MOR Wiki - User contributions [en]

User:Mlinaric

2025-04-29T06:00:15Z

Mlinaric: Update user page

Petar Mlinarić 
https://pmli.github.io/

Comparison of Software

2023-12-07T23:12:09Z

Mlinaric: Update pyMOR version

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.
For more information on MOR software (especially packages listed in the MORwiki), see <ref name="haasdonk21"/>.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.99 (04.2022)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 1.1 (10.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.2 (Matlab), 1.0 (C, Python, Julia)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause] (Matlab). [https://spdx.org/licenses/GPL-2.0.html GPL-2.0] (C, Python, Julia)
| C, Matlab, Python, Julia
|-
! [[MOR Toolbox]]
| Yes
| (on-going)
| Yes
| (Yes)
| No
| Yes
| Yes
| Yes
|
| 1.1 (Dec. 2020)
| [http://mordigitalsystems.fr/en/ MOR Digital Systems]
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [https://www.mw.tum.de/rt/forschung/modellordnungsreduktion/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.3 (07.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (No)
| Yes
| Yes
|
| 2023.2.0 (12.2023)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

----
See also: [[Further Software]]

References:
<references>

<ref name="haasdonk21">B. Haasdonk "[https://doi.org/10.1515/9783110499001-013 MOR Software]", Model Order Reduction, Volume 3: Applications: 431--460, 2021.</ref>

</references>

Earth Atmosphere

2023-09-01T23:53:41Z

Mlinaric: Add categories, add Python code, edit math

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:SISO]]
[[Category:Dense]]

==Description: Model of an Atmospheric Storm Track==

This benchmark models the track of an [[wikipedia:Storm_track|atmospheric storm track]].
More details can be found in <ref name="farrell95"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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/eady.zip eady.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>B</math> and <math>C</math> are stored as arrays of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('eady.mat')
A = mat['A']
B = mat['B'].astype(np.float64)
C = mat['C'].astype(np.float64)
</syntaxhighlight>

==Dimensions==

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{598 \times 598}</math>,
<math>B \in \mathbb{R}^{598 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 598}</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_cdisc,
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{FarI95,
author = <nowiki>{B.F. Farrell and P.J. Ioannou}</nowiki>,
title = {Stochastic Dynamics of the Midlatitude Atmospheric Jet},
journal = {Journal of the Atmospheric Sciences},
volume = {52},
number = {10},
pages = {1642--1656},
year = {1995},
doi = {10.1175/1520-0469(1995)052<1642:SDOTMA>2.0.CO;2}
}

==References==

<references>

<ref name="farrell95"> B.F. Farrell, P.J. Ioannou. [https://doi.org/10.1175/1520-0469(1995)052%3C1642:SDOTMA%3E2.0.CO;2 Stochastic Dynamics of the Midlatitude Atmospheric Jet]. Journal of the Atmospheric Sciences, 52(10): 1642--1656, 1995.</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>

Convection Reaction

2023-09-01T23:49:14Z

Mlinaric: Fix Python

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Sparse]]
[[Category:SISO]]

==Description==

This benchmark models a chemical reaction by a [[wikipedia:Convection-diffusion_equation|convection]]-[[wikipedia:Reaction-diffusion|reaction]] partial differential equation on the unit square,
given by:

:<math>
\frac{\partial x}{\partial t} = \frac{\partial^2 x}{\partial y^2} + \frac{\partial^2 x}{\partial z^2} + 20 \frac{\partial x}{\partial z} - 180 x + f(y,z) x(t),
</math>

with Dirichlet boundary conditions and discretized with centered difference approximation.

The input vector <math>B</math> is composed of random elements and the output vector equals the input vector <math>C = B^T</math>.

More details can be found in <ref name="raschman80"/>, <ref name="saad88"/>, <ref name="grimme97"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/pde.zip pde.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is a sparse matrix of 16-bit integers and <math>B</math> and <math>C</math> are full matrices stored as sparse matrices):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].astype(np.float64)
B = mat['B'].toarray()
C = mat['C'].toarray()
</syntaxhighlight>

==Dimensions==

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{84 \times 84}</math>,
<math>B \in \mathbb{R}^{84 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 84}</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_pde,
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{Saa88,
author = <nowiki>{Y. Saad}</nowiki>,
title = {Projection and deflation method for partial pole assignment in linear state feedback},
journal = {IEEE Transactions on Automatic Control},
volume = {33},
number = {3},
pages = {290--297},
year = {1988},
doi = {10.1109/9.406}
}

==References==

<references>

<ref name="raschman80"> P. Raschman, M. Kuhicek, M. Maros. Waves in distributed chemical systems: Experiments and computations. In: New Approaches to Nonlinear Problems in Dynamics - Proceedings of the Asilomar Conference Ground: 271--288, SIAM, 1980.</ref>

