Rational Krylov methods for optimal ℒ<inf>2</inf> model reduction

Abstract

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 ℒ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (ℝ) to itself and then develop criteria for optimal reduced order approximations to the original (unstable) system with respect to an ℒ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -induced Hilbert-Schmidt norm. Our optimality criteria extend the Meier-Luenberger interpolation conditions for optimal ℋ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> approximation of stable dynamical systems. Based on this interpolation framework, we describe an iteratively corrected rational Krylov algorithm for ℒ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> model reduction. A numerical example involving a hard-to-approximate full-order model illustrates the effectiveness of the proposed approach.