图书馆订阅: Guest
Begell Digital Portal Begell 数字图书馆 电子图书 期刊 参考文献及会议录 研究收集
国际不确定性的量化期刊
影响因子: 3.259 5年影响因子: 2.547 SJR: 0.417 SNIP: 0.8 CiteScore™: 1.52

ISSN 打印: 2152-5080
ISSN 在线: 2152-5099

Open Access

国际不确定性的量化期刊

DOI: 10.1615/Int.J.UncertaintyQuantification.2019027384
pages 365-394

EMBEDDED MODEL ERROR REPRESENTATION FOR BAYESIAN MODEL CALIBRATION

Khachik Sargsyan
Sandia National Laboratories, Livermore, CA, USA
Xun Huan
Sandia National Laboratories, 7011 East Ave, MS 9051, Livermore, CA 94550, USA; Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Habib N. Najm
Sandia National Laboratories P.O. Box 969, MS 9051, Livermore, CA 94551, USA

ABSTRACT

Model error estimation remains one of the key challenges in uncertainty quantification and predictive science. For computational models of complex physical systems, model error, also known as structural error or model inadequacy, is often the largest contributor to the overall predictive uncertainty. This work builds on a recently developed frame-work of embedded, internal model correction, in order to represent and quantify structural errors, together with model parameters, within a Bayesian inference context. We focus specifically on a polynomial chaos representation with additive modification of existing model parameters, enabling a nonintrusive procedure for efficient approximate likelihood construction, model error estimation, and disambiguation of model and data errors' contributions to predictive uncertainty. The framework is demonstrated on several synthetic examples, as well as on a chemical ignition problem.

REFERENCES

  1. Campbell, K., Statistical Calibration of Computer Simulations, Reliab. Eng. System Saf., 91(10):1358-1363,2006.

  2. O'Hagan, A., Bayesian Inference with Misspecified Models: Inference about What?, J. Stat. Plann. Inference, 143(10):1643-1648,2013.

  3. Bojke, L., Claxton, K., Sculpher, M., and Palmer, S., Characterizing Structural Uncertainty in Decision Analytic Models: a Review and Application of Methods, Value Health, 12(5):739-749,2009.

  4. Curry, J.A. and Webster, P.J., Climate Science and the Uncertainty Monster, Bull. Am. Meteorol. Soc, 92(12):1667-1682, 2011.

  5. Gupta, H., Clark, M., Vrugt, J., Abramowitz, G., and Ye, M., Towards a Comprehensive Assessment of Model Structural Adequacy, Water Resour. Res., 48:W08301,2012.

  6. Kennedy, M.C. and O'Hagan, A., Bayesian Calibration of Computer Models, J. R. Stat. Soc, Ser. B, 63(3):425-464,2001.

  7. Strong, M. and Oakley, J., When is a Model Good Enough? Deriving the Expected Value of Model Improvement via Specifying Internal Model Discrepancies, SIAM/ASA J. Uncertainty Quantif., 2:106-125,2014.

  8. Smith, R., Uncertainty Quantification: Theory, Implementation and Applications, Philadelphia, PA: SIAM Computational Science and Engineering, 2013.

  9. Higdon, D., Gattiker, J., Williams, B., and Rightley, M., Computer Model Calibration Using High-Dimensional Output, J. Am. Stat. Assoc, 103(482):570-583,2008.

  10. Brynjarsdottlr, J. and O'Hagan, A., Learning about Physical Parameters: The Importance of Model Discrepancy, Inverse Probl, 30(11):114007,2014.

  11. Sargsyan, K., Najm, H., and Ghanem, R., On the Statistical Calibration of Physical Models, Int. J. Chem. Kinet., 47(4):246-276,2015.

  12. Wang, S., Chen, W., and Tsui, K.L., Bayesian Validation of Computer Models, Technometrics, 51(4):439-451,2009.

  13. Joseph, V.R. and Melkote, S.N., Statistical Adjustments to Engineering Models, J. Qual. Technol, 41(4):362,2009.

  14. Higdon, D., Kennedy, M., Cavendish, J.C., Cafeo, J.A., and Ryne, R.D., Combining Field Data and Computer Simulations for Calibration and Prediction, SIAM J. Sci. Comput., 26(2):448-466,2004.