<ref name="saad88"> Y. Saad. [https://doi.org/10.1109/9.406 Projection and deflation method for partial pole assignment in linear state feedback], IEEE Transactions on Automatic Control, 33(3): 290--297, 1988.</ref>

<ref name="grimme97"> E.J. Grimme. [https://www.proquest.com/dissertations-theses/krylov-projection-methods-model-reduction/docview/304361372/se-2?accountid=14597 Krylov Projection Methods for Model Reduction]. PhD Thesis, University of Illinois at Urbana-Champaign, 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>

Clamped Beam

2023-09-01T23:48:52Z

Mlinaric: Fix Python

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Clamped Beam Model==

This benchmark models a [[wikipedia:Cantilever|cantilever beam]], which is a beam clamped on one end.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

For larger beam-type benchmarks see the [[Linear_1D_Beam|linear 1d beam]] and [[Electrostatic_Beam|electrostatic beam]] benchmarks.

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/beam.zip beam.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('beam.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float64)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + a K_{so} q(t) &= a B_{so} u(t), \\
y(t) &= C_{so} q(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & a I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
C_{so} & 0
\end{pmatrix},
\end{align}
</math>

where <math>a \approx 21.3896</math>.

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 348
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == A[0, n2] * np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

a = A[0, n2]
Eso = -A[n2:, n2:]
Kso = -a * A[n2:, :n2]
Bso = a * B[n2:]
Cso = C[:, :n2]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{348 \times 348}</math>,
<math>B \in \mathbb{R}^{348 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 348}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_p \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{174 \times 174}</math>,
<math>B \in \mathbb{R}^{174 \times 1}</math>,
<math>C_p \in \mathbb{R}^{1 \times 174}</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_beam,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Building Model

2023-09-01T23:47:53Z

Mlinaric: Fix Python

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Motion Problem in a Building==

This benchmark models the displacement of a multi-story building for example during an Earthquake.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Earthquake Model===

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/build.zip build.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float64)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + K_{so} q(t) &= B_{so} u(t), \\
y(t) &= C_{so} \dot{q}(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
0 & C_{so}
\end{pmatrix}
\end{align}
</math>

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 48
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = B[n2:]
Cso = C[:, n2:]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{48 \times 48}</math>,
<math>B \in \mathbb{R}^{48 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 48}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_v \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{24 \times 24}</math>,
<math>B \in \mathbb{R}^{24 \times 1}</math>,
<math>C_v \in \mathbb{R}^{1 \times 24}</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_build,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Coplanar Waveguide

2023-09-01T18:44:23Z

Mlinaric: Text fixes, math edits

[[Category:benchmark]]
[[Category:PDE]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:affine parameter representation]]
[[Category:Stationary]]

==Description==

A '''coplanar waveguide''' (see Fig. 1) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The [[wikipedia:Coplanar_waveguide|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5 \, \text{mm}</math> thickness are buried in a substrate with <math>10 \, \text{mm}</math> thickness and relative permittivity
<math>\epsilon_r = 4.4</math> and relative permeability <math>\mu_r = 1</math>, and low conductivity <math>\sigma = 0.02 \, \text{S/m}</math>.
The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07</math> and <math>\sigma = 0.01 \, \text{S/m}</math>.
The whole structure is enclosed in a metallic box of dimension <math>140 \, \text{mm}</math> by <math>100 \, \text{mm}</math> by <math>50 \, \text{mm}</math>.
The discrete port with <math>50 \, \Omega</math> lumped load imposes <math>1 \, \text{A}</math> current as the input to the one side of the strip.
The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

<figure id="fig:coplanar">
[[File:CoplanarWaveguideScaled.jpg|frame|<caption>Coplanar Waveguide Model<ref name="hess2003"/></caption>]]
</figure>

==Data==

Considered parameters are the frequency <math>\omega</math> and the width <math>\nu</math> of the middle stripline.

The affine form <math>a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) a^q(u, v)</math> can be established using <math>Q = 15</math> affine terms.

The discretized bilinear form is <math>a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) A^q</math>, with matrices <math>A^q</math>.

The matrices corresponding to the bilinear forms <math>a^q(\cdot, \cdot)</math> as well as the input and output forms and H(curl) inner product matrix have been assembled
using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions. The files are numbered according to their
appearance in the summation.

