Taking into account input uncertainties in the Bayesian calibration of time-consuming simulators

Perrin, Guillaume; Durantin, Cédric

Journal de la société française de statistique, Tome 160 (2019) no. 2, pp. 24-46.

La prise en compte des incertitudes expérimentales est un élément clé de la quantification des incertitudes et de la prévision par la simulation. Bien qu’une attention particulière soit accordée aux incertitudes expérimentales sur les sorties de simulation, peu de travaux s’intéressent aux incertitudes concernant les entrées de simulation, sous pretexte qu’elles sont négligeables ou suffisamment petites pour être agrégées avec les incertitudes sur les sorties par développement de Taylor. Toutefois, ces incertitudes sur les entrées ne sont pas toujours faibles et, selon la structure du code, la linéarisation autour de celles-ci n’est pas toujours possible. L’objectif de ce travail est donc double. Premièrement, il introduit un cadre bayésien général permettant l’intégration des incertitudes sur les entrées pour le calage de paramètres du code. Il propose ensuite plusieurs approches pour résoudre efficacement ce problème d’inférence, en fonction de la régularité du code et du type d’entrées considérées. Les avantages et les inconvénients des différentes méthodes sont finalement illustrés sur un exemple analytique, ainsi que sur un problème balistique.

The consideration of experimental uncertainties is a key element in the quantification of uncertainties and prediction by simulation. While particular attention is paid to experimental uncertainties on simulation outputs, little work is done on uncertainties on simulation inputs, arguing that they are negligible or small enough to be aggregated with uncertainties on outputs via Taylor development. However, these uncertainties on inputs are not always low and, depending on the structure of the code, linearization around them is not always possible. The objective of this work is therefore twofold. First, it introduces a general Bayesian framework for integrating input uncertainties into the calibration of code parameters. It then proposes several approaches to effectively solve this inference problem, depending on the regularity of the code and the type of inputs considered. The advantages and disadvantages of the different methods are finally illustrated on an analytical example, as well as on a ballistic problem.

[Berliner, 2001] Berliner, L. M. (2001). Monte carlo based ensemble forecasting. Statistics and Computing, 11. | MR

[Bilionis and Zabaras, 2015] Bilionis, I. and Zabaras, N. (2015). Bayesian uncertainty propagation using gaussian processes. In: Ghanem R., Higdon D., Owhadi H. (eds) Handbook of Uncertainty Quantification. Springer. | MR

[Chen and Schwab, 2015] Chen, P. and Schwab, C. (2015). Sparse-grid, reduced basis bayesian inversion. Computer Methods in Applied Mechanics and Engineering, 297:84–115. | MR

[Conrad et al., 2018] Conrad, P. R., Davis, A., Marzouk, Y. M., Pillai, N. S., and Smith, A. (2018). Parallel local approximation MCMC for expensive models. SIAM/ASA J. Uncertainty Quantification, 6(1):39–373. | MR

[Conrad et al., 2016] Conrad, P. R., Marzouk, Y. M., Pillai, N. S., and Smith, A. (2016). Accelerating asymptotically exact MCMC for computationally intensive models via local approximations. Journal of the American Statistical Association, 111:1591–1607. | MR

[Damblin et al., 2013] Damblin, G., Barbillon, P., Keller, M., Pasanisi, A., and Parent, E. (2013). Adaptive Numerical Designs for the Calibration of Computer Codes. SIAM/ASA J. Uncertainty Quantification, 6(1):151–179. | MR

[Fielding et al., 2011] Fielding, M., Nott, D. J., and Liong, S. Y. (2011). Efficient MCMC schemes for Computationally Expensive Posterior Distributions. Technometrics, 53(1):16–28. | MR

[Higdon et al., 2008] Higdon, D., Gattiker, J., Williams, B., and Rightley, M. (2008). Computer model calibration using high-dimensional output. Journal of the American Statistical Association, 103(482):570–583. | MR

[Higdon et al., 2003] Higdon, D., Lee, H., and Holloman, C. (2003). Markov chain monte carlo based approaches for inference in computationally intensive inverse problems. Bayesian Statistics, 7:181–197. | MR

