MOR Wiki - User contributions [en]

Flexible Space Structures

2026-04-23T11:38:48Z

Schulze: Update category SISO to MIMO

[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:Procedural]]
[[Category:MIMO]]

==Description==

The '''flexible space structure''' benchmark <ref name="gawronski90"/>,<ref name="gawronski91"/>,<ref name="gawronski96"/> is a procedural modal model which represents structural dynamics with a selectable number actuators and sensors. This model is used for truss structures in space environments i.e. the COFS-1 (Control of Flexible Structures) mast flight experiment <ref name="horner86"/>,<ref name="horta86"/>.

===Model===

In modal form the '''flexible space structure''' model for <math>K</math> modes, <math>M</math> actuators and <math>Q</math> sensors is of second order and given by:

:<math>\ddot{\nu}(t) = (2 \xi \circ \omega) \circ \dot{\nu}(t) + (\omega \circ \omega) \circ \nu = Bu(t)</math>

:<math>y(t) = C_r\dot{\nu}(t) + C_d\nu(t)</math>

with the parameters <math>\xi \in \mathbb{R}_{>0}^K</math> (damping ratio), <math>\omega \in \mathbb{R}_{>0}^K</math> (natural frequency) and using the Hadamard product <math>\circ</math>.
The first order representation follows for <math>x(t) = (\dot{\nu}(t), \omega_1\nu_1, \dots, \omega_K\nu_K)</math> by:

:<math>\dot{x}(t) = Ax(t) + Bu(t) </math>

:<math>y(t) = Cx(t)</math>

with the matrices:

:<math>A := \begin{pmatrix} A_1 & & \\ & \ddots & \\ & & A_K \end{pmatrix}, \; B := \begin{pmatrix} B_1 \\ \vdots \\ B_K \end{pmatrix}, \; C := \begin{pmatrix} C_1 & \dots & C_K \end{pmatrix}, </math>

and their components:

:<math>A_k := \begin{pmatrix} -2\xi_k\omega_k & -\omega_k \\ \omega_k & 0 \end{pmatrix}, \; B_k := \begin{pmatrix} b_k \\ 0 \end{pmatrix}, \; C_k := \begin{pmatrix} c_{rk} & \frac{c_{dk}}{\omega_k} \end{pmatrix},</math>

where <math>b_k \in \mathbb{R}^{1 \times M}</math> and <math>c_{rk}, c_{dk} \in \mathbb{R}^{Q \times 1}</math>.

===Benchmark Specifics===

For this benchmark the system matrix is block diagonal and thus chosen to be sparse.
The parameters <math>\xi</math> and <math>\omega</math> are sampled from a uniform random distributions <math>\mathcal{U}_{[0,\frac{1}{1000}]}^K</math> and <math>\mathcal{U}_{[0,100]}^K</math> respectively.
The components of the input matrix <math>b_k</math> are sampled form a uniform random distribution <math>\mathcal{U}_{[0,1]}</math>,
while the output matrix <math>C</math> is sampled from a uniform random distribution <math>\mathcal{U}_{[0,10]}</math> completely w.l.o.g, since if the components of <math>C_d</math> are random their scaling can be ignored.

==Data==

The following Matlab code assembles the above described <math>A</math>, <math>B</math> and <math>C</math> matrix for a given number of modes <math>K</math>, actuators (inputs) <math>M</math> and sensors (outputs) <math>Q</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [A,B,C] = fss(K,M,Q)

rand('seed',1009);
xi = rand(1,K)*0.001; % Sample damping ratio
omega = rand(1,K)*100.0; % Sample natural frequencies

A_k = cellfun(@(p) sparse([-2.0*p(1)*p(2),-p(2);p(2),0]), ...
num2cell([xi;omega],1),'UniformOutput',0);

A = blkdiag(A_k{:});

B = kron(rand(K,M),[1;0]);

C = 10.0*rand(Q,2*K);
end
</source>
</div>

==Dimensions==

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

System dimensions:

<math>A \in \mathbb{R}^{2K \times 2K}</math>,
<math>B \in \mathbb{R}^{2K \times M}</math>,
<math>C \in \mathbb{R}^{Q \times 2K}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Flexible Space Structures'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Flexible_Space_Structures

@MISC{morwiki-flexspacstruc,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Flexible Space Structures},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Flexible_Space_Structures}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGawW91 morGawW91] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGawW91 BibTeX])

==Reference==

<references>

<ref name="gawronski90">W. Gawronski and J.N. Juang. "[https://doi.org/10.1016/B978-0-12-012736-8.50010-3 Model Reduction for Flexible Structures]", Control and Dynamic Systems, 36: 143--222, 1990.</ref>

