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Volume 10, Issue 4
Fast Optimal $\mathcal{H}_2$ Model Reduction Algorithms Based on Grassmann Manifold Optimization

Y. Xu & T. Zeng

Int. J. Numer. Anal. Mod., 10 (2013), pp. 972-991.

Published online: 2013-10

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  • Abstract

The optimal $\mathcal{H}_2$ model reduction is an important tool in studying dynamical systems of a large order and their numerical simulation. We formulate the reduction problem as a minimization problem over the Grassmann manifold. This allows us to develop a fast gradient flow algorithm suitable for large-scale optimal $\mathcal{H}_2$ model reduction problems. The proposed algorithm converges globally and the resulting reduced system preserves stability of the original system. Furthermore, based on the fast gradient flow algorithm, we propose a sequentially quadratic approximation algorithm which converges faster and guarantees the global convergence. Numerical examples are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed algorithms.

  • Keywords

$\mathcal{H}_2$ approximation, gradient flow, Grassmann manifold, model reduction, MIMO system, stability, large-scale sparse system.

  • AMS Subject Headings

65M10, 65M15, 65N10, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-972, author = {Y. Xu and T. Zeng}, title = {Fast Optimal $\mathcal{H}_2$ Model Reduction Algorithms Based on Grassmann Manifold Optimization}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {4}, pages = {972--991}, abstract = {

The optimal $\mathcal{H}_2$ model reduction is an important tool in studying dynamical systems of a large order and their numerical simulation. We formulate the reduction problem as a minimization problem over the Grassmann manifold. This allows us to develop a fast gradient flow algorithm suitable for large-scale optimal $\mathcal{H}_2$ model reduction problems. The proposed algorithm converges globally and the resulting reduced system preserves stability of the original system. Furthermore, based on the fast gradient flow algorithm, we propose a sequentially quadratic approximation algorithm which converges faster and guarantees the global convergence. Numerical examples are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed algorithms.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/606.html} }
TY - JOUR T1 - Fast Optimal $\mathcal{H}_2$ Model Reduction Algorithms Based on Grassmann Manifold Optimization AU - Y. Xu & T. Zeng JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 972 EP - 991 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/606.html KW - $\mathcal{H}_2$ approximation, gradient flow, Grassmann manifold, model reduction, MIMO system, stability, large-scale sparse system. AB -

The optimal $\mathcal{H}_2$ model reduction is an important tool in studying dynamical systems of a large order and their numerical simulation. We formulate the reduction problem as a minimization problem over the Grassmann manifold. This allows us to develop a fast gradient flow algorithm suitable for large-scale optimal $\mathcal{H}_2$ model reduction problems. The proposed algorithm converges globally and the resulting reduced system preserves stability of the original system. Furthermore, based on the fast gradient flow algorithm, we propose a sequentially quadratic approximation algorithm which converges faster and guarantees the global convergence. Numerical examples are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed algorithms.

Y. Xu and T. Zeng. (2013). Fast Optimal $\mathcal{H}_2$ Model Reduction Algorithms Based on Grassmann Manifold Optimization. International Journal of Numerical Analysis and Modeling. 10 (4). 972-991. doi:
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