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Articles

A Generalized Polynomial Chaos-Based Method for Efficient Bayesian Calibration of Uncertain Computational Models

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Pages 602-624 | Received 03 Aug 2012, Accepted 04 Jul 2013, Published online: 06 Aug 2013

References

  • Mehta U. Some aspects of uncertainty in computational fluid dynamics results. J. Fluid Engg. 1991;113:538–543.
  • Mehta U. Guide to credible computer simulations of fluid flows. J. Prop. Power. 1996;12:940–948.
  • Oreskes N, Shrader-Frechett K, Belitz K. Verification, validation and confirmation of numerical models in earth sciences. Science. 1994;263:641–647.
  • Mehta UB. Credible computational fluid dynamics simulations. AIAA J. 1998;36:665–667.
  • Oberkampf W, DeLand S, Rutherford B, Diegert K, Alvin K. Error and uncertainty in modeling and simulation. Rel. Engg. Syst. Safety. 2002;75:335–357.
  • Thunnissen D. Propagating and mitigating uncertainty in the design of complex multidisciplinary systems. California Institute of Technology, California: Ph.D. diss; 2004.
  • Cheung SH, Oliver TA, Prudencio EE, Prudhomme S, Moser RD. Bayesian uncertainty analysis with applications to turbulence modeling. Rel. Engg. Syst. Safety. 2011;96:1137–1149.
  • Trucano T, Swiler L, Igusa T, Oberkampf W, Pilch M. Calibration, validation, and sensitivity analysis: What’s what. Rel. Engg. Syst. Safety. 2006;91:1331–1357.
  • Glimm J, Sharp D. Prediction and the quantification of uncertainty. Physica D. 1999;133:152–170.
  • Kennedy M, O’Hagan A. Bayesian calibration of computer models. J. Royal Stat. Soc. Series B (Stat. Method.). 2001;63:425–464.
  • 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.
  • Goldstein M, Rougier J. Probabilistic formulations for transferring inferences from mathematical models to physical systems. SIAM J. Sci. Comp. 2005;26:467–487.
  • Bayarri M, Berger J, Paulo R, Sacks J, Cafeo J, Cavendish J, Lin C, Tu J. A framework for validation of computer models. Technometrics. 2007;49:138–153.
  • Higdon D, Gattiker J, Williams B, Rightley M. Computer model calibration using high-dimensional output. J. Amer. Stat. Assoc. 2008;103:570–583.
  • Kelly D, Smith C. Bayesian inference in probabilistic risk assessment - the current state of the art. Rel. Engg. Sys. Safety. 2009;94:628–643.
  • Goldstein M, Rougier J. Reified Bayesian modelling and inference for physical systems. J. Stat. Planning Inf. 2009;139:1221–1239.
  • Besag J, Green P, Higdon D, Mengersen K. Bayesian computation and stochastic systems. Stat. Sci. 1995;10:3–41.
  • Gamerman D, Lopes H. Markov chain Monte Carlo: stochastic simulation for Bayesian inference. Boca Raton: Chapman and Hall/CRC; 2006.
  • Marzouk Y, Najm H. Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comp. Phys. 2007;224:560–586.
  • Marzouk Y, Najm H. Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comp. Phys. 2009;228:1862–1902.
  • Walters R, Huyse L. Uncertainty analysis for fluid mechanics with applications. NASA/CR-2002-211449, 2002.
  • Wiener N. The homogeneous chaos. Amer. J. Math. 1938;60:897–936.
  • Wiener N. Nonlinear problems in random theory. New York: John Wiley & Sons; 1958.
  • Cameron R, Martin W. The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. The Annals Math. 1947;48:385–392.
  • Meecham W, Jeng D. Use of the Wiener-Hermite expansion for nearly normal turbulence. J. Fluid Mech. 1968;32:225–249.
  • Orszag S, Bissonnette L. Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids. 1967;10:260–263.
  • Chorin A. Gaussian fields and random flow. J. Fluid Mech. 1974;85:325–347.
  • Ghanem R, Spanos P. Spectral stochastic finite-element formulation for reliability analysis. J. Engg. Mech. 1991;117:2351–2372.
  • Ghanem R, Red-Horse J. Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach. Physica D. 1999;133:137–144.
  • Ghanem R, Spanos P. Stochastic finite elements: a spectral approach. New York (NY): Dover Publications; 2003.
  • Knio O, Maitre O. Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid Dyn. Res. 2006;38:616–640.
  • Maitre O, Knio O, Najm H, Ghanem R. A stochastic projection method for fluid flow. J. Comp. Phys. 2001;173:481–511.
  • Xiu D, Karniadakis G. The Weiner-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comp. 2002;24:619–644.
  • Xiu D, Karniadakis G. Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comp. Phys. 2003;187:137–167.
  • 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.
  • Lucor D, Xiu D, Su C, Karniadakis G. Predictability and uncertainty in CFD. Int. J. Num. Meth. Fluids. 2003;43:483–505.
  • Mathelin L, Hussaini M, Zang T, Bataille F. Uncertainty propagation for a turbulent, compressible nozzle flow using stochastic methods. AIAA J. 2004;42:1669–1676.
  • Narayanan V, Zabaras N. Stochastic inverse heat conduction using spectral approach. Int. J. Num. Methods Engg. 2004;60:1569–1593.
  • Najm H. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Ann. Rev. Fluid Mech. 2009;41:35–52.
  • 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.
  • Sacks J, Welch W, Mitchell T, Wynn H. Design and analysis of computer experiments. Stat. Sci. 1989;4:409–423.
  • Paulo R. Default priors for Gaussian processes. Ann. Stat. 2005;33:556–582.
  • OHagan A. Bayesian analysis of computer code outputs: A tutorial. Rel. Engg. Syst. Safety. 2006;91:1290–1300.
  • Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E. Equation of state calculations by fast computing machines. J. Chem. Phys. 1953;21:1087–1092.
  • Hastings W. Monte Carlo sampling methods using Markov chains and their applications. Biometrika. 1970;57:97–109.
  • Chakrabarti A, Martha S. Approximate solutions of Fredholm integral equations of the second kind. App. Math. Comp. 2009;211:459–466.
  • Huang S, Quek S, Phoon K. Convergence study of the truncated KarhunenLoeve expansion for simulation of stochastic processes. Int. J. Num. Methods Engg. 2001;52:1029–1043.
  • Choi S, Wette R. Maximum likelihood estimation of the parameters of the Gamma distribution and their bias. Technometrics. 1969;11:683–690.

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