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 opens new avenues for related model reduction problems.


  1. model reduction
  2. $\mathcal{H}_2$ approximation
  3. projected nonlinear least squares
  4. rational approximation
  5. transfer function

MSC codes

  1. 41A20
  2. 46E22
  3. 90C53
  4. 93A15
  5. 93C05
  6. 93C15

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A4017 - A4045
ISSN (online): 1095-7197


Submitted: 4 March 2019
Accepted: 21 August 2020
Published online: 17 December 2020


  1. model reduction
  2. $\mathcal{H}_2$ approximation
  3. projected nonlinear least squares
  4. rational approximation
  5. transfer function

MSC codes

  1. 41A20
  2. 46E22
  3. 90C53
  4. 93A15
  5. 93C05
  6. 93C15



Funding Information

Defense Advanced Research Projects Agency https://doi.org/10.13039/100000185

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