Abstract
Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Beck, J.L.: Bayesian system identification based on probability logic. Struct. Control Health Monit. 17(7), 825–847 (2010)
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620 (1957)
Vakilzadeh, M.K., Rahrovani, S., Abrahamsson, T.: Modal reduction based on accurate input-output relation preservation. In: Topics in Modal Analysis, Springer, New York, 7, pp. 333–342 (2014)
Yaghoubi, V., Vakilzadeh, M.K., Abrahamsson, T.: A parallel solution method for structural dynamic response analysis. In: Dynamics of Coupled Structures, Springer International Publishing, New York, 4, pp. 149–161 (2015)
Madarshahian, R., Caicedo, J.M.: Reducing MCMC computational cost with a two layered Bayesian approach. In: Model Validation and Uncertainty Quantification, Springer International Publishing, New York, 3, pp. 291–297 (2015)
Chiachio, M., Beck, J.L., Chiachio, J., Rus, G.: Approximate Bayesian computation by subset simulation. SIAM J. Sci. Comput. 36(3), A1339–A1358 (2014)
Vakilzadeh, M.K., Huang, Y., Beck, J.L., Abrahamsson, T.: Approximate Bayesian computation by subset simulation using hierarchical state-space models. Mech. Syst. Signal Process. 84(part B), 2–20 (2017)
Straub, D., Papaioannou, I.: Bayesian updating with structural reliability methods. J. Eng. Mech. 141(3), 04014134 (2014)
Au, S.-K., DiazDelaO, F.A., Yoshida, I.: Bayesian updating and model class selection with Subset Simulation. arXiv preprint arXiv:1510.06989 (2015)
Green, P.: Bayesian system identification of a nonlinear dynamical system using a novel variant of simulated annealing. Mech. Syst. Signal Process. 52, 133–146 (2015)
Beck, J.L., Au, S.-K.: Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. J. Eng. Mech. 128(4), 380–391 (2002)
Ching, J., Chen, Y.-C.: Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J. Eng. Mech. 133(7), 816–832 (2007)
Gilks, W.R., Berzuini, C.: Following a moving target-Monte Carlo inference for dynamic Bayesian models. J. R. Stat. Soc. Ser. B Stat. Methodol. 63(1), 127–146 (2001)
Smith, A.F, Gelfand, A.E.: Bayesian statistics without tears: a sampling–resampling perspective. Am. Stat. 46(2), 84–88 (1992)
Beck, J.L., Zuev K.M.: Asymptotically independent Markov sampling: a new Markov chain Monte Carlo scheme for Bayesian inference. Int. J. Uncertain. Quantif. 3(5), 445–474 (2013)
Neal, R.M.: Probabilistic inference using Markov chain Monte Carlo methods Technical report CRG-TR-93-1, Department of Computer Science, University of Toronto, Toronto, CA (1993)
Hol, J.D., Schön, T.B., Gustafsson, F.: On resampling algorithms for particle filters. In: 2006 IEEE Nonlinear Statistical Signal Processing Workshop, pp. 79–82. IEEE, Piscataway, NJ (2006)
Douc, R., Cappé, O.: Comparison of resampling schemes for particle filtering. In: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005, pp. 64–69. IEEE, Piscataway, NJ (2005)
Gordon, N.J., Salmond, D.J., Smith, A.F.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In: IEEE Proceedings F Radar and Signal Processing, vol. 140(2), pp. 107–113 (1993). IET
Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T.: A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50(2), 174–188 (2002)
Doucet, A., Johansen, A.M.: A tutorial on particle filtering and smoothing: fifteen years later. In: Handbook of Nonlinear Filtering, vol. 12, pp. 656–704. Oxford University Press, Oxford (2009)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)
Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer Science and Business Media, New York (2013)
Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 73(2), 123–214 (2011)
Nocedal, J., Wright, S.: Numerical Optimization. Springer Science and Business Media, New York (2006)
Vakilzadeh, M.K., Yaghoubi, V., Johansson, A.T., Abrahamsson, T.: Manifold Metropolis adjusted Langevin algorithm for high-dimensional Bayesian FE. In: 9th International Conference on Structural Dynamics (EURODYN), Porto, 30 Jun–02 Jul 2014, pp. 3029–3036 (2014)
Bui-Thanh, T., Ghattas, O., Martin, J., Stadler, G.: A computational framework for infinite-dimensional Bayesian inverse problems Part I: the linearized case, with application to global seismic inversion. SIAM J. Sci. Comput. 35(6), A2494–A2523 (2013)
Martin, J., Wilcox, L.C., Burstedde, C., Ghattas, O.: A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J. Sci. Comput. 34(3), A1460–A1487 (2012)
Haario, H., Saksman, E., Tamminen, J.: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat. 14(3), 375–396 (1999)
Acknowledgements
The first author would like to thank Prof. James L. Beck from California Institute of Technology for his comments and suggestions on the early stages of this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 The Society for Experimental Mechanics, Inc.
About this paper
Cite this paper
Vakilzadeh, M.K., Sjögren, A., Johansson, A.T., Abrahamsson, T.J.S. (2017). Sequential Gauss-Newton MCMC Algorithm for High-Dimensional Bayesian Model Updating. In: Barthorpe, R., Platz, R., Lopez, I., Moaveni, B., Papadimitriou, C. (eds) Model Validation and Uncertainty Quantification, Volume 3. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-54858-6_30
Download citation
DOI: https://doi.org/10.1007/978-3-319-54858-6_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54857-9
Online ISBN: 978-3-319-54858-6
eBook Packages: EngineeringEngineering (R0)