The coefficient functions are given by:

:<math>
\begin{align}
\Theta^1(\omega, \nu) &= 1, \\
\Theta^2(\omega, \nu) &= \omega, \\
\Theta^3(\omega, \nu) &= -\omega^2, \\
\Theta^4(\omega, \nu) &= \frac{\nu}{6}, \\
\Theta^5(\omega, \nu) &= \frac{6}{\nu}, \\
\Theta^6(\omega, \nu) &= \frac{6 \omega}{\nu}, \\
\Theta^7(\omega, \nu) &= -\frac{6 \omega^2}{\nu}, \\
\Theta^8(\omega, \nu) &= \frac{\nu \omega}{6}, \\
\Theta^9(\omega, \nu) &= -\frac{\nu \omega^2}{6}, \\
\Theta^{10}(\omega, \nu) &= \frac{16 - \nu}{10}, \\
\Theta^{11}(\omega, \nu) &= \frac{10}{16 - \nu}, \\
\Theta^{12}(\omega, \nu) &= \frac{10 \omega}{16 - \nu}, \\
\Theta^{13}(\omega, \nu) &= -\frac{10 \omega^2}{16 - \nu}, \\
\Theta^{14}(\omega, \nu) &= \frac{16 - \nu}{10} \omega, \\
\Theta^{15}(\omega, \nu) &= -\frac{16 - \nu}{10} \omega^2.
\end{align}
</math>

The parameter domain of interest is <math>\omega \in [0.6, 3.0] \cdot 10^9 \, \text{Hz}</math>, where the factor of <math>10^9</math> has already been taken into account
while assembling the matrices, while the geometric variation occurs between <math>\nu \in [2.0, 14.0]</math>.
The input functional also has a factor of <math>\omega</math>.

There are two output functionals, which is due to the fact that the complex system has been rewritten as a real symmetric one.
In particular the computation of the output
<math>s(u) = \vert l^T u \vert</math> with complex vector <math>u</math> turns into <math>s(u) = \sqrt{(l_1^T u)^2 + (l_2^T u)^2}</math> with real vector <math>u</math>.

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano] project.

==Data==

The files are numbered according to their appearance in the summation and can be found here: [[Media:Matrices_cp.tar.gz|Matrices_cp.tar.gz]].

==Dimensions==

System structure:

:<math>
\begin{align}
\sum_{q = 1}^{15} \Theta^q(\omega, \nu) A^q u(\omega, \nu) &= b \\
s(\omega, \nu) &= \sqrt{(l_1^T u(\omega, \nu))^2 + (l_2^T u(\omega, \nu))^2}
\end{align}
</math>

System dimensions:

<math>A^q \in \mathbb{R}^{15504 \times 15504}</math>,
<math>b, l_1, l_2 \in \mathbb{R}^{15504}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Coplanar Waveguide'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Coplanar_Waveguide

@MISC{morwiki_waveguide,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Coplanar Waveguide},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Coplanar_Waveguide}</nowiki>,
year = {2018}
}

* For the background on the benchmark:
@ARTICLE{morHesB13,
author = {Hess, M.~W. and Benner, P.},
title = {Fast Evaluation of Time-Harmonic {M}axwell's Equations Using the Reduced Basis Method},
journal = {{IEEE} Trans. Microw. Theory Techn.},
volume = 61,
number = 6,
pages = {2265--2274},
year = 2013,
doi = {10.1109/TMTT.2013.2258167}
}

==References==

<references>

<ref name="hess2003">M.W. Hess, P. Benner, "[https://doi.org/10.1109/TMTT.2013.2258167 Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, 61(6): 2265--2274, 2013.</ref>

</references>

==Contact==

'' [[User:hessm|Martin Hess]]''

Convective Thermal Flow

2023-09-01T18:26:16Z

Mlinaric: Add categories, small fixes

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

==Description: Convective Thermal Flow Problems==
<figure id="fig1">[[File:Convection1.jpg|490px|thumb|right|Figure 1: Convective heat flow example: 2D anemometer model]]</figure>
<figure id="fig2">[[File:Convection2.jpg|490px|thumb|right|Figure 2: Convective heat flow example: 3D cooling structure]]</figure>

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow.
The most elaborate approach to this problem is [[wikipedia:Computational_fluid_dynamics|computational fluid dynamics]] (CFD).
However, CFD is computationally expensive.
A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model.
However, caution has to be taken to select the [[wikipedia:Heat_transfer_coefficient|film coefficient]].

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model.
Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained.
A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation because the solution accuracy when employing upwind finite element schemes depends on the element size.
While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one.
So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an [[Anemometer|anemometer]]-like structure mainly consisting of a tube and a small heat source (Fig. 1) <ref name="ernst2001"/>.
The solid model has been generated and meshed in [http://www.ansys.com ANSYS].
Triangular <tt>PLANE55</tt> elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes.
The second design is a 3D model of a chip cooled by forced convection (Fig. 2) <ref name="harper1997"/>.
In this case, the tetrahedral element type <tt>SOLID70</tt> was used, resulting in 107989 elements and 20542 nodes.
Since the implementation of the convective term in ANSYS does not allow defining the fluid speed on a per-element basis but on a per-region one, the flow profile has to be approximated by piece-wise step functions.
The approximation used for these benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system.
In both models, the reference temperature is set to <math>300 \, \text{K}</math>, and Dirichlet boundary conditions as well as initial conditions are set to <math>0</math> with respect to the reference.
The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection).
Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest.
Hence, a small subset of five nodes is selected as output nodes,
which are filtered from the discretized state by a linear transformation.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38867, see <ref name="moosmann2005"/>.