<ref name="gawronski91">W. Gawronski and T. Williams, "[http://doi.org/10.2514/3.20606 Model Reduction for Flexible Space Structures]", Journal of Guidance 14(1): 68--76, 1991</ref>

<ref name="gawronski96">W. Gawronski. "[https://doi.org/10.1007/3540760172_4 Model reduction]". In: Balanced Control of Flexible Structures. Lecture Notes in Control and Information Sciences, vol 211: 45--106, 1996.</ref>

<ref name="horner86">G.C. Horner. "[https://ntrs.nasa.gov/search.jsp?R=19870006596 COFS-1 Research Overview]". NASA / DOD Control Structures Interaction Technology: 233--251, 1986</ref>

<ref name="horta86">L.G. Horta, J.L. Walsh, G.C. Horner and J.P. Bailey. "[https://ntrs.nasa.gov/search.jsp?R=19870006613 Analysis and simulation of the MAST (COFS-1 flight hardware)]". NASA / DOD Control Structures Interaction Technology: 515--532, 1986.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

RCL Circuit Equations

2026-04-23T11:38:18Z

Schulze: Update category SISO to MIMO

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:MIMO]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = RCL Circuit Equations
|Benchmark ID =
* rclCircuitEquations_n1841m16q16
* rclCircuitEquations_n306m2q2
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 1841
* 306
|ninputs =
* 16
* 2
|noutputs =
* 16
* 2
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description==
These benchmarks originate from [[wikipedia:Very_Large_Scale_Integration|VLSI]] circuits.
Specifically [[wikipedia:RLC_circuit|resistor-capacitor-inductor]] circuits, which can be represented by first order descriptor systems,
following a modeling process based on the two [[wikipedia:Kirchhoff's_circuit_laws|Kirchhoff's circuit laws]] and the [[wikipedia:Modified_nodal_analysis| branch constitutive relations]].

===PEEC Problem===
This RCL circuit is a [[wikipedia:Partial_element_equivalent_circuit|PEEC]] discretization<ref name="ruehli1974"/> and has 2100 capacitors, 172 inductors, 6990 inductive couplings, as well as a resistive source<ref name="feldmann1995"/>,<ref name="freund2005"/>.
The resulting model has 306 states, and two inputs and outputs.

===Package Problem===
The second problem models a 64-pin package of an [[wikipedia:RF_circuit|RF]] circuit.
A subset of eight pins carry signals, which leads to sixteen terminals (eight interior and eight exterior)<ref name="bai1997"/>,<ref name="freund2005"/>.
The resulting model has 1841 states, and sixteen inputs and outputs.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; see <ref name="freund2005"/>.

==Data==

The PEEC problem and package problem are available as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] files, providing the <math>A</math>, <math>B</math>, <math>E</math> matrices,
while <math>C = B^\intercal</math> is assumed:

* [[Media:PEEC.zip|PEEC.zip]] (32.8KB)
* [[Media:Package.zip|Package.zip]] (78.7KB)

==Dimensions==

System structure:

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

System dimensions:

<math>E \in \mathbb{R}^{N \times N}</math>,
<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times M}</math>,
<math>C \in \mathbb{R}^{M \times N}</math>.

System variants:

<math>N = 306</math>, <math>M = 2</math>, for the PEEC problem, and <math>N = 1841</math>, <math>M = 16</math> for the package problem.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''RCL Circuit Equations'''. MORwiki - Model Order Reduction Wiki, 2019. http://modelreduction.org/index.php/RCL_Circuit_Equations

@MISC{morwiki_convection,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {RCL Circuit Equations},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/RCL_Circuit_Equations}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@INCOLLECTION{morFre05,
author = <nowiki>{R.W. Freund}</nowiki>,
title = {RCL Circuit Equations},
booktitle = {Dimension Reduction of Large-Scale Systems},
pages = {367--371),
year = {2005},
doi = {10.1007/3-540-27909-1_22}
}

==References==

<references>

<ref name="ruehli1974">A.E. Ruehli, [https://doi.org/10.1109/TMTT.1974.1128204 Equivalent Circuit Models for Three-Dimensional Multiconductor Systems], IEEE Transactions on Microwave Theory and Techniques 22(1): 216--221, 1974.</ref>

<ref name="feldmann1995"> P. Feldmann, R.W. Freund , [https://doi.org/10.1109/43.384428 Efficient linear circuit analysis by Pade approximation via the Lanczos process], IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 14(5): 639--649, 1995.</ref>

