Kolmogorov n-widths for linear dynamical systems

• Published in: Advances in computational mathematics Vol. 45; no. 5-6; pp. 2273 - 2286
• Main Authors: Unger, Benjamin, Gugercin, Serkan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Dynamical systems; Lower bounds; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Model reduction; Model reduction of parametrized Systems; Subspaces; Visualization; 37M99; 65P99; 34A45; 35A35; 93C05; 47B35; Kolmogorov; width; Active subspaces; Model reduction; Hankel operator; Reduced basis method; Hankel singular values
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09701-0

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.