Bayesian Updating with Structural Reliability Methods
Publication: Journal of Engineering Mechanics
Volume 141, Issue 3
Abstract
Bayesian updating is a powerful method to learn and calibrate models with data and observations. Because of the difficulties involved in computing the high-dimensional integrals necessary for Bayesian updating, Markov chain Monte Carlo (MCMC) sampling methods have been developed and successfully applied for this task. The disadvantage of MCMC methods is the difficulty of ensuring the stationarity of the Markov chain. We present an alternative to MCMC that is particularly effective for updating mechanical and other computational models, termed Bayesian updating with structural reliability methods (BUS). With BUS, structural reliability methods are applied to compute the posterior distribution of uncertain model parameters and model outputs in general. An algorithm for the implementation of BUS is proposed, which can be interpreted as an enhancement of the classic rejection sampling algorithm for Bayesian updating. This algorithm is based on the subset simulation, and its efficiency is not dependent on the number of random variables in the model. The method is demonstrated by application to parameter identification in a dynamic system, Bayesian updating of the material parameters of a structural system, and Bayesian updating of a random field–based finite-element model of a geotechnical site.
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References
Au, S.-K., and Beck, J. L. (2001). “Estimation of small failure probabilities in high dimensions by subset simulation.” Probab. Eng. Mech., 16(4), 263–277.
Beck, J. L., and Au, S.-K. (2002). “Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation.” J. Eng. Mech., 380–391.
Beck, J. L., and Katafygiotis, L. S. (1998). “Updating models and their uncertainties. I: Bayesian statistical framework.” J. Eng. Mech., 455–461.
Benjamin, J., and Cornell, C. A. (1970). Probability, statistics, and decision for civil engineers, McGraw Hill, New York.
Beven, K. (2008). Environmental modelling: An uncertain future? Taylor & Francis, Oxford, U.K.
Bolstad, W. M. (2011). Understanding computational Bayesian statistics, Wiley, Hoboken, NJ.
Bucher, C. G. (1988). “Adaptive sampling—An iterative fast Monte Carlo procedure.” Struct. Saf., 5(2), 119–126.
Cheung, S. H., and Beck, J. L. (2009). “Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters.” J. Eng. Mech., 243–255.
Ching, J., and Chen, Y.-C. (2007). “Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging.” J. Eng. Mech., 816–832.
Chopin, N. (2002). “A sequential particle filter method for static models.” Biometrika, 89(3), 539–552.
Der Kiureghian, A., and Ke, J.-B. (1988). “The stochastic finite element method in structural reliability.” Probab. Eng. Mech., 3(2), 83–91.
Der Kiureghian, A., and Liu, P.-L. (1986). “Structural reliability under incomplete probability information.” J. Eng. Mech., 85–104.
Ditlevsen, O., and Madsen, H. O. (1996). Structural reliability methods, Wiley, Chichester, U.K.
Doucet, A., De Freitas, N., and Gordon, N. (2001). Sequential Monte Carlo methods in practice, Springer, New York.
Fisher, R. A. (1922). “On the mathematical foundations of theoretical statistics.” Phil. Trans. R. Soc. Lond. A, 222(594–604), 309–368.
Gelman, A. (2004). Bayesian data analysis, Chapman & Hall/CRC, Boca Raton, FL.
Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1998). Markov chain Monte Carlo in practice, Chapman & Hall, Boca Raton, FL.
Gilks, W. R., Thomas, A., and Spiegelhalter, D. J. (1994). “A language and program for complex Bayesian modelling.” Statistician, 43(1), 169–177.
Gilks, W. R., and Wild, P. (1992). “Adaptive rejection sampling for Gibbs sampling.” Appl. Stat., 41(2), 337–348.
Hohenbichler, M., and Rackwitz, R. (1981). “Non-normal dependent vectors in structural safety.” J. Engrg. Mech. Div., 107(6), 1227–1238.
