Kolmogorov n-widths for linear dynamical systems
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...
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| Published in | Advances in computational mathematics Vol. 45; no. 5-6; pp. 2273 - 2286 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.12.2019
Springer Nature B.V |
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| 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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Gugercin, Serkan Unger, Benjamin |
| Author_xml | – sequence: 1 givenname: Benjamin orcidid: 0000-0003-4272-1079 surname: Unger fullname: Unger, Benjamin email: unger@math.tu-berlin.de organization: Technische Universität Berlin – sequence: 2 givenname: Serkan surname: Gugercin fullname: Gugercin, Serkan organization: Department of Mathematics, Virginia Polytechnic Institute and State University |
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| Cites_doi | 10.1109/TAC.2016.2521361 10.1137/100795772 10.1137/130932715 10.1007/978-3-319-22470-1 10.1051/m2an:2005006 10.1016/S1631-073X(02)02466-4 10.2307/1968691 10.1051/m2an/2012045 10.1007/BFb0007371 10.1070/SM1971v015n01ABEH001531 10.1137/1.9781611974829 10.1186/2190-5983-1-3 10.1007/978-1-4612-4224-6 10.1023/A:1015145924517 10.1007/978-3-319-15431-2 10.1137/070695332 10.1137/070694855 10.1080/00207178408933239 10.1115/1.1448332 10.1051/m2an:2008001 10.1007/s11831-014-9111-2 10.1007/978-3-319-02090-7_9 10.1137/1.9780898718713 10.1007/978-3-319-78325-3 10.1109/ACC.2010.5530920 10.1109/CDC.2008.4739458 10.1137/1.9781611973860 10.1007/978-3-642-69894-1 |
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-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear... 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... |
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| SubjectTerms | 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 |
| Title | Kolmogorov n-widths for linear dynamical systems |
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