[Kaipio and Somersalo, 2004] Kaipio, J. P. . and Somersalo, E. (2004). Statistics and Computational Inverse Problems. Springer, New York. | MR

[Kennedy and O’Hagan, 2001] Kennedy, M. and O’Hagan, A. (2001). Bayesian calibration of computer models. Journal of the royal statistical society, 63:425–464. | MR | Zbl

[Li and Marzouk, 2014] Li, J. and Marzouk, Y. M. (2014). Adaptive construction of surrogates for the bayesian solution of inverse problems. SIAM Journal on Scientific Computing, 36:A1163-A1186. | MR

[Liu et al., 2009] Liu, F., Bayarri, M., and Berger, J. (2009). Modularization in bayesian analysis, with emphasis on analysis of computer models. Bayesian Analysis, 4(1):119–150. | MR

[Marin and Robert, 2007] Marin, J. M. . and Robert, C. P. (2007). Bayesian core. Springer-Verlag, New York. | MR

[Marzouk and Najm, 2009] Marzouk, Y. M. and Najm, H. N. (2009). Dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems. Journal of Computational Physics, 228 (6):1862–1902. | MR | Zbl

[Marzouk and Xiu, 2009] Marzouk, Y. M. and Xiu, D. (2009). A stochastic collocation approach to bayesian inference in inverse problems. Communications in Computational Physics, 6:826-847. | MR

[Perrin, 2019] Perrin, G. (2019). Adaptive calibration of a computer code with time-series output. submitted to Reliability Engineering and System Safety.

[Perrin and Cannamela, 2017] Perrin, G. and Cannamela, C. (2017). A repulsion-based method for the definition and the enrichment of opotimized space filling designs in constrained input spaces. Journal de la Société Française de Statistique, 158(1):37–67. | Numdam | MR

[Perrin et al., 2018] Perrin, G., Soize, C., and Ouhbi, N. (2018). Data-driven kernel representations for sampling with an unknown block dependence structure under correlation constraints. Journal of Computational Statistics and Data Analysis, 119:139–154. | MR

[Rasmussen, 2003] Rasmussen, C. E. (2003). Gaussian processes to speed up hybrid monte carlo for expensive bayesian integrals. Bayesian Statistics, 7:651-659. | MR

[Rubinstein and Kroese, 2008] Rubinstein, R. T. and Kroese, D. (2008). Simulation and the Monte Carlo method. John Wiley and Sons, Inc., Hoboken, New Jersey. | MR | Zbl

[Santner et al., 2003] Santner, T. J., Williams, B., and Notz, W. (2003). The design and analysis of computer experiments. Springer, New York. | MR | Zbl

[Sargsyan et al., 2018] Sargsyan, K., Huan, X., and Najm, H. (2018). Embedded model error representation for Baysian model calibration. arXiv:1801.06768v1. | MR

[Scott and Sain, 2004] Scott, D. W. and Sain, S. R. (2004). Multidimensional Density Estimation.

[Sinsbeck and Nowak, 2017] Sinsbeck, M. and Nowak, W. (2017). Sequential Design of Computer Experiments for the Solution of Bayesian Inverse. SIAM/ASA J. Uncertainty Quantification, 5:640–664. | MR

[Tian et al., 2016] Tian, M., Li, D., Cao, Z., Phoon, K., and Wang, Y. (2016). Bayesian identification of random field model using indirect test data. Engineering Geology, 210:197–211.

[Tsilifis et al., 2017] Tsilifis, P., Ghanem, R. G., and Hajali, P. (2017). Efficient Bayesian Experimentation Using an Expected Information Gain Lower Bound. SIAM/ASA J. Uncertainty Quantification, 5:30-62. | MR

[Wan and Zabaras, 2011] Wan, J. and Zabaras, N. (2011). A Bayesian approach to multiscale inverse problems using the sequential Monte Carlo method. Inverse Problems, 27. | MR | Zbl

[Wand and Jones, 1995] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Encyclopedia of Statistics in Behavioral Science, 60(60):212. | MR | Zbl