Kolmogorov n-widths for linear dynamical systems

Abstract

Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear time-invariant (LTI) dynamical systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov n-width of an LTI system equals its (n + 1)st Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width.

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Acknowledgments

We thank Profs. Christopher Beattie, Volker Mehrmann, and Karen Willcox for their valuable comments.

Funding

The work of the first author was funded by the DFG Collaborative Research Center 910 Control of self-organizing nonlinear systems: theoretical methods and concepts of application. The work of the second author was financially supported in parts by NSF through Grant DMS-1720257 and by the Alexander von Humboldt Foundation.

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Correspondence to Benjamin Unger.

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Communicated by: Anthony Nouy

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Unger, B., Gugercin, S. Kolmogorov n-widths for linear dynamical systems. Adv Comput Math 45, 2273–2286 (2019). https://doi.org/10.1007/s10444-019-09701-0

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Keywords

  • Model reduction
  • Hankel singular values
  • Kolmogorov n-width
  • Hankel operator
  • Reduced basis method
  • Active subspaces

Mathematics Subject Classification (2010)

  • 37M99
  • 47B35
  • 65P99
  • 34A45
  • 35A35
  • 93C05