$$\mathcal {H}_2$$ optimal rational approximation on general domains

Abstract

Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space \(\varvec{\mathcal {H}}_{\varvec{2}}\), a new \(\varvec{\mathcal {H}}_{\varvec{2}}\)-like optimal model reduction problem is introduced and first-order optimality conditions are derived. As in the classical \(\varvec{\mathcal {H}}_{\varvec{2}}\) case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.