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
The matrix name is used as an extension of the matrix file.
<tt>*.C.names</tt> contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Output nodes for the two models.''
|-
!Model
!Number
!Code
!Comment
|-
|Flow Meter
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|SenL
|left sensor position
|-
|
|4
|Heater
|within the heater
|-
|
|5
|SenR
|right sensor position
|-
|Cooling Structure
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|out3
|outlet position
|-
|
|4
|out4
|outlet position
|-
|
|5
|Heater
|within the heater
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Provided files.''
|-
!Model
!Fluid Speed (m/s)
!Link
!Size
|-
|Flow Meter
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.tgz Convection-dim1e4-flow_meter_model_v0.tgz]
|649.4 kB
|-
|
|<math>0.5</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.5.tgz Convection-dim1e4-flow_meter_model_v0.5.tgz]
|757.8 kB
|-
|Cooling Structure
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.tgz Convection-dim1e4-chip_cooling_model_v0.tgz]
|3.9 MB
|-
|
|<math>0.1</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.1.tgz Convection-dim1e4-chip_cooling_model_v0.1.tgz]
|4.0 MB
|}

Further information on the models can be found in <ref name="moosmann2004"/>,
where model reduction by means of the [[wikipedia:Arnoldi_iteration|Arnoldi algorithm]] is also presented.

==Dimensions==

System structure:

:<math>
\begin{align}
E \dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(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 1}</math>,
<math>C \in \mathbb{R}^{5 \times n}</math>.

System variants:

<tt>flow_v0</tt>: <math>n = 9669</math>,
<tt>flow_v0.5</tt>: <math>n = 9669</math>,
<tt>chip_v0</tt>: <math>n = 20082</math>,
<tt>chip_v0.1</tt>: <math>n = 20082</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Convective Thermal Flow'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Convective_Thermal_Flow

@MISC{morwiki_convection,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Convective Thermal Flow},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Convection}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
author = <nowiki>{C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink}</nowiki>,
title = {Model Order Reduction for Linear Convective Thermal Flow},
booktitle = {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}</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], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="ernst2001">H. Ernst, [https://freidok.uni-freiburg.de/data/201 High-Resolution Thermal Measurements in Fluids], PhD thesis, University of Freiburg, Germany, 2001.</ref>

<ref name="harper1997">C.A. Harper, [https://doi.org/10.1036/0071430482 Electronic packaging and interconnection handbook], New York McGraw- Hill, USA, 1997</ref>

<ref name="moosmann2004">C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink, [http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf Model Order Reduction for Linear Convective Thermal Flow], Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems, THERMINIC2004, Sophia Antipolis, France, 2004.</ref>

<ref name="moosmann2005">C. Moosmann, A. Greiner, [https://doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems], In: Dimension Reduction of Large-Scale Systems. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, vol 45: 341--343, 2005.</ref>

</references>

Convection-Diffusion

2023-09-01T18:07:14Z

Mlinaric: Add categories, small fixes

[[Category:benchmark]]
[[Category:procedural]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==

The '''Convection-Diffusion''' benchmark is given by a finite-difference discretization of a two-dimensional dynamic convection-diffusion heat equation on a unit square domain, which was introduced as an example in <ref name="lyapack"/>:

:<math>
\begin{align}
\Delta u - f_x \partial_x u - f_y \partial_y u - g u &= f \\
u &= 0
\end{align}
</math>

For this procedural partial differential equation benchmark, Dirichlet boundary conditions are used.
The setup of this benchmark is rather generic, in that it allows based on the configuration of <math>f_x</math>, <math>f_y</math>, <math>g</math>, and <math>f</math> to tune the influence of the diffusive, convective and reactive differential operators.

==Origin==

This benchmark is part of LYAPACK <ref name="lyapack"/>.

==Data==

In the folder <tt>examples</tt> of LYAPCK the MATLAB functions <tt>fdm_2d_matrix</tt> and <tt>fdm_2d_vector</tt> are utilized to generate the system, input matrices:

:<syntaxhighlight lang="matlab">
A = fdm_2d_matrix(n0, fx_str, fy_str, g_str);
B = fdm_2d_vector(n0, f_str);
</syntaxhighlight>

The <tt>n0</tt> argument determines the common spatial resolution, hence the discretized problem has dimension <math>N = n_0^2</math>.
The arguments <tt>fx_str</tt>, <tt>fy_str</tt>, <tt>g_str</tt>, and <tt>f_str</tt> are strings (see documentation) describing the convection in the first dimension, the second dimension, the reaction coefficient, and the source term, respectively.
The output matrix can be generated in the same fashion as the input matrix.

For example, in <ref name="lyapack"/>, the examples are constructed with:

:<syntaxhighlight lang="matlab">
A = fdm_2d_matrix(20, '10*x', '100*y', '0');
B = fdm_2d_vector(20, '.1<x<=.3');
C = fdm_2d_vector(20, '.7<x<=.9')';
</syntaxhighlight>

resulting in a SISO system of order <math>400</math>.

