematical Engineering, Springer, Cham, 2015, pp. 23–29.zbMATHGoogle Scholar
  29. [29]
    R. Pulch: Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations. J. Comput. Appl. Math. 262 (2014), 281–291.MathSciNetCrossRefGoogle Scholar
  30. [30]
    G. Stefanou: The stochastic finite element method: Past, present and future. Comput. Methods Appl. Mech. Eng. 198 (2009), 1031–1051.CrossRefGoogle Scholar
  31. [31]
    N. Wiener: The homogeneous chaos. Am. J. Math. 60 (1938), 897–936.MathSciNetCrossRefGoogle Scholar
  32. [32]
    D. Xiu: Fast numerical methods for stochastic computations: A review. Commun. Comput. Phys. 5 (2009), 242–272.MathSciNetzbMATHGoogle Scholar
  33. [33]
    D. Xiu: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton, 2010.CrossRefGoogle Scholar
  34. [34]
    D. Xiu, J. S. Hesthaven: High–order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005), 1118–1139.MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  • Eliška Janouchová
    • 1
    Email author
  • Jan Sýkora
    • 1
  • Anna Kučerová
    • 1
  1. 1.Faculty of Civil Engineering, Department of MechanicsCzech Technical University in PraguePraha 6Czech Republic

Personalised recommendations