MOR Wiki - User contributions [en]

Penzl's FOM

2018-11-21T15:31:12Z

Yue: /* Parametric Variant */

{{preliminary}} 

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:MIMO]]
[[Category:Sparse]]

'''This is a stub. Please expand.'''

==Description: Full Order Model==

This benchmark is an artificial example system of order <math>1006</math> from <ref name="penzl06"/> also listed in <ref name="chahlaoui02"/>.
The benchmark system consists of the following system components:

<math>
\begin{array}{rcl}
A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \;
A_1 = \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \;
A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \;
A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \;
A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\
B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\
C &=& B^T.
\end{array}
</math>

===MIMO Variant===

In <ref name="heyouni08"/> a MIMO variant of this benchmark is utilized by adding random vectors to <math>B</math> and <math>C</math>.

===Parametric Variant===

In <ref name="Ionita14"/>, a parametric variant of this benchmark is formulated by redefining
<math>
A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.
</math>

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/fom.zip fom.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{1006 \times 1006}</math>,
<math>B \in \mathbb{R}^{1006 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 1006}</math>,

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_fom,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.}
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the background on the benchmark:

@ARTICLE{morPen06,
author = <nowiki>{T. Penzl}</nowiki>,
title = {Algorithms for Model Reduction of Large Dynamical Systems},
journal = {Linear Algebra and its Application},
volume = {415},
number = {2--3},
pages = {322--343},
year = {2006},
doi = {10.1016/j.laa.2006.01.007}
}

==References==

<references>

<ref name="penzl06"> T. Penzl. [https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for Model Reduction of Large Dynamical Systems]. Linear Algebra and its Application 415(2--3): 322--343, 2006.</ref>

<ref name="heyouni08"> M. Heyouni, K. Jbilou, A. Messaoudi, K. Tabaa. [https://doi.org/10.1590/S0101-82052008000200006 Model Reduction in Large-Scale MIMO Dynamical Systems via the Block Lanczos Method]. Computational & Applied Mathematics 27(11): 211--236, 2008.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

<ref name="Ionita14"> A. C. Ionita,A. C. Antoulas, [https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework], SIAM J. Sci. Comput. 36(3): A984–A1007, 2014.</ref>

</references>

Penzl's FOM

2018-11-16T12:14:47Z

Yue: /* Description: Full Order Model */

{{preliminary}} 

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:MIMO]]
[[Category:Sparse]]

'''This is a stub. Please expand.'''

==Description: Full Order Model==

This benchmark is an artificial example system of order <math>1006</math> from <ref name="penzl06"/> also listed in <ref name="chahlaoui02"/>.
The benchmark system consists of the following system components:

<math>
\begin{array}{rcl}
A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \;
A_1 = \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \;
A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \;
A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \;
A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\
B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\
C &=& B^T.
\end{array}
</math>

===MIMO Variant===

In <ref name="heyouni08"/> a MIMO variant of this benchmark is utilized by adding random vectors to <math>B</math> and <math>C</math>.

===Parametric Variant===

In <ref name="Ionita14"/>, a parametric variant of this benchmark is used by redefining
<math>
A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.
</math>

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/fom.zip fom.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{1006 \times 1006}</math>,
<math>B \in \mathbb{R}^{1006 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 1006}</math>,

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_fom,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.}
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the background on the benchmark:

@ARTICLE{morPen06,
author = <nowiki>{T. Penzl}</nowiki>,
title = {Algorithms for Model Reduction of Large Dynamical Systems},
journal = {Linear Algebra and its Application},
volume = {415},
number = {2--3},
pages = {322--343},
year = {2006},
doi = {10.1016/j.laa.2006.01.007}
}

==References==

<references>

<ref name="penzl06"> T. Penzl. [https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for Model Reduction of Large Dynamical Systems]. Linear Algebra and its Application 415(2--3): 322--343, 2006.</ref>

<ref name="heyouni08"> M. Heyouni, K. Jbilou, A. Messaoudi, K. Tabaa. [https://doi.org/10.1590/S0101-82052008000200006 Model Reduction in Large-Scale MIMO Dynamical Systems via the Block Lanczos Method]. Computational & Applied Mathematics 27(11): 211--236, 2008.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

<ref name="Ionita14"> A. C. Ionita,A. C. Antoulas, [https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework], SIAM J. Sci. Comput. 36(3): A984–A1007, 2014.</ref>

</references>

Penzl's FOM

2018-11-16T12:13:42Z

Yue: /* References */

{{preliminary}} 

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:MIMO]]
[[Category:Sparse]]

'''This is a stub. Please expand.'''

