Fast Optimal $\mathcal{H}_2$ Model Reduction Algorithms Based on Grassmann Manifold Optimization

Abstract

The optimal $\mathcal{H}_2$ model reduction is an important tool in studying dynamical systems of a large order and their numerical simulation. We formulate the reduction problem as a minimization problem over the Grassmann manifold. This allows us to develop a fast gradient flow algorithm suitable for large-scale optimal $\mathcal{H}_2$ model reduction problems. The proposed algorithm converges globally and the resulting reduced system preserves stability of the original system. Furthermore, based on the fast gradient flow algorithm, we propose a sequentially quadratic approximation algorithm which converges faster and guarantees the global convergence. Numerical examples are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed algorithms.