Kavetski, D., Kuczera, G., and Franks, S. W. (2006). “Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory.” Water Resour. Res., 42(3), W03407.
Kennedy, M. C., and O’Hagan, A. (2001). “Bayesian calibration of computer models.” J. R. Stat. Soc. Series B Stat. Methodol., 63(3), 425–464.
Kurtz, N., and Song, J. (2013). “Cross-entropy-based adaptive importance sampling using Gaussian mixture.” Struct. Saf., 42, 35–44.
Madsen, H. O., Krenk, S., and Lind, N. C. (1985). Methods of structural safety, Prentice Hall, Englewood Cliffs, NJ.
Mahadevan, S., Zhang, R., and Smith, N. (2001). “Bayesian networks for system reliability reassessment.” Struct. Saf., 23(3), 231–251.
Natke, H. G. (1988). “Updating computational models in the frequency domain based on measured data: A survey.” Probab. Eng. Mech., 3(1), 28–35.
Papadimitriou, C., Beck, J. L., and Katafygiotis, L. S. (2001). “Updating robust reliability using structural test data.” Probab. Eng. Mech., 16(2), 103–113.
Papaioannou, I., Betz, W., Zwirglmaier, K., and Straub, D. (2014). “MCMC algorithms for subset simulation.” 〈http://www.era.bgu.tum.de/fileadmin/w00bkd/www/Papers/MCMC_SubS.pdf〉 (Aug. 20, 2014).
Papaioannou, I., and Straub, D. (2012). “Reliability updating in geotechnical engineering including spatial variability of soil.” Comput. Geotech., 42, 44–51.
Plummer, M., Best, N., Cowles, K., and Vines, K. (2006). “CODA: Convergence diagnosis and output analysis for MCMC.” R News 6(1), 7–11.
Rackwitz, R. (2001). “Reliability analysis—A review and some perspectives.” Struct. Saf., 23(4), 365–395.
Smith, A. F. M., and Gelfand, A. E. (1992). “Bayesian statistics without tears: A sampling-resampling perspective.” Am. Stat., 46(2), 84–88.
SOFiSTiK 2012 [Computer software]. Oberschleissheim, Germany, SOFiSTiK AG.
Straub, D. (2009). “Stochastic modeling of deterioration processes through dynamic Bayesian networks.” J. Eng. Mech., 1089–1099.
Straub, D. (2011). “Reliability updating with equality information.” Probab. Eng. Mech., 26(2), 254–258.
Straub, D., and Der Kiureghian, A. (2010a). “Bayesian network enhanced with structural reliability methods: Application.” J. Eng. Mech., 1259–1270.
Straub, D., and Der Kiureghian, A. (2010b). “Bayesian network enhanced with structural reliability methods: Methodology.” J. Eng. Mech., 1248–1258.
Straub, D., and Faber, M. H. (2005). “Risk based inspection planning for structural systems.” Struct. Saf., 27(4), 335–355.
Sundar, V. S., and Manohar, C. S. (2013). “Updating reliability models of statically loaded instrumented structures.” Struct. Saf., 40(0), 21–30.
Tang, W. H. (1973). “Probabilistic updating of flaw information.” J. Test. Eval., 1(6), 459–467.
Turlach, B. A. (1993). “Bandwidth selection in kernel density estimation: A review.” 〈http://staffhome.ecm.uwa.edu.au/∼00043886/cv/rep.html〉 (Aug. 20, 2014).
Yang, J., and Trapp, W. (1974). “Reliability analysis of aircraft structures under random loading and periodic inspection.” AIAA J., 12(12), 1623–1630.
Zuev, K. M., Beck, J. L., Au, S.-K., and Katafygiotis, L. S. (2012). “Bayesian post-processor and other enhancements of subset simulation for estimating failure probabilities in high dimensions.” Comp. Struct., 92–93, 283–296.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 3 • March 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Apr 15, 2013
Accepted: May 20, 2014
Published online: Sep 3, 2014
Published in print: Mar 1, 2015
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