<ref name="bai1997">Z. Bai, P. Feldmann, R.W. Freund, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.7763&rep=rep1&type=pdf Equivalent Stable and Passive Reduced-Order Models Based on Partial Pade Approximation Via the Lanczos Process], Numerical Analysis Manuscript 97(3): 1--17, 1997.</ref>

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

<ref name="freund2005"> R.W. Freund, [https://doi.org/10.1007/3-540-27909-1_22 RCL Circuit Equations], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 367--371, 2005.</ref>

</references>

Transmission Lines (SLICOT)

2026-04-23T11:15:47Z

Schulze: Add infobox

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:MIMO]]
[[Category:Dense]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = Transmission Lines (SLICOT)
|Benchmark ID = transmissionLines_n256m2q2.mat
|Category = slicot
|System-Class = LTI-FOS
|nstates = 256
|ninputs = 2
|noutputs = 2
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator =
|Editor =
|Zenodo-link = NA
}}

==Description: Model of a Transmission Line Model==

This benchmark models impedence of interconnected structures of a [[wikipedia:Transmission_line|transmission line]].
More details can be found in <ref name="li99"/>, <ref name="marques98"/> and <ref name="chahlaoui02"/>.

For a larger transmission lines model see the alternative [[Transmission_Lines]] benchmark.

==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>, <math>E</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/tline.zip tline.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}
E\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{256 \times 256}</math>,
<math>B \in \mathbb{R}^{256 \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times 256}</math>,
<math>E \in \mathbb{R}^{256 \times 256}</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_tline,
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:

@INPROCEEDINGS{MarKWetal98,
author = <nowiki>{N. Marques and M. Kamon and J. White and L.M. Silveira}</nowiki>,
title = {A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects},
booktitle = {Proceedings of the 1998 Design and Automation Conference},
pages = {297--302},
year = {1998},
doi = {10.1145/277044.277132}
}

==References==

<references>

<ref name="li99"> J.R. Li, J. White. [https://doi.org/10.1109/ICCAD.1999.810679 Efficient model reduction of interconnect via approximate system gramians]. 1999 IEEE/ACM International Conference on Computer-Aided Design: 380--383, 1999.</ref>

<ref name="marques98"> N. Marques, M. Kamon, J. White, L.M. Silveira. [https://doi.org/10.1145/277044.277132 A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects]. Proceedings of the 1998 Design and Automation Conference: 297--302, 1998.</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>

Modified Nodal Analysis

2026-04-23T11:12:59Z

Schulze: Delete gyroscope infobox

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

{{Infobox
|Title = Modified Nodal Analysis
|Benchmark ID =
* mna_n10913m9q9
* mna_n4863m22q22
* mna_n578m9q9
* mna_n9223m18q18
* mna_n980m4q4
|Category = slicot
|System-Class = LTI-FOS
|nstates =
* 10913
* 4863
* 578
* 9223
* 980
|ninputs =
* 9
* 22
* 9
* 18
* 4
|noutputs =
* 9
* 22
* 9
* 18
* 4
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Modified Nodal Analysis Model==

This set of benchmark uses models resulting from [[wikipedia:Modified_nodal_analysis|modified nodal analysis]].
More details can be found in <ref name="odabasioglu98"/> and <ref name="chahlaoui02"/>.

==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>, <math>E</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Benchmark files and dimensions.''
|-
!
!Inputs
!States
!Outputs
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_1.zip MNA_1.zip]
|<math>M = 9</math>
|<math>N = 578</math>
|<math>Q = 9</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_2.zip MNA_2.zip]
|<math>M = 18</math>
|<math>N = 9223</math>
|<math>Q = 18</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_3.zip MNA_3.zip]
|<math>M = 22</math>
|<math>N = 4863</math>
|<math>Q = 22</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_4.zip MNA_4.zip]
|<math>M = 4</math>
|<math>N = 980</math>
|<math>Q = 4</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_5.zip MNA_5.zip]
|<math>M = 9</math>
|<math>N = 10913</math>
|<math>Q = 9</math>
|}

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

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times M}</math>,
<math>C \in \mathbb{R}^{Q \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</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_mna,
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{morOda98,
author = <nowiki>{A. Odabasioglu and M. Celik and L.T. Pileggi}</nowiki>,
title = {{PRIMA}: Passive Reduced-order Interconnect Macromodeling Algorithm},
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems},
volume = {17},
number = {8},
pages = {645--654},
year = {1998},
doi = {10.1109/43.712097}
}

==References==

<references>

<ref name="odabasioglu98">A. Odabasioglu, M. Celik, L.T. Pileggi, [https://doi.org/10.1109/43.712097 PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 17(8): 645--654, 1998.</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>

