Probabilistic Methods for Sensitivity Analysis and Calibration in the NASA Challenge Problem
Abstract
In this paper, a series of algorithms are proposed to address the problems in the NASA Langley Research Center Multidisciplinary Uncertainty Quantification Challenge. A Bayesian approach is employed to characterize and calibrate the epistemic parameters based on the available data, whereas a variance-based global sensitivity analysis is used to rank the epistemic and aleatory model parameters. A nested sampling of the aleatory–epistemic space is proposed to propagate uncertainties from model parameters to output quantities of interest.
References
[1] , “The NASA Langley Multidisciplinary Uncertainty Quantification Challenge,” 16th AIAA Non-Deterministic Approaches Conference, AIAA Paper 2014-1347, Jan. 2014.
[2] , Data Analysis: A Bayesian Tutorial, Oxford Science, New York, 1996.
[3] , Bayesian Data Analysis, 2nd ed., Chapman and Hall/CRC Press, Boca Raton, FL, 2003.
[4] , “Likelihood-Free Markov Chain Monte Carlo,” Handbook of Markov Chain Monte Carlo, Chapman and Hall/CRC Press, Boca Raton, FL, 2011, pp. 313–333.
[5] , “Sensitivity Estimates for Nonlinear Mathematical Models,” Mathematical Modeling and Computational Experiment, John Wiley & Sons, New York, 1993, pp. 407–414.
[6] , “Strictly Proper Scoring Rules, Prediction, and Estimation,” Journal of the American Statistical Association, Vol. 102, No. 477, 2007, pp. 359–378. doi:https://doi.org/10.1198/016214506000001437
[7] , Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986, pp. 75–94.
[8] , Multivariate Density Estimation. Theory, Practice and Visualization, Wiley, New York, 1992, pp. 125–194.
[9] , “An Adaptive Metropolis Algorithm,” Bernoulli, Vol. 7, No. 2, 2001, pp. 223–242. doi:https://doi.org/10.2307/3318737
[10] , “How Many Iterations in the Gibbs Sampler?,” Bayesian Statistics 4, Oxford Univ. Press, New York, 1992, pp. 763–773.
[11] , “Approximate Bayesian Computation in Population Genetics,” Genetics, Vol. 162, No. 4, 2002, pp. 2025–2035. GENTAE 0016-6731
[12] , “Decomposition of the Continuous Ranked Probability Score for Ensemble Prediction Systems,” Weather and Forecasting, Vol. 15, No. 5, 2000, pp. 559–570. doi:10.1175/1520-0434(2000)015<0559:DOTCRP>2.0.CO;2 WEFOE3 0882-8156
[13] , “Hitchhiker’s Guide to Sensitivity Analysis,” Sensitivity Analysis, edited by Saltelli A., Chan K. and Scott E., Wiley, Chicester, England, U.K., 2000, pp. 15–50.
[14] , “Estimation of Global Sensitivity Indices for Models with Dependent Variables,” Computer Physics Communications, Vol. 183, No. 4, 2012, pp. 937–946. doi:https://doi.org/10.1016/j.cpc.2011.12.020 CPHCBZ 0010-4655
[15] , “Making Best Use of Model Evaluations to Compute Sensitivity Indices,” Computer Physics Communications, Vol. 145, No. 2, 2002, pp. 280–297. doi:https://doi.org/10.1016/S0010-4655(02)00280-1 CPHCBZ 0010-4655
[16] , “Measuring and Testing Dependence by Correlation of Distances,” Annals of Statistics, Vol. 35, No. 6, 2007, pp. 2769–2794. doi:https://doi.org/10.1214/009053607000000505 ASTSC7 0090-5364
[17] , “Remarks on a Multivariate Transformation,” Annals of Mathematical Statistics, Vol. 23, No. 3, 1952, pp. 470–472. doi:https://doi.org/10.1214/aoms/1177729394 AASTAD 0003-4851
[18] , “General Foundations of High-Dimensional Model Representations,” Journal of Mathematical Chemistry, Vol. 25, Nos. 2–3, 1999, pp. 197–233. doi:https://doi.org/10.1023/A:1019188517934 JMCHEG 0259-9791
[19] , “High Dimensional Model Representations,” Journal of Physical Chemistry A, Vol. 105, No. 33, 2001, pp. 7765–7777. doi:https://doi.org/10.1021/jp010450t JPCAFH 1089-5639
[20] , “The Homogeneous Chaos,” American Journal of Mathematics, Vol. 60, No. 4, Oct. 1938, pp. 897–936. doi:https://doi.org/10.2307/2371268
[21] , Stochastic Finite Elements: A Spectral Approach, Springer–Verlag, New York, 1991.
[22] , “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM Journal on Scientific Computing, Vol. 24, No. 2, 2002, pp. 619–644. doi:https://doi.org/10.1137/S1064827501387826 SJOCE3 1064-8275
[23] , “Sensitivity Analysis Techniques Applied to a System of Hyperbolic Conservation Laws,” Reliability Engineering & System Safety, Vol. 107, Nov. 2012, pp. 157–170. doi:https://doi.org/10.1016/j.ress.2011.12.008 SJOCE3 1064-8275


