Time-limited Balanced Truncation within Incremental Four-Dimensional Variational Data Assimilation

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

1. König, J. & Freitag, M. A. "Time-limited balanced truncation within incremental four-dimensional variational data assimilation". Proceedings in Applied Mathematics and Mechanics, e202300019 (2023).

Summary

The work in [1] applies time-limited balanced truncation within the inner loop of incremental four-dimensional variational data assimilation. Results and code for (time-limited) balanced truncation for Bayesian inference from [2] (to be found at https://github.com/joskoUP/TLBTforDA) and [3] (to be found at https://github.com/elizqian/balancing-bayesian-inference) allow to simplify the problem and use a Quasi-Newton minimisation with a constant Hessian.

Examples

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

To generate the plots comparing the full incremental 4D-Var method to time-limited balanced truncation and $\alpha$-bounded balanced truncation, run the Comparison_lorenzN_4dvar.m script.
There will be a pop-up menu. Choose the observation points and prior covariance as follows:

• Figure 1: Observations in all N variables, prior covariance matrix Gaussian exponential.
• Figure 2: Observations every s points, in command window: Take observations every ... points. - enter 9, prior covariance matrix scaled identity.

References

2. König, J., Freitag, M. A. "Time-limited Balanced Truncation for Data Assimilation Problems." Journal of Scientific Computing 97.47 (2023).
3. Qian, E., Tabeart, J. M., Beattie, C., Gugercin, S., Jiang, J., Kramer, P. R., & Narayan, A. "Model reduction for linear dynamical systems via balancing for Bayesian inference." Journal of Scientific Computing 91.29 (2022).

Contact

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