==Description: Full Order Model==

This benchmark is an artificial example system of order <math>1006</math> from <ref name="penzl06"/> also listed in <ref name="chahlaoui02"/>.
The benchmark system consists of the following system components:

<math>
\begin{array}{rcl}
A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \;
A_1 = \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \;
A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \;
A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \;
A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\
B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\
C &=& B^T.
\end{array}
</math>

===MIMO Variant===

In <ref name="heyouni08"/> a MIMO variant of this benchmark is utilized by adding random vectors to <math>B</math> and <math>C</math>.

===Parametric Variant===

In, a parametric variant of this benchmark is used by redefining
<math>
A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.
</math>

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/fom.zip fom.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{1006 \times 1006}</math>,
<math>B \in \mathbb{R}^{1006 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 1006}</math>,

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_fom,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.}
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the background on the benchmark:

@ARTICLE{morPen06,
author = <nowiki>{T. Penzl}</nowiki>,
title = {Algorithms for Model Reduction of Large Dynamical Systems},
journal = {Linear Algebra and its Application},
volume = {415},
number = {2--3},
pages = {322--343},
year = {2006},
doi = {10.1016/j.laa.2006.01.007}
}

==References==

<references>

<ref name="penzl06"> T. Penzl. [https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for Model Reduction of Large Dynamical Systems]. Linear Algebra and its Application 415(2--3): 322--343, 2006.</ref>

<ref name="heyouni08"> M. Heyouni, K. Jbilou, A. Messaoudi, K. Tabaa. [https://doi.org/10.1590/S0101-82052008000200006 Model Reduction in Large-Scale MIMO Dynamical Systems via the Block Lanczos Method]. Computational & Applied Mathematics 27(11): 211--236, 2008.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

<ref name="Ionita14"> A. C. Ionita,A. C. Antoulas, [https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework], SIAM J. Sci. Comput. 36(3): A984–A1007, 2014.</ref>

</references>

Penzl's FOM

2018-11-16T11:57:36Z

Yue: /* Parametric Variant */

{{preliminary}} 

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:MIMO]]
[[Category:Sparse]]

'''This is a stub. Please expand.'''

==Description: Full Order Model==

This benchmark is an artificial example system of order <math>1006</math> from <ref name="penzl06"/> also listed in <ref name="chahlaoui02"/>.
The benchmark system consists of the following system components:

<math>
\begin{array}{rcl}
A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \;
A_1 = \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \;
A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \;
A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \;
A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\
B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\
C &=& B^T.
\end{array}
</math>

===MIMO Variant===

In <ref name="heyouni08"/> a MIMO variant of this benchmark is utilized by adding random vectors to <math>B</math> and <math>C</math>.

===Parametric Variant===

In, a parametric variant of this benchmark is used by redefining
<math>
A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.
</math>

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/fom.zip fom.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{1006 \times 1006}</math>,
<math>B \in \mathbb{R}^{1006 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 1006}</math>,

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_fom,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.}
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the background on the benchmark:

@ARTICLE{morPen06,
author = <nowiki>{T. Penzl}</nowiki>,
title = {Algorithms for Model Reduction of Large Dynamical Systems},
journal = {Linear Algebra and its Application},
volume = {415},
number = {2--3},
pages = {322--343},
year = {2006},
doi = {10.1016/j.laa.2006.01.007}
}

