A Data-Driven Krylov Model Order Reduction for Large-Scale Dynamical Systems

Dynamical systems that involve non-linearity of the dynamics is a major challenge encountered in learning these systems. Similarly, the lack of adequate models for phenomena that reflect the governing physics can be an obstacle to an appropriate analysis. Nonetheless, some numerically or experimentally measured data can be found. Based on this data, and using a data-driven method such as the Loewner framework, it is possible to manage this data to derive a high fidelity reduced dynamical system that mimics the behaviour of the original data. In this paper, we tackle the issue of large amount of data presented by samples of transfer functions in a frequency-domain. The main step in this framework consists in computing singular value decomposition (SVD) of the Loewner matrix which provides accurate reduced systems. However, the large amount of data prevents this decomposition from being computed properly. We exploit the fact that the Loewner and shifted Loewner matrices, the key tools of Loewner framework, satisfy certain large scale Sylvester matrix equations. Using an extended block Krylov subspace method, a good approximation in a factored form of the Loewner and shifted Loewner matrices can be obtained and a minimal computation cost of the SVD is ensured. This method facilitates the process of a large amount of data and guarantees a good quality of the inferred model at the end of the process. Accuracy and efficiency of our method are assessed in the final section.