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A generalized polynomial chaos-based method for efficient Bayesian calibration of uncertain computational models. (English) Zbl 1308.62048

Summary: This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent stationary Gaussian processes with uncertain hyper-parameters describe uncertainties of the model structure and parameters, while Karhunen-Loeve expansion is adopted to spectrally represent these Gaussian processes. The Karhunen-Loeve expansion of a prior Gaussian process is projected on a generalized Polynomial Chaos basis, whereas intrusive Galerkin projection is utilized to calculate the associated coefficients of the simulator output. Bayesian inference is used to update the prior probability distribution of the generalized Polynomial Chaos basis, which along with the chaos expansion coefficients represent the posterior probability distribution. The proposed method is demonstrated for calibration of a simulator of quasi-one-dimensional flow through a divergent nozzle with uncertain nozzle area profile. The posterior distribution of the nozzle area profile and the hyper-parameters obtained using the proposed method are found to match closely with the direct Markov Chain Monte Carlo-based implementation of the Bayesian framework. Efficacy of the proposed method is demonstrated for various choices of prior. Posterior hyper-parameters of the model structural uncertainty are shown to quantify acceptability of the simulator model.

MSC:

62F15 Bayesian inference
60G15 Gaussian processes
65C20 Probabilistic models, generic numerical methods in probability and statistics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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References:

[1] DOI: 10.1115/1.2926512 · doi:10.1115/1.2926512
[2] DOI: 10.2514/3.24126 · doi:10.2514/3.24126
[3] DOI: 10.1126/science.263.5147.641 · doi:10.1126/science.263.5147.641
[4] DOI: 10.2514/2.431 · doi:10.2514/2.431
[5] Oberkampf W, Rel. Engg. Syst. Safety 75 pp 335– (2002)
[6] Thunnissen D, Propagating and mitigating uncertainty in the design of complex multidisciplinary systems (2004)
[7] DOI: 10.1016/j.ress.2010.09.013 · doi:10.1016/j.ress.2010.09.013
[8] DOI: 10.1016/j.ress.2005.11.031 · doi:10.1016/j.ress.2005.11.031
[9] DOI: 10.1016/S0167-2789(99)00103-7 · Zbl 1194.76243 · doi:10.1016/S0167-2789(99)00103-7
[10] DOI: 10.1111/1467-9868.00294 · Zbl 1007.62021 · doi:10.1111/1467-9868.00294
[11] Higdon D, Kennedy M, Cavendish J, Cafeo J, Ryne R. Combining field data and computer simulations for calibration and prediction. SIAM J. Sci. Comp. 2005;26:448–466. · Zbl 1072.62018
[12] DOI: 10.1137/S106482750342670X · Zbl 1138.62375 · doi:10.1137/S106482750342670X
[13] DOI: 10.1198/004017007000000092 · doi:10.1198/004017007000000092
[14] DOI: 10.1198/016214507000000888 · Zbl 1469.62414 · doi:10.1198/016214507000000888
[15] Kelly D, Smith C. Bayesian inference in probabilistic risk assessment - the current state of the art. Rel. Engg. Sys. Safety. 2009;94:628–643.
[16] DOI: 10.1016/j.jspi.2008.07.019 · Zbl 1156.62316 · doi:10.1016/j.jspi.2008.07.019
[17] DOI: 10.1214/ss/1177010123 · Zbl 0955.62552 · doi:10.1214/ss/1177010123
[18] Gamerman D, Markov chain Monte Carlo: stochastic simulation for Bayesian inference (2006)
[19] Marzouk Y, Najm H. Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comp. Phys. 2007;224:560–586. · Zbl 1120.65306
[20] Marzouk Y, Najm H. Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comp. Phys. 2009;228:1862–1902. · Zbl 1161.65308
[21] Walters R, Huyse L. Uncertainty analysis for fluid mechanics with applications. NASA/CR-2002-211449, 2002.
[22] DOI: 10.2307/2371268 · Zbl 0019.35406 · doi:10.2307/2371268
[23] Wiener N, Nonlinear problems in random theory (1958)
[24] DOI: 10.2307/1969178 · Zbl 0029.14302 · doi:10.2307/1969178
[25] Meecham W, Jeng D. Use of the Wiener-Hermite expansion for nearly normal turbulence. J. Fluid Mech. 1968;32:225–249. · Zbl 0155.55705 · doi:10.1017/S0022112068000698
[26] Orszag S, Phys. Fluids 10 pp 260– (1967)
[27] Chorin A, J. Fluid Mech 85 pp 325– (1974)
[28] Ghanem R, Spanos P. Spectral stochastic finite-element formulation for reliability analysis. J. Engg. Mech. 1991;117:2351–2372.
[29] DOI: 10.1016/S0167-2789(99)00102-5 · Zbl 1194.74400 · doi:10.1016/S0167-2789(99)00102-5
[30] Ghanem R, Spanos P. Stochastic finite elements: a spectral approach. New York (NY): Dover Publications; 2003. · Zbl 0953.74608
[31] DOI: 10.1016/j.fluiddyn.2005.12.003 · Zbl 1178.76297 · doi:10.1016/j.fluiddyn.2005.12.003
[32] DOI: 10.1006/jcph.2001.6889 · Zbl 1051.76056 · doi:10.1006/jcph.2001.6889
[33] DOI: 10.1137/S1064827501387826 · Zbl 1014.65004 · doi:10.1137/S1064827501387826
[34] DOI: 10.1016/S0021-9991(03)00092-5 · Zbl 1047.76111 · doi:10.1016/S0021-9991(03)00092-5
[35] Koekoek R, Swarttouw R. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Department of Technical Mathematics and Informatics, Delft University of Technology, Report no. 98–17, 1998.
[36] Lucor D, Int. J. Num. Meth. Fluids 43 pp 483– (2003)
[37] DOI: 10.2514/1.5674 · doi:10.2514/1.5674
[38] Narayanan V, Zabaras N. Stochastic inverse heat conduction using spectral approach. Int. J. Num. Methods Engg. 2004;60:1569–1593. · Zbl 1098.80008
[39] DOI: 10.1146/annurev.fluid.010908.165248 · Zbl 1168.76041 · doi:10.1146/annurev.fluid.010908.165248
[40] Tagade P, Choi HL. A polynomial chaos based Bayesian inference method with uncertain hyperparameters. In ASME International Design Engineering Technical Conference and Computers and Information in Engineering Conference. DC; Washington; 2011.
[41] DOI: 10.1214/ss/1177012413 · Zbl 0955.62619 · doi:10.1214/ss/1177012413
[42] DOI: 10.1214/009053604000001264 · Zbl 1069.62030 · doi:10.1214/009053604000001264
[43] DOI: 10.1016/j.ress.2005.11.025 · doi:10.1016/j.ress.2005.11.025
[44] DOI: 10.1063/1.1699114 · doi:10.1063/1.1699114
[45] DOI: 10.1093/biomet/57.1.97 · Zbl 0219.65008 · doi:10.1093/biomet/57.1.97
[46] DOI: 10.1016/j.amc.2009.01.088 · Zbl 1162.65416 · doi:10.1016/j.amc.2009.01.088
[47] DOI: 10.1002/nme.255 · Zbl 0994.65004 · doi:10.1002/nme.255
[48] DOI: 10.1080/00401706.1969.10490731 · doi:10.1080/00401706.1969.10490731
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