Time-limited Balanced Truncation for unstable Bayesian Inference

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

1. König, J., Freitag, M. A. "Time-limited Balanced Truncation for Data Assimilation Problems." Journal of Scientific Computing 97.47 (2023).

Summary

The work in [1] generalizes the concept of balancing for Bayesian inference from [2] to arbitrary prior covariances and unstable system matrices. For this purpose, the concept of time-limited balanced truncation is used. Time-limited Gramians are efficiently computed by rational Krylov methods from [4]. Numerical examples compare the performance with the original approach from [2] and the optimal approach from [3]. This work uses the code from [2] (to be found at https://github.com/elizqian/balancing-bayesian-inference) and [4] (to be found at https://zenodo.org/record/7366026).

Examples

To run this code, you need the MATLAB Control System Toolbox.

To generate the plots comparing computations with a compatible and non-compatible prior from the paper (Figure 1), run the heat_incompatible_prior.m script.

To generate the TLBT plots from the paper, run the compare_*.m scripts, corresponding to:

• compare_heat.m: The heat equation example for end times T = 1, 3, 10 with measurements spaced 0.005 seconds apart (Figure 2)
• compare_ISS.m: The ISS example for end times T = 1, 3, 10 with measurements spaced 1 seconds apart (Figure 3)
• compare_unstable.m: The advection-diffusion equation example for end times T = 0.1, 0.5, 1 with measurements spaced 0.001 seconds apart (Figure 4)

References

2. 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." Journal of Scientific Computing 91.29 (2022).
3. 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. 6 (2015): A2451-A2487.
4. Kürschner, P. "Balanced truncation model order reduction in limited time intervals for large systems." Advances in Computational Mathematics 44.6 (2018): 1821–1844.

Contact

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