==References==

<references>

<ref name="penzl06"> T. Penzl. [https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for Model Reduction of Large Dynamical Systems]. Linear Algebra and its Application 415(2--3): 322--343, 2006.</ref>

<ref name="heyouni08"> M. Heyouni, K. Jbilou, A. Messaoudi, K. Tabaa. [https://doi.org/10.1590/S0101-82052008000200006 Model Reduction in Large-Scale MIMO Dynamical Systems via the Block Lanczos Method]. Computational & Applied Mathematics 27(11): 211--236, 2008.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

</references>

Penzl's FOM

2018-11-16T11:57:18Z

Yue: /* Description: Full Order Model */

{{preliminary}} 

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:MIMO]]
[[Category:Sparse]]

'''This is a stub. Please expand.'''

==Description: Full Order Model==

This benchmark is an artificial example system of order <math>1006</math> from <ref name="penzl06"/> also listed in <ref name="chahlaoui02"/>.
The benchmark system consists of the following system components:

<math>
\begin{array}{rcl}
A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \;
A_1 = \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \;
A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \;
A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \;
A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\
B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\
C &=& B^T.
\end{array}
</math>

===MIMO Variant===

In <ref name="heyouni08"/> a MIMO variant of this benchmark is utilized by adding random vectors to <math>B</math> and <math>C</math>.

===Parametric Variant===

In, a parametric variant of this benchmark is used by defining
<math>
A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.
</math>

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/fom.zip fom.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{1006 \times 1006}</math>,
<math>B \in \mathbb{R}^{1006 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 1006}</math>,

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_fom,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.}
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the background on the benchmark:

@ARTICLE{morPen06,
author = <nowiki>{T. Penzl}</nowiki>,
title = {Algorithms for Model Reduction of Large Dynamical Systems},
journal = {Linear Algebra and its Application},
volume = {415},
number = {2--3},
pages = {322--343},
year = {2006},
doi = {10.1016/j.laa.2006.01.007}
}

==References==

<references>

<ref name="penzl06"> T. Penzl. [https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for Model Reduction of Large Dynamical Systems]. Linear Algebra and its Application 415(2--3): 322--343, 2006.</ref>

<ref name="heyouni08"> M. Heyouni, K. Jbilou, A. Messaoudi, K. Tabaa. [https://doi.org/10.1590/S0101-82052008000200006 Model Reduction in Large-Scale MIMO Dynamical Systems via the Block Lanczos Method]. Computational & Applied Mathematics 27(11): 211--236, 2008.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

</references>

Windscreen

2018-04-01T11:28:43Z

Yue: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:Parametric 1 parameter]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K_d x - \omega^2 M x & = f \\
y & = f^* x
\end{align}
</math>

where <math>f</math> represents a unit point load in one unknown of the state vector.
<math>M</math> is a symmetric positive-definite matrix and <math>K_d = (1+i\gamma) K</math> where <math>K</math> is symmetric positive semi-definite.

The test problem is a structural model of a car windscreen. <ref name="meerbergen2007"/>
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in <xr id="fig1"/>.
The material is glass with the following properties:
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>.
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

<xr id="fig1"/> shows the mesh of the car windscreen and <xr id="fig2"/> the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Windscreen%20%2838886%29/files/fileinnercontentproxy.2010-02-26.3407583176 windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K_d</math>, <math>-M</math> and <math>f</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = B^{\mathrm{T}} x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = {Oberwolfach Benchmark Collection},
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Windscreen</nowiki>},
year = 2004
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058 Fast frequency response computation for Rayleigh damping], International Journal for Numerical Methods in Engineering, '''73'''(1): 96--106, 2007.</ref>

</references>

Windscreen

2018-04-01T11:25:46Z

Yue: /* References */

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:Parametric 1 parameter]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K_d x - \omega^2 M x & = f \\
y & = f^* x
\end{align}
</math>

where <math>f</math> represents a unit point load in one unknown of the state vector.
<math>M</math> is a symmetric positive-definite matrix and <math>K_d = (1+i\gamma) K</math> where <math>K</math> is symmetric positive semi-definite.