FEniCS Rail

2024-04-19T10:39:29Z

Schulze: Add missing dimension 300k

{{preliminary}} 

[[Category:benchmark]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = FEniCS Rail
|Benchmark ID =
* fenicsRail_n371m7q6
* fenicsRail_n1357m7q6
* fenicsRail_n5177m7q6
* fenicsRail_n20209m7q6
* fenicsRail_n79841m7q6
* fenicsRail_n317377m7q6
* fenicsRail_n1265537m7q6
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
* 1357
* 5177
* 20209
* 79841
* 317377
* 1265537
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License = CC-BY-4.0
|Creator =
* [[User:Saak]]
* Maximilian Behr
|Editor =
|Zenodo-link = https://zenodo.org/record/5113560 Zenodo
}}

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

FEniCS Rail

2024-04-19T10:38:12Z

Schulze: Fix infobox title

{{preliminary}} 

[[Category:benchmark]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = FEniCS Rail
|Benchmark ID =
* fenicsRail_n371m7q6
* fenicsRail_n1357m7q6
* fenicsRail_n5177m7q6
* fenicsRail_n20209m7q6
* fenicsRail_n79841m7q6
* fenicsRail_n1265537m7q6
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
* 1357
* 5177
* 20209
* 79841
* 1265537
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License = CC-BY-4.0
|Creator =
* [[User:Saak]]
* Maximilian Behr
|Editor =
|Zenodo-link = https://zenodo.org/record/5113560 Zenodo
}}

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

FEniCS Rail

2024-04-18T15:29:08Z

Schulze: Fix list of creators/editors

{{preliminary}} 

[[Category:benchmark]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Rail
|Benchmark ID =
* fenicsRail_n371m7q6
* fenicsRail_n1357m7q6
* fenicsRail_n5177m7q6
* fenicsRail_n20209m7q6
* fenicsRail_n79841m7q6
* fenicsRail_n1265537m7q6
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
* 1357
* 5177
* 20209
* 79841
* 1265537
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License = CC-BY-4.0
|Creator =
* [[User:Saak]]
* Maximilian Behr
|Editor =
|Zenodo-link = https://zenodo.org/record/5113560 Zenodo
}}

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

FEniCS Rail

2024-04-18T15:22:10Z

Schulze: Fix model IDs

{{preliminary}} 

[[Category:benchmark]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Rail
|Benchmark ID =
* fenicsRail_n371m7q6
* fenicsRail_n1357m7q6
* fenicsRail_n5177m7q6
* fenicsRail_n20209m7q6
* fenicsRail_n79841m7q6
* fenicsRail_n1265537m7q6
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
* 1357
* 5177
* 20209
* 79841
* 1265537
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License = CC-BY-4.0
|Creator =
* [[User:Saak]]
* Maximilian Behr
* Martin Köhler
|Editor = [[User:Saak]]
|Zenodo-link = https://zenodo.org/record/5113560 Zenodo
}}

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

FEniCS Rail

2024-04-18T15:07:00Z

Schulze: Add preliminary benchmark IDs

{{preliminary}} 

[[Category:benchmark]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Rail
|Benchmark ID =
* steelProfile_n371m7q6_v2
* steelProfile_n1357m7q6_v2
* steelProfile_n5177m7q6_v2
* steelProfile_n20209m7q6_v2
* steelProfile_n79841m7q6_v2
* steelProfile_n1265537m7q6_v2
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
* 1357
* 5177
* 20209
* 79841
* 1265537
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License = CC-BY-4.0
|Creator =
* [[User:Saak]]
* Maximilian Behr
* Martin Köhler
|Editor = [[User:Saak]]
|Zenodo-link = https://zenodo.org/record/5113560 Zenodo
}}

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

ALBERTA Rail 371

2024-04-11T14:04:09Z

Schulze: Remove math environments from numbers, so they become searchable

{{preliminary}} 

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Steel Profile 
|Benchmark ID =
* steelProfile_n371m7q6 
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

This is a small configuration of the original [[Steel Profile]],
that was not published in the [[:Category:Oberwolfach|Oberwolfach Collection]].
There are several publications that used this benchmark despite citing [[Steel Profile]]:

* Köhler, Martin, Norman Lang, and Jens Saak. “Solving Differential Matrix Equations Using Parareal.” PAMM 16, no. 1 (October 2016): 847–48. https://doi.org/10.1002/pamm.201610412. (@[[User:Saak]]: please confirm)
* Schulze, Jonas. “A Low-Rank Parareal Solver for Differential Riccati Equations Written in Julia.” Otto-von-Guericke-Universität Magdeburg, 2022. https://doi.org/10.5281/zenodo.7843197.
* ...

