Iterative Rational Krylov Algorithm - Revision history

Lund: remove link to list of abbrev

2023-05-11T12:21:27Z

remove link to list of abbrev

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	==Description==		==Description==

	'''IRKA''' (iterative rational Krylov algorithm) is an interpolation-based model reduction method for ~~[[List_of_abbreviations#~~SISO~~\|SISO]]~~ linear time invariant systems		'''IRKA''' (iterative rational Krylov algorithm) is an interpolation-based model reduction method for SISO linear time invariant systems.

	:<equation id="gensys" shownumber>		:<equation id="gensys" shownumber>

Saak: /* References */

2019-10-17T16:48:18Z

References

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	<ref name="FlaBG12"> G. Flagg, C. Beattie, S. Gugercin "<span class="plainlinks">[http://www.sciencedirect.com/science/article/pii/S0167691112000576 Convergence of the Iterative Rational Krylov Algorithm]</span>", Systems & Control Letters, vol.61, no.6, pp.688-691, 2012.</ref>		<ref name="FlaBG12"> G. Flagg, C. Beattie, S. Gugercin "<span class="plainlinks">[http://www.sciencedirect.com/science/article/pii/S0167691112000576 Convergence of the Iterative Rational Krylov Algorithm]</span>", Systems & Control Letters, vol.61, no.6, pp.688-691, 2012.</ref>

	</ references>		</references>

Himpe: Himpe moved page IRKA to Iterative Rational Krylov Algorithm

2018-02-15T09:37:14Z

Himpe moved page IRKA to Iterative Rational Krylov Algorithm

← Older revision	Revision as of 09:37, 15 February 2018
(No difference)

Fehr: /* Description */

2016-03-17T15:38:34Z

Description

← Older revision		Revision as of 15:38, 17 March 2016
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	Although a rigorous convergence proof so far has only be given for symmetric state space systems <ref name="FlaBG12"></ref>, numerous experiments have shown that the algorithm often converges rapidly. Moreover, the algorithm has been extended to, e.g., multiple-input-multiple output, discrete time and differential algebraic systems.		Although a rigorous convergence proof so far has only be given for symmetric state space systems <ref name="FlaBG12"></ref>, numerous experiments have shown that the algorithm often converges rapidly. Moreover, the algorithm has been extended to, e.g., multiple-input-multiple output, discrete time and differential algebraic systems.

			== A minimal example ==

			For the lecture Model [http://www.itm.uni-stuttgart.de/courses/model_reduction/model_reduction_en.php MOR] of Mechanical System from the Institute of Engineering and Computational Mechanics University of Stuttgart a very simple example of the IRKA Algorithm were written.

			The implementation with two basic exampless can be found here

			[[Media:IRKA_Example.zip]]

			This code is published under the BSD3-Clause License
			All rights reserved. (c) 2015, Joerg.Fehr, University of Stuttgart

	==References==		==References==

Himpe at 08:27, 6 June 2013

2013-06-06T08:27:20Z

← Older revision		Revision as of 08:27, 6 June 2013
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	==Description==		==Description==

	~~The~~ iterative rational Krylov algorithm ~~(IRKA~~) is an interpolation-based model reduction method for ~~single-input-single-output~~ linear time invariant systems		'''IRKA''' (iterative rational Krylov algorithm) is an interpolation-based model reduction method for [[List_of_abbreviations#SISO\|SISO]] linear time invariant systems

	:<equation id="gensys" shownumber>		:<equation id="gensys" shownumber>
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	G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,		G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,
	</math>		</math>
	where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation at the reflected reduced system poles is ensured. In pseudocode, the classical algorithm ~~(IRKA)~~ from <ref name="GugAB08"></ref> looks like		where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation at the reflected reduced system poles is ensured. In pseudocode, the classical '''IRKA''' algorithm from <ref name="GugAB08"></ref> looks like

	1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.		1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.