The test problem is a structural model of a car windscreen.
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in <xr id="fig1"/>.
The material is glass with the following properties:
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>. <ref name="meerbergen2007"/>
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

<xr id="fig1"/> shows the mesh of the car windscreen and <xr id="fig2"/> the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Windscreen%20%2838886%29/files/fileinnercontentproxy.2010-02-26.3407583176 windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K_d</math>, <math>-M</math> and <math>f</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = B^{\mathrm{T}} x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = {Oberwolfach Benchmark Collection},
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Windscreen</nowiki>},
year = 2004
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058 Fast frequency response computation for Rayleigh damping], International Journal for Numerical Methods in Engineering, '''73'''(1): 96--106, 2007.</ref>

</references>

Windscreen

2018-04-01T11:24:56Z

Yue: /* References */

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:Parametric 1 parameter]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K_d x - \omega^2 M x & = f \\
y & = f^* x
\end{align}
</math>

where <math>f</math> represents a unit point load in one unknown of the state vector.
<math>M</math> is a symmetric positive-definite matrix and <math>K_d = (1+i\gamma) K</math> where <math>K</math> is symmetric positive semi-definite.

The test problem is a structural model of a car windscreen.
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in <xr id="fig1"/>.
The material is glass with the following properties:
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>. <ref name="meerbergen2007"/>
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

<xr id="fig1"/> shows the mesh of the car windscreen and <xr id="fig2"/> the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Windscreen%20%2838886%29/files/fileinnercontentproxy.2010-02-26.3407583176 windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K_d</math>, <math>-M</math> and <math>f</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = B^{\mathrm{T}} x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = {Oberwolfach Benchmark Collection},
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Windscreen</nowiki>},
year = 2004
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058 ], International Journal for Numerical Methods in Engineering, '''73'''(1): 96--106, 2007.</ref>

</references>

Windscreen

2018-04-01T11:23:51Z

Yue: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:Parametric 1 parameter]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K_d x - \omega^2 M x & = f \\
y & = f^* x
\end{align}
</math>

where <math>f</math> represents a unit point load in one unknown of the state vector.
<math>M</math> is a symmetric positive-definite matrix and <math>K_d = (1+i\gamma) K</math> where <math>K</math> is symmetric positive semi-definite.

The test problem is a structural model of a car windscreen.
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in <xr id="fig1"/>.
The material is glass with the following properties:
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>. <ref name="meerbergen2007"/>
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

<xr id="fig1"/> shows the mesh of the car windscreen and <xr id="fig2"/> the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Windscreen%20%2838886%29/files/fileinnercontentproxy.2010-02-26.3407583176 windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K_d</math>, <math>-M</math> and <math>f</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = B^{\mathrm{T}} x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = {Oberwolfach Benchmark Collection},
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Windscreen</nowiki>},
year = 2004
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058], International Journal for Numerical Methods in Engineering, '''73'''(1): 96--106, 2007.</ref>

</references>

Windscreen

2018-04-01T11:21:13Z

Yue: /* References */ Add a reference

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:Parametric 1 parameter]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K_d x - \omega^2 M x & = f \\
y & = f^* x
\end{align}
</math>

where <math>f</math> represents a unit point load in one unknown of the state vector.
<math>M</math> is a symmetric positive-definite matrix and <math>K_d = (1+i\gamma) K</math> where <math>K</math> is symmetric positive semi-definite.

The test problem is a structural model of a car windscreen.
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in <xr id="fig1"/>.
The material is glass with the following properties:
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>.
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

<xr id="fig1"/> shows the mesh of the car windscreen and <xr id="fig2"/> the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Windscreen%20%2838886%29/files/fileinnercontentproxy.2010-02-26.3407583176 windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K_d</math>, <math>-M</math> and <math>f</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = B^{\mathrm{T}} x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = {Oberwolfach Benchmark Collection},
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Windscreen</nowiki>},
year = 2004
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058], International Journal for Numerical Methods in Engineering, '''73'''(1): 96--106, 2007.</ref>

</references>