== Data ==

{| class="wikitable" style="margin: auto; text-align:right;"
|+ style="caption-side:bottom;"|
|-
|
|# nonzeros in A
|# nonzeros in E
|max. mesh width
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_371.zip SteelProfile-dim1e3-rail_371.zip] (32kB)
|2341
|2343
|
|}

== Related Variants ==

* [[Steel Profile]]: dimensions 1357, 5177, 20209, 79841
* [[FEniCS Rail]]: dimensions 371, 1357, 5177, 20209, 79841, 317377, 1265537

FEniCS Rail

2024-04-11T13:53:27Z

Schulze: Add some categories

{{preliminary}} 

[[Category:benchmark]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

ALBERTA Rail 371

2024-04-11T13:52:11Z

Schulze: Add some categories

{{preliminary}} 

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Steel Profile 
|Benchmark ID =
* steelProfile_n371m7q6 
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

This is a small configuration of the original [[Steel Profile]],
that was not published in the [[:Category:Oberwolfach|Oberwolfach Collection]].
There are several publications that used this benchmark despite citing [[Steel Profile]]:

* Köhler, Martin, Norman Lang, and Jens Saak. “Solving Differential Matrix Equations Using Parareal.” PAMM 16, no. 1 (October 2016): 847–48. https://doi.org/10.1002/pamm.201610412. (@[[User:Saak]]: please confirm)
* Schulze, Jonas. “A Low-Rank Parareal Solver for Differential Riccati Equations Written in Julia.” Otto-von-Guericke-Universität Magdeburg, 2022. https://doi.org/10.5281/zenodo.7843197.
* ...

== Data ==

{| class="wikitable" style="margin: auto; text-align:right;"
|+ style="caption-side:bottom;"|
|-
|
|# nonzeros in A
|# nonzeros in E
|max. mesh width
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_371.zip SteelProfile-dim1e3-rail_371.zip] (32kB)
|<math>2\,341</math>
|<math>2\,343</math>
|
|}

== Related Variants ==

* [[Steel Profile]]: dimensions 1357, 5177, 20209, 79841
* [[FEniCS Rail]]: dimensions 371, 1357, 5177, 20209, 79841, 317377, 1265537

ALBERTA Rail 371

2024-04-11T13:08:29Z

Schulze: Add hint...

{{preliminary}} 

{{Infobox
|Title = Steel Profile 
|Benchmark ID =
* steelProfile_n371m7q6 
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

This is a small configuration of the original [[Steel Profile]],
that was not published in the [[:Category:Oberwolfach|Oberwolfach Collection]].
There are several publications that used this benchmark despite citing [[Steel Profile]]:

* Köhler, Martin, Norman Lang, and Jens Saak. “Solving Differential Matrix Equations Using Parareal.” PAMM 16, no. 1 (October 2016): 847–48. https://doi.org/10.1002/pamm.201610412. (@[[User:Saak]]: please confirm)
* Schulze, Jonas. “A Low-Rank Parareal Solver for Differential Riccati Equations Written in Julia.” Otto-von-Guericke-Universität Magdeburg, 2022. https://doi.org/10.5281/zenodo.7843197.
* ...

== Data ==

{| class="wikitable" style="margin: auto; text-align:right;"
|+ style="caption-side:bottom;"|
|-
|
|# nonzeros in A
|# nonzeros in E
|max. mesh width
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_371.zip SteelProfile-dim1e3-rail_371.zip] (32kB)
|<math>2\,341</math>
|<math>2\,343</math>
|
|}

== Related Variants ==

* [[Steel Profile]]: dimensions 1357, 5177, 20209, 79841
* [[FEniCS Rail]]: dimensions 371, 1357, 5177, 20209, 79841, 317377, 1265537

ALBERTA Rail 371

2024-04-11T13:07:25Z

Schulze: Add data

{{preliminary}} 

{{Infobox
|Title = Steel Profile 
|Benchmark ID =
* steelProfile_n371m7q6 
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

This is a small configuration of the original [[Steel Profile]],
that was not published in the [[:Category:Oberwolfach|Oberwolfach Collection]].
There are several publications that used this benchmark despite citing [[Steel Profile]]:

* Köhler, Martin, Norman Lang, and Jens Saak. “Solving Differential Matrix Equations Using Parareal.” PAMM 16, no. 1 (October 2016): 847–48. https://doi.org/10.1002/pamm.201610412. (@[[User:Saak]]: please confirm)
* Schulze, Jonas. “A Low-Rank Parareal Solver for Differential Riccati Equations Written in Julia.” Otto-von-Guericke-Universität Magdeburg, 2022. https://doi.org/10.5281/zenodo.7843197.