==Dimensions==

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

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

==Citation==

To cite this benchmark, use the following references:

* For the benchmark:
::The MORwiki Community, '''LYAPACK - A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear–Quadratic Optimal Control Problems'''. Version 1.0, netlib, 1999. http://www.netlib.org/lyapack

@MISC{lyapack,
author = {T. Penzl},
title = {LYAPACK - A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear–Quadratic Optimal Control Problems},
howpublished = {netlib},
note = {Version 1.0},
url = <nowiki>{http://www.netlib.org/lyapack}</nowiki>,
year = {1999}
}

==References==

<references>

<ref name="lyapack">T. Penzl. [http://www.netlib.org/lyapack/README LYAPACK - A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems], Version 1.0, netlib: 1999. [http://www.netlib.org/lyapack/README netlib.org/lyapack/README]</ref>

</references>

Convection Reaction

2023-08-30T13:01:32Z

Mlinaric: Add categories, edit SLICOT links, add Python code, edit math

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Sparse]]
[[Category:SISO]]

==Description==

This benchmark models a chemical reaction by a [[wikipedia:Convection-diffusion_equation|convection]]-[[wikipedia:Reaction-diffusion|reaction]] partial differential equation on the unit square,
given by:

:<math>
\frac{\partial x}{\partial t} = \frac{\partial^2 x}{\partial y^2} + \frac{\partial^2 x}{\partial z^2} + 20 \frac{\partial x}{\partial z} - 180 x + f(y,z) x(t),
</math>

with Dirichlet boundary conditions and discretized with centered difference approximation.

The input vector <math>B</math> is composed of random elements and the output vector equals the input vector <math>C = B^T</math>.

More details can be found in <ref name="raschman80"/>, <ref name="saad88"/>, <ref name="grimme97"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/pde.zip pde.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is a sparse matrix of 16-bit integers and <math>B</math> and <math>C</math> are full matrices stored as sparse matrices):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].astype(np.float_)
B = mat['B'].toarray()
C = mat['C'].toarray()
</syntaxhighlight>

==Dimensions==

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{84 \times 84}</math>,
<math>B \in \mathbb{R}^{84 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 84}</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_pde,
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{Saa88,
author = <nowiki>{Y. Saad}</nowiki>,
title = {Projection and deflation method for partial pole assignment in linear state feedback},
journal = {IEEE Transactions on Automatic Control},
volume = {33},
number = {3},
pages = {290--297},
year = {1988},
doi = {10.1109/9.406}
}

==References==

<references>

<ref name="raschman80"> P. Raschman, M. Kuhicek, M. Maros. Waves in distributed chemical systems: Experiments and computations. In: New Approaches to Nonlinear Problems in Dynamics - Proceedings of the Asilomar Conference Ground: 271--288, SIAM, 1980.</ref>

<ref name="saad88"> Y. Saad. [https://doi.org/10.1109/9.406 Projection and deflation method for partial pole assignment in linear state feedback], IEEE Transactions on Automatic Control, 33(3): 290--297, 1988.</ref>

<ref name="grimme97"> E.J. Grimme. [https://www.proquest.com/dissertations-theses/krylov-projection-methods-model-reduction/docview/304361372/se-2?accountid=14597 Krylov Projection Methods for Model Reduction]. PhD Thesis, University of Illinois at Urbana-Champaign, 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>

Clamped Beam

2023-08-30T12:50:55Z

Mlinaric: Edit SLICOT links

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Clamped Beam Model==

This benchmark models a [[wikipedia:Cantilever|cantilever beam]], which is a beam clamped on one end.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

For larger beam-type benchmarks see the [[Linear_1D_Beam|linear 1d beam]] and [[Electrostatic_Beam|electrostatic beam]] benchmarks.

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/beam.zip beam.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('beam.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + a K_{so} q(t) &= a B_{so} u(t), \\
y(t) &= C_{so} q(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & a I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
C_{so} & 0
\end{pmatrix},
\end{align}
</math>

where <math>a \approx 21.3896</math>.

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 348
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == A[0, n2] * np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

a = A[0, n2]
Eso = -A[n2:, n2:]
Kso = -a * A[n2:, :n2]
Bso = a * B[n2:]
Cso = C[:, :n2]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{348 \times 348}</math>,
<math>B \in \mathbb{R}^{348 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 348}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_p \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{174 \times 174}</math>,
<math>B \in \mathbb{R}^{174 \times 1}</math>,
<math>C_p \in \mathbb{R}^{1 \times 174}</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_beam,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

