Advertisement

The Use of Radial Basis Function Surrogate Models for Sampling Process Acceleration in Bayesian Inversion

  • Simona DomesováEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 554)

Abstract

The Bayesian approach provides a natural way of solving engineering inverse problems including uncertainties. The objective is to describe unknown parameters of a mathematical model based on noisy measurements. Using the Bayesian approach, the vector of unknown parameters is described by its joint probability distribution, i.e. the posterior distribution. To provide samples, Markov Chain Monte Carlo methods can be used. Their disadvantage lies in the need of repeated evaluations of the mathematical model that are computationally expensive in the case of practical problems.

This paper focuses on the reduction of the number of these evaluations. Specifically, it explores possibilities of the use of radial basis function surrogate models in sampling methods based on the Metropolis-Hastings algorithm. Furthermore, updates of the surrogate model during the sampling process are suggested. The procedure of surrogate model updates and its integration into the sampling algorithm is implemented and supported by numerical experiments.

Keywords

Bayesian inversion Metropolis-Hastings Radial basis functions Surrogate model Uncertainty quantification 

Notes

Acknowledgement

This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPS II) project “IT4Innovations excellence in science - LQ1602”. The work was also partially supported by Grant of SGS No. SP2018/68 and by Grant of SGS No. SP2018/161, VŠB - Technical University of Ostrava, Czech Republic.

References

  1. 1.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003).  https://doi.org/10.1017/CBO9780511543241CrossRefzbMATHGoogle Scholar
  2. 2.
    Christen, J.A., Fox, C.: Markov chain Monte Carlo using an approximation. J. Comput. Graph. Stat. 14(4), 795–810 (2005).  https://doi.org/10.1198/106186005X76983MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cui, T., Fox, C., O’Sullivan, M.J.: Bayesian calibration of a large-scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm: adaptive delayed acceptance Metropolis-hastings algorithm. Water Resour. Res. 47(10) (2011).  https://doi.org/10.1029/2010WR010352
  4. 4.
    Cui, T., Marzouk, Y.M., Willcox, K.E.: Data-driven model reduction for the Bayesian solution of inverse problems: Data-driven Model Reduction for Inverse Problems. Int. J. Numer. Methods Eng. 102(5), 966–990 (2015).  https://doi.org/10.1002/nme.4748CrossRefzbMATHGoogle Scholar
  5. 5.
    Dodwell, T.J., Ketelsen, C., Scheichl, R., Teckentrup, A.L.: A hierarchical multilevel Markov Chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow. SIAM/ASA J. Uncertain. Quantif. 3(1), 1075–1108 (2015).  https://doi.org/10.1137/130915005MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Domesová, S., Béreš, M.: A bayesian approach to the identification problem with given material interfaces in the darcy flow. In: High Performance Computing in Science and Engineering, vol. 11087, pp. 203–216. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-97136-0_15
  7. 7.
    Domesova, S., Beres, M.: Inverse problem solution using bayesian approach with application to darcy flow material parameters estimation. Adv. Electr. Electron. Eng. 15(2), 258–266 (2017).  https://doi.org/10.15598/aeee.v15i2.2236CrossRefGoogle Scholar
  8. 8.
    Dostert, P., Efendiev, Y., Mohanty, B.: Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models. Adv. Water Resour. 32(3), 329–339 (2009).  https://doi.org/10.1016/j.advwatres.2008.11.009CrossRefGoogle Scholar
  9. 9.
    Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov Chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28(2), 776–803 (2006).  https://doi.org/10.1137/050628568MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kaipio, J.P., Fox, C.: The Bayesian framework for inverse problems in heat transfer. Heat Transfer Eng. 32(9), 718–753 (2011).  https://doi.org/10.1080/01457632.2011.525137CrossRefGoogle Scholar
  11. 11.
    Moulton, J.D., Fox, C., Svyatskiy, D.: Multilevel approximations in sample-based inversion from the Dirichlet-to-Neumann map. J. Phys. Conf. Ser. 124, 012035 (2008).  https://doi.org/10.1088/1742-6596/124/1/012035CrossRefGoogle Scholar
  12. 12.
    Robert, C.P.: The Bayesian choice: from decision-theoretic foundations to computational implementation, 2nd (edn.) Springer Texts in Statistics. Springer, New York (2007). OCLC: 255965262Google Scholar
  13. 13.
    Sherlock, C., Golightly, A., Henderson, D.A.: Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods. J. Comput. Graph. Stat. 26(2), 434–444 (2017).  https://doi.org/10.1080/10618600.2016.1231064MathSciNetCrossRefGoogle Scholar
  14. 14.
    Stuart, A.M.: Inverse problems: a bayesian perspective. Acta Numerica 19, 451–559 (2010).  https://doi.org/10.1017/S0962492910000061MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhang, G., Lu, D., Ye, M., Gunzburger, M., Webster, C.: An adaptive sparse-grid high-order stochastic collocation method for Bayesian inference in groundwater reactive transport modeling: sparse-grid method for bayesian inference. Water Resour. Res. 49(10), 6871–6892 (2013).  https://doi.org/10.1002/wrcr.20467CrossRefGoogle Scholar
  16. 16.
    Zhang, W., Liu, J., Cho, C., Han, X.: A fast Bayesian approach using adaptive densifying approximation technique accelerated MCMC. Inverse Prob. Sci. Eng. 24(2), 247–264 (2016).  https://doi.org/10.1080/17415977.2015.1017488MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.FEECS, Department of Applied MathematicsVŠB - Technical University of OstravaOstrava-PorubaCzech Republic
  2. 2.IT4Innovations National Supercomputing CenterVŠB - Technical University of OstravaOstrava-PorubaCzech Republic
  3. 3.Institute of Geonics of the CASOstrava-PorubaCzech Republic

Personalised recommendations