The Use of Radial Basis Function Surrogate Models for Sampling Process Acceleration in Bayesian Inversion
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.
KeywordsBayesian inversion Metropolis-Hastings Radial basis functions Surrogate model Uncertainty quantification
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.
- 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
- 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.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
- 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
- 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