CD Player

2023-08-30T12:50:10Z

Mlinaric: Edit SLICOT links

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

==Description: Classical CD Player Model==

This benchmark models the swing arm of a [[wikipedia:CD_Player|CD Player]] holding a lens which can be moved in the horizontal plain.
More details can be found in <ref name="draijer92"/>, <ref name="wortelboer96"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/CDplayer.zip CDplayer.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{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{120 \times 120}</math>,
<math>B \in \mathbb{R}^{120 \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times 120}</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_cdisc,
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{DraSB92,
author = <nowiki>{W. Draijer and M. Steinbuch and O. Bosgra}</nowiki>,
title = {Adaptive Control of the Radial Servo System of a Compact Disc Player},
journal = {Automatica},
volume = {28},
number = {3},
pages = {455--462},
year = {1992},
doi = {10.1016/0005-1098(92)90171-B}
}

==References==

<references>

<ref name="draijer92"> W. Draijer, M. Steinbuch, O. Bosgra. [https://doi.org/10.1016/0005-1098(92)90171-B Adaptive Control of the Radial Servo System of a Compact Disc Player]. Automatica, 28(3): 455--462, 1992.</ref>

<ref name="wortelboer96"> P. Wortelboer, M. Steinbuch, O. Bosgra. [https://research.tue.nl/en/publications/closed-loop-balanced-reduction-with-cation-to-a-compact-disc-mech Closed-Loop Balanced Reduction with Application to a Compact Disc Mechanism]. Selected Topics in Identification, Modeling and Control, 9: 47--58, 1996.</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>

Building Model

2023-08-30T12:49:33Z

Mlinaric: Edit SLICOT link

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Motion Problem in a Building==

This benchmark models the displacement of a multi-story building for example during an Earthquake.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Earthquake Model===

==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 [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/build.zip build.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + K_{so} q(t) &= B_{so} u(t), \\
y(t) &= C_{so} \dot{q}(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
0 & C_{so}
\end{pmatrix}
\end{align}
</math>

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 48
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = B[n2:]
Cso = C[:, n2:]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{48 \times 48}</math>,
<math>B \in \mathbb{R}^{48 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 48}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_v \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{24 \times 24}</math>,
<math>B \in \mathbb{R}^{24 \times 1}</math>,
<math>C_v \in \mathbb{R}^{1 \times 24}</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_build,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Building Model

2023-08-30T12:48:31Z

Mlinaric: Edit SLICOT link

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Motion Problem in a Building==

This benchmark models the displacement of a multi-story building for example during an Earthquake.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Earthquake Model===

==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: [https://www.slicot.org/objects/software/shared/bench-data/build.zip build.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + K_{so} q(t) &= B_{so} u(t), \\
y(t) &= C_{so} \dot{q}(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
0 & C_{so}
\end{pmatrix}
\end{align}
</math>

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 48
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = B[n2:]
Cso = C[:, n2:]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{48 \times 48}</math>,
<math>B \in \mathbb{R}^{48 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 48}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_v \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{24 \times 24}</math>,
<math>B \in \mathbb{R}^{24 \times 1}</math>,
<math>C_v \in \mathbb{R}^{1 \times 24}</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_build,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Clamped Beam

2023-08-30T12:42:20Z

Mlinaric: Edit categories, add Python code, add second-order structure

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Clamped Beam Model==

This benchmark models a [[wikipedia:Cantilever|cantilever beam]], which is a beam clamped on one end.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

For larger beam-type benchmarks see the [[Linear_1D_Beam|linear 1d beam]] and [[Electrostatic_Beam|electrostatic beam]] benchmarks.

==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/beam.zip beam.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('beam.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + a K_{so} q(t) &= a B_{so} u(t), \\
y(t) &= C_{so} q(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & a I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
C_{so} & 0
\end{pmatrix},
\end{align}
</math>

where <math>a \approx 21.3896</math>.

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 348
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == A[0, n2] * np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

a = A[0, n2]
Eso = -A[n2:, n2:]
Kso = -a * A[n2:, :n2]
Bso = a * B[n2:]
Cso = C[:, :n2]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{348 \times 348}</math>,
<math>B \in \mathbb{R}^{348 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 348}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_p \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{174 \times 174}</math>,
<math>B \in \mathbb{R}^{174 \times 1}</math>,
<math>C_p \in \mathbb{R}^{1 \times 174}</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_beam,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Circular Piston

2023-08-29T17:26:15Z

Mlinaric: Edits, add categories

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

==Description: Axi-Symmetric Infinite Element Model for Circular Piston==
This example is a model of the form

:<math>
M \ddot{x}(t) + E \dot{x}(t) + K x(t) = B,
</math>

with <math>M</math>, <math>E</math>, and <math>K</math> non-symmetric matrices and <math>M</math> singular.
This is thus a differential algebraic equation.
It is shown that it has index 1<ref name="coyette2003"/>.
The input of the system is <math>B</math> and the output is the state vector <math>x</math>.
The motivation for using model reduction for this type of problem is the reduction of the computation time of a simulation.