Breiten at 07:14, 30 May 2013

2013-05-30T07:14:55Z

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	<references>		<references>

	<ref name="MeiL67"> L. Meier, D.G. Luenberger, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1098680&tag=1 Approximation of linear constant systems]</span>", IEEE Transactions on Automatic Control, vol.12, no.5, pp.585-588 1967</ref>		<ref name="MeiL67"> L. Meier, D.G. Luenberger, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1098680&tag=1 Approximation of linear constant systems]</span>", IEEE Transactions on Automatic Control, vol.12, no.5, pp.585-588, 1967.</ref>

	<ref name="GugAB08"> S. Gugercin, A.C. Antoulas, C. Beattie "<span class="plainlinks">[http://epubs.siam.org/doi/abs/10.1137/060666123 H2 Model Reduction for Large-Scale Linear Dynamical Systems]</span>", SIAM. J. Matrix Anal. & Appl., vol.30, no.2, pp.609-638 2008</ref>		<ref name="GugAB08"> S. Gugercin, A.C. Antoulas, C. Beattie "<span class="plainlinks">[http://epubs.siam.org/doi/abs/10.1137/060666123 H2 Model Reduction for Large-Scale Linear Dynamical Systems]</span>", SIAM. J. Matrix Anal. & Appl., vol.30, no.2, pp.609-638, 2008.</ref>


	<ref name="FlaBG12"> G. Flagg, C. Beattie, S. Gugercin "<span class="plainlinks">[http://www.sciencedirect.com/science/article/pii/S0167691112000576 Convergence of the Iterative Rational Krylov Algorithm]</span>", Systems & Control Letters, vol.61, no.6, pp.688-691 2012</ref>		<ref name="FlaBG12"> G. Flagg, C. Beattie, S. Gugercin "<span class="plainlinks">[http://www.sciencedirect.com/science/article/pii/S0167691112000576 Convergence of the Iterative Rational Krylov Algorithm]</span>", Systems & Control Letters, vol.61, no.6, pp.688-691, 2012.</ref>

	</ references>		</ references>

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2013-05-30T07:12:32Z

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	:<math>		:<math>
	\|\|G-\hat{G} \|\|_{H_2} = \min_{\text{dim}(\tilde{G})=r} \|\|G-\~~hat~~{G}\|\|_{H_2}.		\|\|G-\hat{G} \|\|_{H_2} = \min_{\text{dim}(\tilde{G})=r} \|\|G-\tilde{G}\|\|_{H_2}.
	</math>		</math>

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2013-05-30T07:12:07Z

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	<math>		<math>
	\dot{x}(t)=A x(t)+b u(t), \quad		\dot{x}(t)=A x(t)+b u(t), \quad
	y(t)=c^Tx(t),\quad E,A\in\mathbb{R}^{n\times n},~b\in\mathbb{R}^{n},~c\in\mathbb{R}^{n}.\qquad (1)		y(t)=c^Tx(t),\quad A\in\mathbb{R}^{n\times n},~b\in\mathbb{R}^{n},~c\in\mathbb{R}^{n}.\qquad (1)
	</math>		</math>
	</equation>		</equation>

Breiten at 07:11, 30 May 2013

2013-05-30T07:11:38Z

← Older revision		Revision as of 07:11, 30 May 2013
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	G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,		G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,
	</math>		</math>
	where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation is ensured. In pseudocode, the classical algorithm (IRKA) from <ref name="GugAB08"></ref> looks like		where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation at the reflected reduced system poles is ensured. In pseudocode, the classical algorithm (IRKA) from <ref name="GugAB08"></ref> looks like

	1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.		1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.
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	4. <math>\hat{A} = W_r^TAV_r, \hat{b}= W_r^Tb, \hat{c}^T = c^TV_r.</math>		4. <math>\hat{A} = W_r^TAV_r, \hat{b}= W_r^Tb, \hat{c}^T = c^TV_r.</math>

	Although a rigorous convergence proof so far has only be given for symmetric state space systems <ref name="FlaBG12"></ref>, numerous experiments have shown that the algorithm often converges rapidly.		Although a rigorous convergence proof so far has only be given for symmetric state space systems <ref name="FlaBG12"></ref>, numerous experiments have shown that the algorithm often converges rapidly. Moreover, the algorithm has been extended to, e.g., multiple-input-multiple output, discrete time and differential algebraic systems.

Breiten at 07:10, 30 May 2013

2013-05-30T07:10:21Z

← Older revision		Revision as of 07:10, 30 May 2013
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	G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,		G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,
	</math>		</math>
	where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct ~~the~~ projection subspaces. In pseudocode, the classical algorithm (IRKA) from <ref name="GugAB08"></ref> looks like		where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation is ensured. In pseudocode, the classical algorithm (IRKA) from <ref name="GugAB08"></ref> looks like

	1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.		1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.