Balanced truncation for Bayesian inference

This repository contains code for the numerical results of the following paper:

1. Qian, E., Tabeart, J. M., Beattie, C., Gugercin, S., Jiang, J., Kramer, P. R., and Narayan, A. Model reduction for linear dynamical systems via balancing for Bayesian inference. arXiv Journal of Scientific Computing 91:29, 2022.
   BibTeX

   @article{Qian2021Balancing,
   title   = {Model reduction for linear dynamical systems via balancing for Bayesian inference},
   author  = {Qian, E. and Tabeart, J. M. and Beattie, C. and Gugercin, S. and Jiang, J. and Kramer, P. R. and Narayan, A.},
   journal = {Journal of Scientific Computing},
   url     = {https://link.springer.com/article/10.1007/s10915-022-01798-8},
   year    = {2022},
   volume  = 91,
   issue   = 29
   }

Summary

The work in [1] considers a Bayesian approach to the task of inferring the unknown initial state of a linear dynamical system based on noisy linear measurements taken after the initial time. The initial state is endowed with a Gaussian prior and measurement noise is also assumed to be Gaussian, so that the Bayesian posterior is also Gaussian. We define a balanced truncation for Bayesian inference model reduction approach and show that the resulting reduced models inherit stability and error guarantees from the system-theoretic setting, and in some settings yield an optimal posterior covariance approximation as defined in [2]. See [3] for a generalization of the present method to new settings.

Examples

To generate the plots from the paper, run the *_plot{1,2}.m scripts, corresponding to:

• heat_plot1.m: The heat equation example with 500,000 measurements spaced 10^-4 seconds apart (Figure 5.1) -- this example may take a minute or two to run.
• heat_plot2.m: The heat equation example with 100 measurements spaced 0.1 seconds apart (Figure 5.2)
• iss_plot1.m: The ISS example with 3000 measurements spaced 0.1 seconds apart (Figure 5.3)
• iss_plot2.m: The ISS example with 10 measurements spaced 1 second apart (Figure 5.4)

See also

2. Spantini, A., Solonen, A., Cui, T., Martin, J., Tenorio, L., and Marzouk, Y. "Optimal low-rank approximations of Bayesian linear inverse problems." SIAM Journal on Scientific Computing 37, no. 6 (2015): A2451-A2487.

3. König, J. and Freitag, M. "Time-limited balanced truncation for data assimilation problems." arXiv preprint, 2022. arXiv:2212.07719. Related GitHub repo

Contact

Please feel free to contact Elizabeth Qian with any questions about this repository or the associated paper.

Acknowledgments

Thanks to Josie König for pointing out a bug.