== Data ==

{| class="wikitable" style="margin: auto; text-align:right;"
|+ style="caption-side:bottom;"|
|-
|
|# nonzeros in A
|# nonzeros in E
|max. mesh width
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_371.zip SteelProfile-dim1e3-rail_371.zip] (32kB)
|<math>2\,341</math>
|<math>2\,343</math>
|
|}

== Related Variants ==

* [[Steel Profile]]: dimensions 1357, 5177, 20209, 79841
* [[FEniCS Rail]]: dimensions 371, 1357, 5177, 20209, 79841, 317377, 1265537

ALBERTA Rail 371

2024-04-11T12:49:31Z

Schulze: Add related variants

{{preliminary}} 

{{Infobox
|Title = Steel Profile 
|Benchmark ID =
* steelProfile_n371m7q6 
|Category =
|System-Class = LTI-FOS
|nstates =
* 371
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

This is a small configuration of the original [[Steel Profile]],
that was not published in the [[:Category:Oberwolfach|Oberwolfach Collection]].
There are several publications that used this benchmark despite citing [[Steel Profile]]:

* Köhler, Martin, Norman Lang, and Jens Saak. “Solving Differential Matrix Equations Using Parareal.” PAMM 16, no. 1 (October 2016): 847–48. https://doi.org/10.1002/pamm.201610412. (@[[User:Saak]]: please confirm)
* Schulze, Jonas. “A Low-Rank Parareal Solver for Differential Riccati Equations Written in Julia.” Otto-von-Guericke-Universität Magdeburg, 2022. https://doi.org/10.5281/zenodo.7843197.

== Related Variants ==

* [[Steel Profile]]: dimensions 1357, 5177, 20209, 79841
* [[FEniCS Rail]]: dimensions 371, 1357, 5177, 20209, 79841, 317377, 1265537

ALBERTA Rail 371

2024-04-11T12:38:02Z

Schulze: Add ALBERTA Rail 371

Steel Profile

2024-04-11T12:35:51Z

Schulze: Fix file names within data sets

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Steel Profile
|Benchmark ID =
* steelProfile_n1357m7q6
* steelProfile_n5177m7q6
* steelProfile_n20209m7q6
* steelProfile_n79841m7q6
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 1357
* 5177
* 20209
* 79841
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

==Description: A Semi-discretized Heat Transfer Problem for Optimal Cooling of Steel Profiles==
<figure id="fig1">[[File:Steelprofile1.jpg|490px|thumb|right|<caption>Initial mesh and partition of the boundary.</caption>]]</figure>
<figure id="fig2">[[File:Steelprofile2.jpg|490px|thumb|right|<caption>Cooling plant.</caption>]]</figure>

A Semi-discretized heat transfer problem for optimal cooling of steel profiles.
Several generalized state-space models arising from a semi-discretization of a controlled heat transfer process for optimal cooling of steel profiles are presented. The models order differs due to different refinements applied to the computational mesh.

===Model Equations===

We consider the problem of optimal cooling of steel profiles.
This problem arises in a [[wikipedia:Rolling_(metalworking)#Mills|rolling mill]] when different steps in the production process require different temperatures of the raw material.
To achieve a high production rate, economical interests suggest to reduce the temperature as fast as possible to the required level before entering the next production phase.
At the same time, the cooling process, which is realized by spraying cooling fluids onto the surface, has to be controlled so that material properties, such as durability or porosity, achieve given quality standards.
Large gradients in the temperature distributions of the steel profile may lead to unwanted deformations, brittleness, loss of rigidity, and other undesirable material properties.
It is therefore the engineer's goal to have a preferably even temperature distribution.

The scientific challenge here is to give the engineers a tool to precalculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose.

We can only briefly introduce the model here for details we refer to <ref name="Saa03"/>, <ref name="BenS05b"/>, or <ref name="bs04"/>.
We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model.
Exploiting the symmetry of the workpiece, the computational domain <math>\Omega \subset \mathbb{R}^2</math> is chosen as half a cross section of the rail profile.
The heat distribution is modeled by the unsteady linear heat equation on <math>\Omega</math>:

:<math id="eq1">
\begin{align}
c \rho \partial_t x(t,\chi) - \lambda \Delta x(t,\chi) &= 0 \in \mathbb{R}_{>0} \times \Omega, \\
x(0,\chi) &= x_0(\chi) \in \Omega, & (1)\\
\lambda \partial_\nu x(t,\chi) &= g_i \in \mathbb{R}_{>0} \times \Gamma_i,~ \partial \Omega = \bigcup_i \Gamma_i,
\end{align}
</math>

where <math>x</math> is the temperature distribution (<math>x \in H^1([0,\infty],X)</math> with <math>X:=H^1(\Omega)</math> the state space), <math>c=7\,620</math> the specific heat capacity, <math>\lambda=26.4</math> the heat conductivity and <math>\rho=654.0</math> the density of the rail profile.
We split the boundary into several parts <math>\Gamma_i</math> on which we have different boundary functions <math>g_i</math>,
allowing us to vary controls on different parts of the surface.
By <math>\nu</math> we denote the outer normal on the boundary.