This is an example from an acoustic radiation problem discussed in <ref name="pinsky1991"/>.
Consider a circular piston subtending a polar angle <math>0<\theta<\theta_p</math> on a submerged massless and rigid sphere of radius <math>\delta</math>.
The piston vibrates harmonically with a uniform radial acceleration.
The surrounding acoustic domain is unbounded and is characterized by its density <math>\rho</math> and sound speed <math>c</math>.

We denote by <math>p</math> and <math>a_r</math> the prescribed pressure and normal acceleration respectively.
In order to have a steady state solution <math>\tilde{p}(r,\theta,t)</math> verifying

:<math>
\tilde{p}(r,\theta,t) = \mathcal{R} e (p(r,\theta) e^{i\omega t})
</math>

the transient boundary condition is chosen as:

:<math>
a_r
= \frac{-1}{\rho} \frac{\partial p(r,\theta)}{\partial r} \big|_{r=a}
=
\begin{cases}
a_0 \sin(\omega t) & 0 < \theta < \theta_p \\
0 & \theta > \theta_p
\end{cases}
</math>

The axi-symmetric discrete finite-infinite element model relies on a mesh of linear quadrangle finite elements for the inner domain
(region between spherical surfaces <math>r=\delta</math> and <math>r=1.5\delta</math>).
The numbers of divisions along radial and circumferential directions are <math>5</math> and <math>80</math>, respectively.
The outer domain relies on conjugated infinite elements of order <math>5</math>.
For this example we used <math>\delta=1 [m]</math>, <math>\rho=1.225 [kg/m^3]</math>, <math>c=340 [m/s]</math>, <math>a_0 = 0.001 [m/s^2]</math> and <math>\omega = 1000 [rad/s]</math>.

The matrices <math>K</math>, <math>E</math>, <math>M</math> and the right-hand side <math>f</math> are computed by [http://www.fft.be/ Free Field Technologies].
The dimension of the second-order system is <math>N=2025</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38890, see <ref name="bai2005"/>.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/CircularPiston-dim1e3-piston.tar.gz CircularPiston-dim1e3-piston.tar.gz] (1.9 MB).

Extracting will produce

* <tt>piston.M</tt>
* <tt>piston.E</tt>
* <tt>piston.K</tt>
* <tt>piston.B</tt>

Note that for <tt>piston.B</tt>, loading with <tt>scipy.io.mmread</tt> will not work because the number of nonzeros is specified in the dense Matrix Market format.
Replacing the line with "2025 1 2025" by "2025 1" will make it work.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B \\
y(t) &= x(t)
\end{align}
</math>

System dimensions:

<math>M, E, K \in \mathbb{R}^{2025 \times 2025}</math>,
<math>B \in \mathbb{R}^{2025 \times 1}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Circular Piston'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Circular_Piston

@MISC{morwiki_piston,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Circular Piston},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Circular_Piston}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@ARTICLE{PinA91,
author = <nowiki>{P.M. Pinsky and N.N. Abboud}</nowiki>,
title = {Finite element solution of the transient exterior structural acoustics problem based on the use of radially asymptotic boundary conditions},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {85},
pages = {311--348},
year = {1991},
doi = {10.1016/0045-7825(91)90101-B}
}

==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="bai2005">Z. Bai, K. Meerbergen, Y. Su, [https://doi.org/10.1007/3-540-27909-1_21 Second Order Models: Linear-Drive Multi-Mode Resonator and Axi Symmetric Model of a Circular Piston]. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 363--365, 2005.</ref>

<ref name="coyette2003">J.-P. Coyette, K. Meerbergen, M. Robbé, [https://doi.org/10.1002/nme.1419 Time integration for spherical acoustic finite-infinite element models], Numerical Methods in Engineering 64(13): 1752--1768, 2003.</ref>

<ref name="pinsky1991">P.M. Pinsky and N.N. Abboud, [https://doi.org/10.1016/0045-7825(91)90101-B Finite element solution of the transient exterior structural acoustics problem based on the use of radially asymptotic boundary conditions], Computer Methods in Applied Mechanics and Engineering, 85: 311--348, 1991.</ref>

</references>

Artificial Fishtail

2023-08-29T17:04:15Z

Mlinaric: Add MIMO category

[[Category:benchmark]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:Sparse]]
[[Category:MIMO]]

<figure id="fig:plot1">
[[File:Fishtail.png|4380px|thumb|right|<caption>Schematic 3D-Model-Fishtail</caption>]]
</figure>
<figure id="fig:plot2">
[[File:fishtail_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp</caption>]]
</figure>
<figure id="fig:plot3">
[[File:fishtail_ext_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp_ext</caption>]]
</figure>

==Description==
Today's [[wikipedia:Autonomous_underwater_vehicle|autonomous underwater vehicles]] (AUVs) are a source of noise pollution and inefficiency due to their screw propeller-driven design.
The evolution of fish has, on the other hand, optimized their underwater efficiency and agility over millennia.
The adaption of fish-like drive systems for AUVs is therefore a promising choice.

