  In many applications throughout science and engineering, model reduction
plays an important role replacing expensive large-scale linear dynamical
systems by inexpensive reduced order models that capture key features of the
original, full order model. One approach to model reduction finds reduced order
models that are locally optimal approximations in the $\mathcal{H}_2$ norm, an
approach taken by the Iterative Rational Krylov Algorithm (IRKA) among others.
Here we introduce a new approach for $\mathcal{H}_2$-optimal model reduction
using the projected nonlinear least squares framework previously introduced in
[J. M. Hokanson, SIAM J. Sci. Comput. 39 (2017), pp. A3107--A3128]. At each
iteration, we project the $\mathcal{H}_2$ optimization problem onto a
finite-dimensional subspace yielding a weighted least squares rational
approximation problem. Subsequent iterations append this subspace such that the
least squares rational approximant asymptotically satisfies the first order
necessary conditions of the original, $\mathcal{H}_2$ optimization problem.
This enables us to build reduced order models with similar error in the
$\mathcal{H}_2$ norm but using far fewer evaluations of the expensive, full
order model compared to competing methods. Moreover, our new algorithm only
requires access to the transfer function of the full order model, unlike IRKA
which requires a state-space representation or TF-IRKA which requires both the
transfer function and its derivative. Applying the projected nonlinear least
squares framework to the $\mathcal{H}_2$-optimal model reduction problem open
new avenues for related model reduction problems.