  15. Bayarri, M.J., Berger, J.O., Paulo, R., Sacks, J., Cafeo, J.A., Cavendish, J., Lin, C.H., and Tu, J., A Framework for Validation of Computer Models, Technometrics, 49(2):138-154,2007.

  16. Oliver, T.A., Terejanu, G., Simmons, C.S., and Moser, R.D., Validating Predictions of Unobserved Quantities, Comput. Methods Appl. Mech. Eng., 283:1310-1335,2015.

  17. Pernot, P., The Parameter Uncertainty Inflation Fallacy, J. Chem. Phys, 147(10):104102,2017.

  18. He, Y. and Xiu, D., Numerical Strategy for Model Correction Using Physical Constraints, J. Comput. Phys, 313:617-634, 2016.

  19. Strong, M., Oakley, J., and Chilcott, J., Managing Structural Uncertainty in Health Economic Decision Models: A Discrepancy Approach, J. Royal Stat. Soc.: Series C (Appl. Stat.), 61(1):25-45,2012.

  20. Oliver, T.A. andMoser, R.D.,Bayesian Uncertainty Quantification Applied to RANS Turbulence Models, J. Phys.: Conf. Ser., 318(4):042032,2011.

  21. Emory, M., Pecnik, R., and Iaccarino, G., Modeling Structural Uncertainties in Reynolds-Averaged Computations of Shock/Boundary Layer Interactions, in 49th Proc. of AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, p. 479,2011.

  22. Huan, X., Safta, C., Sargsyan, K., Geraci, G., Eldred, M.S., Vane, Z.P., Lacaze, G., Oefelein, J.C., and Najm, H.N., Global Sensitivity Analysis and Quantification of Model Error for Large Eddy Simulation in Scramjet Design, in 19th Proc. of AIAA Non-Deterministic Approaches Conference, Grapevine, TX, paper no. 2017-1089,2017.

  23. Pernot, P. and Cailliez, F., A Critical Review of Statistical Calibration/Prediction Models Handling Data Inconsistency and Model Inadequacy, AIChEJ, 63(10):4642-4665,2017.

  24. Zio, S., da Costa, H.F., Guerra, G.M., Paraizo, P.L., Camata, J.J., Elias, R.N., Coutinho, A.L., and Rochinha, F.A., Bayesian Assessment of Uncertainty in Viscosity Closure Models for Turbidity Currents Computations, Comput. Methods Appl. Mech. Eng., 342:653-673,2018.

  25. Hakim, L., Lacaze, G., Khalil, M., Sargsyan, K., Najm, H., and Oefelein, J., Probabilistic Parameter Estimation in a Two-Step Chemical Kinetics Model forN-Dodecane Jet Autoignition, Combust. TheoryModell, 22(3):446-466,2018.

  26. Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach,New York: Springer Verlag, 1991.

  27. Xiu, D. and Karniadakis, G., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput., 24(2):619-644,2002.

  28. Najm, H., Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics, Ann. Rev. Fluid Mech., 41(1):35-52,2009.

  29. Le Maitre, O. andKnio, O., Spectral Methods for Uncertainty Quantification, New York, NY: Springer, 2010.

  30. Bayarri, M.J. and Castellanos, M.E., Bayesian Checking of the Second Level of Hierarchical Models (Discussion Paper), Stat. Sci, 22(3):322-367,2007.

  31. Bernardo, J. and Smith, A., Bayesian Theory, Wiley Series in Probability and Statistics, Chichester, UK: John Wiley & Sons Ld., 2000.

  32. Sivia, D.S. and Skilling, J., Data Analysis: A Bayesian Tutorial, Second Edition, 2nd ed., Oxford: Oxford University Press, 2006.

  33. Carlin, B.P. and Louis, T.A., Bayesian Methods for Data Analysis, Boca Raton, FL: Chapman andHall/CRC, 2011.

  34. Gamerman, D. and Lopes, H.F., Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Boca Raton, FL: Chapman and Hall/CRC, 2006.

  35. Gilks, W.R., Markov Chain Monte Carlo, New York: Wiley Online Library, 2005.

  36. Scott, D., Multivariate Density Estimation. Theory, Practice and Visualization, New York: Wiley, 1992.

  37. Silverman, B., Density Estimation for Statistics and Data Analysis, London: Chapman and Hall, 1986.

  38. Arnst, M., Ghanem, R., and Soize, C., Identification of Bayesian Posteriors for Coefficients of Chaos Expansions, J. Comput. Phys, 229(9):3134-3154,2010.