We want to establish the control by a feedback law, i.e., we define the boundary functions <math>g_i</math> to be functions of the state <math>x</math> and the control <math>u_i</math>, where <math>(u_i)_i =: u = Fx</math> for a linear operator <math>F</math> which is chosen such that the cost functional

:<math id="eq2">
\begin{align}
J(x_0,u) & := \int_0^\infty (Qy,y)_Y + (Ru,u)_U \operatorname{d}t, & (2)
\end{align}
</math>

with <math>y=Cx</math> is minimized.
Here, <math>Q</math> and <math>R</math> are linear self-adjoint operators on the output space <math>Y</math> and the control space <math>U</math> with <math>Q \geq 0,~ R > 0</math> and <math>C \in L(X,Y)</math>.
The variational formulation of [[#eq1|(1)]] with <math>g_i(t,\chi) = q_i(u_i- x(t,\chi))</math> leads to:

:<math>
(\partial_t x,v) = -\int_\Omega \alpha \nabla x \nabla v \operatorname{d}\chi + \sum_k \Big(q_k u_k \int_{\Gamma_k} (c \rho)^{-1} v \operatorname{d}\sigma - \int_{\Gamma_k} q_k(c\rho)^{-1} xv d\sigma\Big)
</math>

for all <math>v \in C_0^\infty(\Omega)</math>. Here the <math>u_k</math> are the exterior (cooling fluid) temperatures used as the controls,
<math>q_k</math> are constant heat transfer coefficients (i.e., parameters for the spraying intensity of the cooling nozzles) and <math>\alpha := \lambda /(c\rho)</math>.
Note that <math>q_0 = 0</math> gives the Neumann isolation boundary condition on the artificial inner boundary on the symmetry axis.
In view of this weak formulation, we can apply a standard Galerkin approach for discretizing the heat transfer model in space, resulting in a first-order ordinary differential equation. This is described in the following section.

===Discretized Model===

For the discretization we use the <tt>ALBERTA-1.2 fem-toolbox</tt> (see <ref name="SS00"/> for details).
We applied linear Lagrange elements and used a projection method for the curved boundaries.
The initial mesh (see Fig. 1) was produced by MATLABs <tt>pdetool</tt>, which implements a [[wikipedia:Delaunay_triangulation|Delaunay triangulation]] algorithm.
The finer discretizations were produced by global mesh refinement using a bisection refinement method.
The discrete [[wikipedia:Linear–quadratic_regulator|LQR]] problem is then: minimize [[#eq2|(2)]] with respect to:

:<math>
\begin{align}
E \frac{\partial}{\partial t} x(t) &= A x(t) + B u(t), \\
y(t) &= C x(t), \\
x(0) &= x_0,
\end{align}
</math>

with <math>t>0</math>.

==Acknowledgements==

This benchmark example serves as a model problem for the project '''A15: Efficient numerical solution of optimal control problems for instationary convection-diffusion-reaction-equations''' of the Sonderforschungsbereich [https://www.tu-chemnitz.de/sfb393/ SFB393 Parallel Numerical Simulation for Physics and Continuum Mechanics], supported by the [http://www.dfg.de/en/index.jsp Deutsche Forschungsgemeinschaft].
It was motivated by the model described in <ref name="TU01"/>. A very similar problem is used as model problem in the [https://www.tu-chemnitz.de/sfb393/lyapack/ LYAPACK] software package <ref name="Pen00"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38881, see <ref name="benner2005"/>.

