On-the-Fly Bayesian Data Assimilation Using Transport Map Sampling and PGD Reduced Models

  • Paul-Baptiste Rubio
  • Ludovic ChamoinEmail author
  • François Louf
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 93)


The motivation of this research work is to address real-time sequential data assimilation and inference of model parameters within a full Bayesian formulation. For that purpose, we couple two advanced numerical approaches. First, the Transport Map sampling is used as an alternative to classical Markov Chain approaches in order to facilitate the sampling of posterior densities resulting from Bayesian inference. It builds a deterministic mapping, obtained from a minimization problem, between the posterior probability measure of interest and a simple reference measure. Second, the Proper Generalized Decomposition (PGD) is implemented in order to reduce the computational effort for the evaluation of the multi-parametric numerical model in the online phase, and therefore for uncertainty quantification on outputs of interest of the model. The PGD also speeds up the minimization algorithm of the Transport Map method, as derivatives with respect to model parameters can then be computed in a straightforward manner. The performance of the approach is illustrated on a fusion welding application.


  1. 1.
    Arulampalam, M. S., Maskell, S., Gordon, N., & Clapp, T. (2002). A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing, 50(2), 174–188.CrossRefGoogle Scholar
  2. 2.
    Berger, J., Orlande, H. R. B., & Mendes, N. (2017). Proper generalized decomposition model reduction in the Bayesian framework for solving inverse heat transfer problems. Inverse Problems in Science and Engineering, 25(2), 260–278.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chinesta, F., Keunings, R., & Leygue, A. (2014). The proper generalized decomposition for advanced numerical simulations: A primer. Springer Briefs in Applied Sciences and Technology.Google Scholar
  4. 4.
    Chinesta, F., Ladevèze, P., & Cueto, E. (2011). A short review on model order reduction based on proper generalized decomposition. Archives of Computational Methods in Engineering, 18(4), 395–404.CrossRefGoogle Scholar
  5. 5.
    Darema, F. (2004). Dynamic data driven applications systems: A new paradigm for application simulations and measurements. In Computational Science—ICCS (pp. 662–669).Google Scholar
  6. 6.
    El Moselhy, T. A., & Marzouk, Y. (2012). Bayesian inference with optimal maps. Journal of Computational Physics, 231(23), 7815–7850.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gamerman, D., & Lopes, H. F. (2006). Markov Chain Monte Carlo-stochastic simulation for Bayesian inference. Boca Raton: CRC Press.Google Scholar
  8. 8.
    Grepl, M. A. (2005). Reduced-basis approximation and a posteriori error estimation for parabolic partial differential equations (Ph.D. thesis). Massachusetts Institute of Technology.Google Scholar
  9. 9.
    Kaipio, J., & Somersalo, E. (2004). Statistical and computational inverse problems. New York: Springer.zbMATHGoogle Scholar
  10. 10.
    Marzouk, Y., Moselhy, T., Parno, M., & Spantini, A. (2016). Sampling via measure transport: An introduction. Handbook of Uncertainty Quantification, 1–41.Google Scholar
  11. 11.
    Robert, C. P., & Casella, G. (2004). Monte Carlo statistical methods, Springer texts in statistics. New York: Springer.CrossRefGoogle Scholar
  12. 12.
    Rubio, P. B., Louf, F., & Chamoin, L. (2018). Fast model updating coupling Bayesian inference and PGD model reduction. Computational Mechanics, 62(6), 1485–1509.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rubio, P. B., Louf, F., & Chamoin, L. (2019). Transport Map sampling with PGD model reduction for fast dynamical Bayesian data assimilation. International Journal in Numerical Methods in Engineering, 120(4), 447–472.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Spantini, A., Bigoni, D., & Marzouk, Y. (2018). Inference via low-dimensional couplings. Journal of Machine Learning Research, 19, 1–71.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Stuart, A. M. (2010). Inverse problems: A Bayesian perspective. Acta Numerica, 19, 451–559.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tarantola, A. (2005). Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics.Google Scholar
  17. 17.
    Villani, C. (2008). Optimal transport: Old and new. Berlin: Springer.Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Paul-Baptiste Rubio
    • 1
  • Ludovic Chamoin
    • 1
    Email author
  • François Louf
    • 1
  1. 1.LMT ENS Paris-SaclayCachanFrance

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