  39. Beaumont, M., Zhang, W., and Balding, D.J., Approximate Bayesian Computation in Population Genetics, Genetics, 162(4):2025-2035,2002.

  40. Marjoram, P., Molitor, J., Plagnol, V., and Tavare, S., Markov Chain Monte Carlo without Likelihoods, Proc. Natl. Acad. Sci. U.S.A., 100(26): 15324-15328,2003.

  41. Sisson, S.A. and Fan, Y., Handbook of Markov Chain Monte Carlo, in Likelihood-Free Markov Chain Monte Carlo, Boca Raton, FL: Chapman and Hall/CRC Press, pp. 313-338,2011.

  42. Leonard, T. and Hsu, J.S., Bayesian Inference for a Covariance Matrix, Ann. Stat., 20(4):1669-1696,1992.

  43. Alvarez, I., Niemi, J., and Simpson, M., Bayesian Inference for a Covariance Matrix, in Proc. of 26th Conf. on Applied Statistics in Agriculture, 2014.

  44. Daniels, M. and Kass, R., Nonconjugate Bayesian Estimation of Covariance Matrices and Its Use in Hierarchical Models, J. Am. Stat. Assoc, 94(448):1254-1263,1999.

  45. Chung, Y, Gelman, A., Rabe-Hesketh, S.,Liu, J., andDorie, V., Weakly Informative Prior for Point Estimation of Covariance Matrices in Hierarchical Models, J. Educ. Behav. Stat., 40(2):136-157,2015.

  46. Kass, R.E. and Natarajan, R., A Default Conjugate Prior for Variance Components in Generalized Linear Mixed Models (Comment on Article By Browne and Draper), Bayesian Anal., 1(3):535-542,2006.

  47. Smith, M. and Kohn, R., Parsimonious Covariance Matrix Estimation for Longitudinal Data, J. Aim. Stat. Assoc., 97(460): 1141-1153,2002.

  48. Yang, R. andBerger, J., Estimation of a Covariance Matrix Using the Reference Prior, Ann. Stat., 22(3): 1195-1211, 1994.

  49. Wang, H. and Pillai, N.S., On a Class of Shrinkage Priors for Covariance Matrix Estimation, J. Comput. Graph. Stat., 22(3):689-707,2013.

  50. Huang, A. and Wand, M.P., Simple Marginally Noninformative Prior Distributions for Covariance Matrices, Bayesian Anal., 8(2):439-452,2013.

  51. Gelman, A., Prior Distributions for Variance Parameters in Hierarchical Models, Bayesian Anal., 1(3): 515-534,2006.

  52. Barnard, J., McCulloch, R., andMeng, X.L., Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage, Stat. Sin, 10(4): 1281-1311,2000.

  53. Hsu, C.W., Sinay, M.S., and Hsu, J.S., Bayesian Estimation of a Covariance Matrix with Flexible Prior Specification, Ann. Inst. Stat. Math., 64(2):319-342,2012.

  54. Gelman, A., Meng, X.L., and Stern, H., Posterior Predictive Assessment of Model Fitness via Realized Discrepancies, Stat. Sin, 6:733-807,1996.

  55. Eldred, M.S., Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design, in Proc. of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf. 17th AIAA/ASME/AHS Adaptive Structures Conf. 11th AIAA, p. 2274,2009.

  56. Sargsyan, K., Surrogate Models for Uncertainty Propagation and Sensitivity Analysis, in Handbook of Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Berlin: Springer, 2017.

  57. Haario, H., Saksman, E., and Tamminen, J., An Adaptive Metropolis Algorithm, Bernoulli, 7:223-242,2001.

  58. Westbrook, C. and Dryer, F., Simplified Reaction Mechanisms for the Oxidation of Hydrocarbon Fuels in Flames, Combust. Sci. Technol, 27:31-43,1981.

  59. Dryer, F.L. and Glassman, I., High-Temperature Oxidation of CO and CH4, Proc. Combust. Inst, 14:987-1003,1973.

  60. Leenson, I.A. and Sergeev, G.B., Negative Temperature Coefficient in Chemical Reactions, Russ. Chem. Rev., 53(5):417, 1984.

