A Method of Parameter Calibration with Hybrid Uncertainty

  • Liu Bo
  • Shang XiaoBing
  • Wang Songyan
  • Chao TaoEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 946)


A method, which combines the cumulative distribution function and modified Kolmogorov–Smirnov test, is proposed to solve parameter calibration problem with genetic algorithm seeking the optimal result, due to the hybrid uncertainty in model. The framework is built on comparing the difference between cumulative distribution functions of some target observed values and that of sample values. First, an auxiliary variable method is used to decomposition hybrid parameters into sub-parameters with only one kind of uncertainty, which is aleatory or epistemic, because only epistemic uncertainty can be calibrated. Then we find optimal matching values with genetic algorithm according to the index of difference of joint cumulative distribution functions. Finally, we demonstrate that the proposed model calibration method is able to get the approximation values of the unknown true value of epistemic parameters, in mars entry dynamics profile. The example illustrates the rationality and efficiency of the method of this paper.


Hybrid uncertainty Parameter calibration Auxiliary variable 



This work was supported by National Natural Science Foundation of China (No. 61790562, 61627810, 61403096).


  1. 1.
    Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 63(3), 425–464 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bayarri, M.J., Berger, J.O., Paulo, R., Sacks, J.: A framework for validation of computer models. Technometrics 49(2), 138–154 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arendt, P.D., Apley, D.W., Chen, W., Lamb, D., Gorsich, D.: Improving identifiability in model calibration using multiple responses. J. Mech. Des. 134(10), 100909 (2012)CrossRefGoogle Scholar
  4. 4.
    Arendt, P.D., Apley, D.W., Chen, W.: Quantification of model uncertainty: calibration, model discrepancy, and identifiability. J. Mech. Des. 134(10), 100908 (2012)CrossRefGoogle Scholar
  5. 5.
    Oberkampf, W.L., Helton, J.C., Joslyn, C.A., Wojtkiewicz, S.F., Ferson, S.: Challenge problems: uncertainty in system response given uncertain parameters. Reliab. Eng. Syst. Saf. 85(1–3), 11–19 (2004)CrossRefGoogle Scholar
  6. 6.
    Gogu, C., Qiu, Y., Segonds, S., Bes, C.: Optimization based algorithms for uncertainty propagation through functions with multidimensional output within evidence theory. J. Mech. Des. 134(10), 100914 (2012)CrossRefGoogle Scholar
  7. 7.
    Xia, X., Wang, Z., Gao, Y.: Estimation of non-statistical uncertainty using fuzzy-set theory. Meas. Sci. Technol. 11(4), 430 (2000)CrossRefGoogle Scholar
  8. 8.
    Du, L., Choi, K.K., Youn, B.D., Gorsich, D.: Possibility-based design optimization method for design problems with both statistical and fuzzy input data. J. Mech. Des. 128(4), 928–935 (2006)CrossRefGoogle Scholar
  9. 9.
    Nikolaidis, E., Chen, S., Cudney, H., Haftka, R.T., Rosca, R.: Comparison of probability and possibility for design against catastrophic failure under uncertainty. J. Mech. Des. 126(3), 386–394 (2004)CrossRefGoogle Scholar
  10. 10.
    Jiang, C., Ni, B.Y., Han, X., Tao, Y.R.: Non-probabilistic convex model process: a new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems. Comput. Methods Appl. Mech. Eng. 268, 656–676 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Crespo, L.G., Kenny, S.P., Giesy, D.P.: The NASA Langley multidisciplinary uncertainty quantification challenge. In: 16th AIAA Non-Deterministic Approaches Conference, p. 1347 (2014)Google Scholar
  12. 12.
    Bruns, M., Paredis, C.J.: Numerical methods for propagating imprecise uncertainty. In: ASME 2006 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, January, pp. 1077–1091 (2006)Google Scholar
  13. 13.
    Li, S., Jiang, X.: Review and prospect of guidance and control for Mars atmospheric entry. Prog. Aerosp. Sci. 69, 40–57 (2014)CrossRefGoogle Scholar
  14. 14.
    Chaudhuri, A., Waycaster, G., Price, N., Matsumura, T., Haftka, R.T.: NASA uncertainty quantification challenge: an optimization-based methodology and validation. J. Aerosp. Inf. Syst. 12(1), 10–34 (2015)Google Scholar
  15. 15.
    Ghanem, R., et al.: Probabilistic approach to NASA Langley research center multidisciplinary uncertainty quantification challenge problem. J. Aerosp. Inf. Syst. 12(1), 170–188 (2014)MathSciNetGoogle Scholar
  16. 16.
    Brune, A.J., West, T., Hosder, S., Edquist, K.T.: Uncertainty analysis of mars entry flows over hypersonic inflatable aerodynamic decelerators. In: 11th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, p. 2672 (2014)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Liu Bo
    • 1
    • 2
  • Shang XiaoBing
    • 1
  • Wang Songyan
    • 1
  • Chao Tao
    • 1
    Email author
  1. 1.Harbin Institute of TechnologyHarbinChina
  2. 2.China Shipbuilding Industry CorporationXi’anChina

Personalised recommendations