$\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems

The optimal $\mathcal{H}_2$ model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal $\mathcal{H}_2$ approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov- and interpolation-based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation-based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for $\mathcal{H}_2$ model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based first-order necessary conditions for $\cHtwo$ optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.