  61. Fernandez-Tarrazo, E., Sanchez, A.L., Linan, A., and Williams, F.A., A Simple One-Step Chemistry Model for Partially Premixed Hydrocarbon Combustion, Combust. Flame, 147(1):32-38,2006.

  62. Franzelli, B., Riber, E., Sanjose, M., and Poinsot, T., A Two-Step Chemical Scheme for Kerosene-Air Premixed Flames, Combust. Flame, 157(7):1364-1373,2010.

  63. Misdariis, A., Vermorel, O., and Poinsot, T., A Methodology based on Reduced Schemes to Compute Autoignition and Propagation in Internal Combustion Engines, Proc. Combust. Inst., 35(3):3001-3008,2015.

  64. Hakim, L., Lacaze, G., Khalil, M., Najm, H., and Oefelein, J., Modeling Auto-Ignition Transients in Reacting Diesel Jets, J. Eng. Gas Turbines Power, 138(11):112806-1-112806-8,2016.

  65. Safta, C., Najm, H., and Knio, O., TChem-A Software Toolkit for the Analysis of Complex Kinetic Models, Sandia Rep. SAND2011-3282, from http://www.sandia.gov/tchem, 2011.

  66. Vasu, S.S., Davidson, D.F., Hong, Z., Vasudevan, V., and Hanson, R.K.,N-Dodecane Oxidation At High-Pressures: Measurements of Ignition Delay Times and OH Concentration Time-Histories, Proc. Combust. Inst., 32(1):173-180,2009.

  67. Narayanaswamy, K., Pepiot, P., and Pitsch, H., A Chemical Mechanism for Low to High Temperature Oxidation of N-Dodecane as a Component of Transportation Fuel Surrogates, Comb. Flame, 161:867-884,2014.

  68. Christensen, R., Plane Answers to Complex Questions: The Theory of Linear Models, 3rd ed., New York, NY: Springer-Verlag, 2002.

  69. Huan, X., Safta, C., Sargsyan, K., Geraci, G., Eldred, M.S., Vane, Z.P., Lacaze, G., Oefelein, J.C., and Najm, H.N., Global Sensitivity Analysis and Estimation of Model Error, Toward Uncertainty Quantification in Scramjet Computations, AIAA J, 56(3): 1170-1184,2018.

  70. Arendt, P.D., Apley, D.W., and Chen, W., Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability, J. Mech. Des., 134(10):100908,2012.

  71. Debusschere, B., Sargsyan, K., Safta, C., and Chowdhary, K., UQ Toolkit, from http://www.sandia.gov/UQToolkit,2017.

  72. Debusschere, B., Sargsyan, K., Safta, C., and Chowdhary, K., The Uncertainty Quantification Toolkit (UQTk), in Handbook ofUncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Berlin: Springer, 2017.

  73. Sobol, I.M., Theorems and Examples on High Dimensional Model Representation, Reliab. Eng. Syst. Saf, 79:187-193,2003.

  74. Saltelli, A., Tarantola, S., Campolongo, F., and Ratto, M., Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, New York: John Wiley & Sons, 2004.


Articles with similar content:

A NEW INVERSE METHOD FOR THE UNCERTAINTY QUANTIFICATION OF SPATIALLY VARYING RANDOM MATERIAL PROPERTIES
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 6
Gun Jin Yun, Shen Shang
BIAS MINIMIZATION IN GAUSSIAN PROCESS SURROGATE MODELING FOR UNCERTAINTY QUANTIFICATION
International Journal for Uncertainty Quantification, Vol.1, 2011, issue 4
Vadiraj Hombal, Sankaran Mahadevan
A NEW GIBBS SAMPLING BASED BAYESIAN MODEL UPDATING APPROACH USING MODAL DATA FROM MULTIPLE SETUPS
International Journal for Uncertainty Quantification, Vol.5, 2015, issue 4
Sahil Bansal
A MATHEMATICAL AND COMPUTATIONAL FRAMEWORK FOR MULTIFIDELITY DESIGN AND ANALYSIS WITH COMPUTER MODELS
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 1
Karen Willcox, Douglas Allaire
INFERENCE AND UNCERTAINTY PROPAGATION OF ATOMISTICALLY-INFORMED CONTINUUM CONSTITUTIVE LAWS, PART 1: BAYESIAN INFERENCE OF FIXED MODEL FORMS
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 2
Jeremy A. Templeton, Maher Salloum