Bayesian Analysis

Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Bani K. Mallick, Keren Yang, Nilabja Guha, and Yalchin Efendiev

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Abstract

This note is a discussion of the article “Bayesian Solution Uncertainty Quantification for Differential Equations” by Chkrebtii, Campbell, Calderhead, and Girolami. The authors propose stochastic models for differential equation discretizations. While appreciating the main concepts, we point out some possible extensions and modifications.

Article information

Source
Bayesian Anal., Volume 11, Number 4 (2016), 1279-1284.

Dates
First available in Project Euclid: 22 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ba/1479805385

Digital Object Identifier
doi:10.1214/16-BA1038

Mathematical Reviews number (MathSciNet)
MR3577381

Zentralblatt MATH identifier
1357.62120

Keywords
Bayesian uncertainty quantification discretization errors multiscale variational formulation

Citation

Mallick, Bani K.; Yang, Keren; Guha, Nilabja; Efendiev, Yalchin. Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. 11 (2016), no. 4, 1279--1284. doi:10.1214/16-BA1038. https://projecteuclid.org/euclid.ba/1479805385


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References

  • Barron, A., Schervish, M. J., Wasserman, L., et al. (1999). “The consistency of posterior distributions in nonparametric problems.” The Annals of Statistics, 27(2): 536–561.
  • Calo, V. M., Efendiev, Y., Galvis, J., and Li, G. (2016). “Randomized oversampling for generalized multiscale finite element methods.” Multiscale Modeling & Simulation, 14(1): 482–501.
  • Chakraborty, A., Bingham, D., Dhavala, S. S., Kuranz, C. C., Drake, R. P., Grosskopf, M. J., Rutter, E. M., Torralva, B. R., Holloway, J. P., McClarren, R. G., et al. (2016). “Emulation of numerical models with over-specified basis functions.” Technometrics,
  • Chung, E., Efendiev, Y., and Hou, T. Y. (2016a). “Adaptive multiscale model reduction with generalized multiscale finite element methods.” Journal of Computational Physics, 320: 69–95.
  • Chung, E., Efendiev, Y., Leung, W. T., and Li, G. (2016b). “Sparse Generalized Multiscale Finite Element Methods and their applications.” International Journal for Multiscale Computational Engineering, 14(1).
  • Efendiev, Y., Datta-Gupta, A., Hwang, K., Ma, X., and Mallick, B. (2008a). “Bayesian partition models for subsurface characterization.” Large-Scale Inverse Problems and Quantification of Uncertainty, 107–122.
  • Efendiev, Y., Datta-Gupta, A., Ma, X., and Mallick, B. (2008b). “Modified Markov Chain Monte Carlo method for dynamic data integration using streamline approach.” Mathematical Geosciences, 40(2): 213–232.
  • Efendiev, Y. and Hou, T. Y. (2009). “Multiscale finite element methods: Theory and Applications.” Springer Science & Business Media, 2009,
  • Guha, N., Wu, X., Efendiev, Y., Jin, B., and Mallick, B. K. (2015). “A variational Bayesian approach for inverse problems with skew-t error distributions.” Journal of Computational Physics, 301: 377–393.
  • Hoang, V. H., Schwab, C., and Stuart, A. M. (2013). “Complexity analysis of accelerated MCMC methods for Bayesian inversion.” Inverse Problems, 29(8): 085010.
  • Holmes, C., Denison, D. T., Ray, S., and Mallick, B. (2012). “Bayesian prediction via partitioning.” Journal of Computational and Graphical Statistics. 14(4): 811–830,
  • Kennedy, M. C. and O’Hagan, A. (2001). “Bayesian calibration of computer models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(3): 425–464.
  • Konomi, B. A., Sang, H., and Mallick, B. K. (2014). “Adaptive Bayesian nonstationary modeling for large spatial datasets using covariance approximations.” Journal of Computational and Graphical Statistics, 23(3): 802–829.
  • Mondal, A., Mallick, B., Efendiev, Y., and Datta-Gupta, A. (2014). “Bayesian Uncertainty Quantification for Subsurface Inversion Using a Multiscale Hierarchical Model.” Technometrics, 56(3): 381–392.
  • Stuart, A. (2014). “Uncertainty quantification in Bayesian inversion.” ICM2014. Invited Lecture.
  • Vollmer, S. J. (2013). “Posterior consistency for Bayesian inverse problems through stability and regression results.” Inverse Problems, 29(12): 125011.
  • Yang, K., Guha, N., Efendiev, Y., and Mallick, B. (2016). “Bayesian and Variational Bayesian approaches for flows in heterogenous random media.” arXiv:1611.01213.

See also

  • Related item: Oksana A. Chkrebtii, David A. Campbell, Ben Calderhead, Mark A. Girolami (2016). Bayesian Solution Uncertainty Quantification for Differential Equations. Bayesian Anal. Vol. 11, Iss. 4, 1239–1267.