==Data==

This benchmark includes four different mesh resolutions.
The best FEM-approximation error that one can expect (under suitable smoothness assumptions on the solution) is of order <math>O(h^2)</math> where <math>h</math> is the maximum edge size in the corresponding mesh.
This order should be matched in a model reduction approach.
The following table lists some relevant quantities for the provided models:

{| class="wikitable" style="margin: auto; text-align:right;"
|+ style="caption-side:bottom;"|
|-
|
|# nonzeros in A
|# nonzeros in E
|max. mesh width
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_1357.zip SteelProfile-dim1e3-rail_1357.zip] (95kB)
|<math>8\,985</math>
|<math>8\,997</math>
|<math>5.5280 \cdot 10^{-2}</math>
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_5177.zip SteelProfile-dim1e4-rail_5177.zip] (299kB)
|<math>35\,185</math>
|<math>35\,241</math>
|<math>2.7640 \cdot 10^{-2}</math>
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_20209.zip SteelProfile-dim1e4-rail_20209.zip] (1011kB)
|<math>139\,233</math>
|<math>139\,473</math>
|<math>1.3820 \cdot 10^{-2}</math>
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e5-rail_79841.zip SteelProfile-dim1e5-rail_79841.zip] (3.7MB)
|<math>553\,921</math>
|<math>554\,913</math>
|<math>6.9100 \cdot 10^{-3}</math>
|}

Note that <math>A</math> is negative definite while <math>E</math> is positive definite, so that the resulting linear time-invariant system is stable.

The data sets are named <tt>rail_(problem dimension)_c60.(matrix name)</tt>.
Here, <tt>c60</tt> refers to a specific output matrix <math>C</math>,
which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in Fig. 1) and keep temperature gradients small.
The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g., temperature differences between nodes 83 and 34.
Again refer to Fig. 1 for the nodes used.
The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.
The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500-700</math> degrees depending on calculation time.
The state values are scaled to <math>1\,000</math> degrees centigrade being equivalent to <math>1.000</math>.
This, together with the scaling of the domain, results in a scaling of the time line with factor <math>100</math>, meaning that calculated times have to be divided by <math>100</math> to get the real time in seconds.

==Dimensions==

System structure:
:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{N \times N}</math>,
<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 7}</math>,
<math>C \in \mathbb{R}^{6 \times N}</math>

System variants:

<tt>rail1357</tt>: <math>N = 1\,357</math>,
<tt>rail5177</tt>: <math>N = 5\,177</math>,
<tt>rail20209</tt>: <math>N = 20\,209</math>,
<tt>rail79841</tt>: <math>N = 79\,841</math>

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

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Steel Profile'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Steel_Profile

@MISC{morwiki_steel,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Steel Profile},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Steel_Profile}</nowiki>,
year = 2005
}

* For the background on the benchmark:

@TechReport{BenS05b,
title = {Linear-Quadratic Regulator Design for Optimal Cooling of Steel Profiles},
author = {P. Benner and J. Saak},
institution = {Sonderforschungsbereich 393 {\itshape Parallele Numerische Simulation f\"ur Physik und Kontinuumsmechanik}, TU Chem\-nitz},
address = {D-09107 Chem\-nitz (Germany)},
number = {SFB393/05-05},
year = {2005},
url = {<nowiki>http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597</nowiki>}
}

==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="benner2005">P. Benner, J. Saak, [https://doi.org/10.1007/3-540-27909-1_19 A Semi-Discretized Heat Transfer Model for Optimal Cooling of Steel Profiles], In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 353--356, 2005.</ref>

<ref name="bs04"> P. Benner, J. Saak, [https://doi.org/10.1002/pamm.200410305 Efficient Numerical Solution of the LQR-problem for the Heat Equation], Proceedings in Applied Mathematics and Mechanics, 4(1): 648--649, 2004.</ref>

<ref name="BenS05b"> P. Benner, J. Saak, [http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597 Linear-Quadratic Regulator Design for Optimal Cooling of Steel Profiles], Sonderforschungsbereich 393: Parallele Numerische Simulation für Physik und Kontinuumsmechanik, Technical Report SFB393/05-05, TU Chemnitz, 2005.</ref>

<ref name="Pen00"> T. Penzl, [https://www.tu-chemnitz.de/sfb393/Files/PDF/sfb00-33.pdf LYAPACK Users Guide], Sonderforschungsbereich 393: Numerische Simulation auf massiv parallelen Rechnern, Technical Report SFB393/00-33, TU Chemnitz, 2000.</ref>

<ref name="Saa03"> J. Saak, [https://doi.org/10.5281/zenodo.1187041 Effiziente numerische Lösung eines Optimalsteuerungsproblems für die Abkühlung von Stahlprofilen], Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universität Bremen, 2003.</ref>

<ref name="SS00"> A. Schmidt, K. Siebert, [https://doi.org/10.1007/b138692 Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA], Lecture Notes in Computational Science and Engineering, vol 42, 2005. (See also: [http://www.alberta-fem.de ALBERTA])</ref>

<ref name="TU01"> F. Tröltzsch, A. Unger, [https://doi.org/b2h3hr Fast Solution of Optimal Control Problems in the Selective Cooling of Steel], ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 81(7): 447--456, 2001.</ref>

</references>

==Contact==
'' [[User:Saak]]''