Rational Krylov methods for optimal ℒ2 model reduction

Unstable dynamical systems can be viewed from a variety of perspectives. We discuss the potential of an input-output map associated with an unstable system to represent a bounded map from ℒ₂(ℝ) to itself and then develop criteria for optimal reduced order approximations to the original (unstable) system with respect to an ℒ₂-induced Hilbert-Schmidt norm. Our optimality criteria extend the Meier-Luenberger interpolation conditions for optimal ℋ₂ approximation of stable dynamical systems. Based on this interpolation framework, we describe an iteratively corrected rational Krylov algorithm for ℒ₂ model reduction. A numerical example involving a hard-to-approximate full-order model illustrates the effectiveness of the proposed approach.