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On convergence of moments in uncertainty quantification based on direct quadrature

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  • Attar, Peter J.
  • Vedula, Prakash

Abstract

Theoretical results for the convergence of statistical moments in numerical quadrature based polynomial chaos computational uncertainty quantification are presented in this work. This is accomplished by considering the computation of the moments through a direct numerical quadrature method, which is shown to be equivalent to stochastic collocation. For problems which involve output variables which have a polynomial dependence on the random input variables, lower bound expressions are derived for the number of quadrature points required for convergence of arbitrary order moments. In addition, an error expression is derived for when this lower bound is used for problems which have a higher degree of continuity than what was assumed when the bounds are computed. The theoretical results are demonstrated through a simple random algebraic problem and a nonlinear plate problem. The results presented in this work provide further insight into the widely used polynomial chaos expansion method of uncertainty quantification along with presenting simple expressions which can be used for uncertainty quantification code verification.

Suggested Citation

  • Attar, Peter J. & Vedula, Prakash, 2013. "On convergence of moments in uncertainty quantification based on direct quadrature," Reliability Engineering and System Safety, Elsevier, vol. 111(C), pages 119-125.
    • Handle: RePEc:eee:reensy:v:111:y:2013:i:c:p:119-125
    DOI: 10.1016/j.ress.2012.11.003
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    References listed on IDEAS

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    1. Crestaux, Thierry & Le Maıˆtre, Olivier & Martinez, Jean-Marc, 2009. "Polynomial chaos expansion for sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 94(7), pages 1161-1172.
    2. Borgonovo, E., 2007. "A new uncertainty importance measure," Reliability Engineering and System Safety, Elsevier, vol. 92(6), pages 771-784.
    3. Buzzard, Gregery T., 2012. "Global sensitivity analysis using sparse grid interpolation and polynomial chaos," Reliability Engineering and System Safety, Elsevier, vol. 107(C), pages 82-89.
    4. Ratto, M. & Pagano, A. & Young, P.C., 2009. "Non-parametric estimation of conditional moments for sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 237-243.
    5. Prempraneerach, P. & Hover, F.S. & Triantafyllou, M.S. & Karniadakis, G.E., 2010. "Uncertainty quantification in simulations of power systems: Multi-element polynomial chaos methods," Reliability Engineering and System Safety, Elsevier, vol. 95(6), pages 632-646.
    6. Sudret, Bruno, 2008. "Global sensitivity analysis using polynomial chaos expansions," Reliability Engineering and System Safety, Elsevier, vol. 93(7), pages 964-979.
    7. Durga Rao, K. & Kushwaha, H.S. & Verma, A.K. & Srividya, A., 2007. "Quantification of epistemic and aleatory uncertainties in level-1 probabilistic safety assessment studies," Reliability Engineering and System Safety, Elsevier, vol. 92(7), pages 947-956.
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    Cited by:

    1. Oladyshkin, Sergey & Nowak, Wolfgang, 2018. "Incomplete statistical information limits the utility of high-order polynomial chaos expansions," Reliability Engineering and System Safety, Elsevier, vol. 169(C), pages 137-148.

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