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

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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.

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Data Availability

The code used for this paper is available in https://github.com/aaborghi/H2-arbitrary-domains.

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Acknowledgements

We would like to thank Olivier Sète, Jan Zur, and Mathias Oster for their insightful and helpful discussions.

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 384950143 as part of GRK2433 DAEDALUS.

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Author notes

  1. Alessandro Borghi and Tobias Breiten contributed equally to this work.

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  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Alessandro Borghi & Tobias Breiten

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  1. Alessandro Borghi
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Correspondence to Alessandro Borghi.

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Borghi, A., Breiten, T. \(\mathcal {H}_2\) optimal rational approximation on general domains. Adv Comput Math 50, 28 (2024). https://doi.org/10.1007/s10444-024-10125-8

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