==Model Description==
This model describes the silicon body of an artificial fishtail supported by a central carbon beam.
The rear part of the fish body without the fins is modeled as a 3D FEM model using linear elasticity.
In the current stage of modeling the tail is rigidly mounted in the front, the states in <math>x</math> represent the displacements of the finite element degrees of freedom.
The fish-like locomotion is enabled by pumping air between two sets of pressure chambers in the left and right halves of the tail.
The single input <math>u</math> of the system is thus the pumping pressure.
The outputs are displacements of certain surface points.
There are two variants of the model.
The first has three outputs representing the displacements of the point of interest, the rear tip of the carbon beam, in the three spatial directions.
For the second variant, six additional points <math>(z_1,z_2,z_3)</math> on the flank are added as outputs, yielding a total of 21 outputs.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! scope="col" style="width: 12ex;" | <math>z_1</math>
! scope="col" style="width: 12ex;" | <math>z_2</math>
! scope="col" style="width: 12ex;" | <math>z_3</math>
|-
| 0.05
| 0.0
| 0.0
|-
| 0.0474526
| 0.0
| 0.0599584
|-
| 0.04032111
| 0.0
| 0.105274
|-
| 0.0326229
| 0.0
| 0.136726
|-
| 0.0250675
| 0.0
| 0.16107
|-
| 0.0168069
| 0.0
| 0.183588
|-
| 0.0
| 0.0
| 0.21
|}

Note that the POI (Point of Interest) is the last row in this table and in Cp_ext in the data files (see below).
The additional outputs show two effects.
On the one hand, for purely input-output-related reduction methods they avoid drastic deviations on the interior states.
On the other hand, they show a smoothing effect for the model's transfer function.

==Origin==
The model was set up and computed at the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Artificial_Fishtail chair of automatic control] at CAU Kiel and first presented in <ref name="SieKM18" />.

==Data==
The model is based on the finite element package [https://www.firedrakeproject.org Firedrake] and uses the material parameters:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|
|<math>\varrho_1</math>
|<math>1.07 \cdot 10^{-3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-style="background-color:#FFFFFF;"
|Hull
|<math>E_1</math>
|<math>0.025 \cdot 10^6</math>
|<math>\frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu_1</math>
|<math>0.48</math>
|
|-
|
|<math>\varrho_2</math>
|<math>1.4 \cdot 10^{3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-
| Beam
|<math> E_2</math>
|<math> 2.96 \cdot 10^{10}</math>
|<math> \frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-
|
|<math>\nu_2</math>
|<math> 0.3</math>
|
|-style="background-color:#FFFFFF;"
| Rayleigh damping
|<math>\alpha_r</math>
|<math> 1.0 \cdot 10^{-4}</math>
|<math>\frac{1}{s}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\beta_r</math>
|<math> 2.0 \cdot 10^{-4}</math>
|<math>\text{s}</math>
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M, E, K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{p \times n}</math>,
with <math>n = 779\,232</math> and <math>m = 1</math>.

The internal damping is modeled as Rayleigh damping <math>E = \alpha_r M + \beta_r K</math> using the coefficients from the table above.

System variants:

* <math>p = 3</math>: <math>C=</math><tt>Cp</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files],
* <math>p = 21</math>: <math>C=</math><tt>Cp_ext</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files].

== Remarks ==
* Physically meaningful inputs are of dimension <math>u(t) = \mathcal{O}(10^3)</math>. As an example, a step signal with around <math>5\,000</math>Pa leads to a horizontal POI displacement of about <math>3</math>cm.
* The interesting operation frequencies are in the range between <math>0</math>Hz and <math>10</math>Hz.
* If required, the finite element mesh behind the model and a CSV file with the output locations are available [https://doi.org/10.5281/zenodo.2565173 separately].
* Warning: the data files are quite large and may exceed the RAM of a typical machine if the user is also running MATLAB.

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

* For the benchmark itself and its data:
@Misc{SieKM19,
author = {Siebelts, D. and Kater, A. and Meurer, T. and Andrej, J.},
title = {Matrices for an Artificial Fishtail},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2019,
doi = {10.5281/zenodo.2558728}
}

* For the background on the benchmark:

@Article{SieKM18,
author = {Siebelts, D. and Kater, A. and Meurer, T.},
title = {Modeling and Motion Planning for an Artificial Fishtail},
journal = {IFAC-PapersOnLine},
volume = 51,
number = 2,
year = 2018,
pages = {319--324},
doi = {10.1016/j.ifacol.2018.03.055},
publisher = {Elsevier {BV}}
}

==References==
<references>
<ref name="SieKM18">D. Siebelts, A. Kater, T. Meurer, [https://doi.org/doi:10.1016/j.ifacol.2018.03.055 Modeling and Motion Planning for an Artificial Fishtail], IFAC-PapersOnLine (9th Vienna International Conference on Mathematical Modelling) 51(2): 319--324, 2018.</ref>
</references>

CD Player

2023-08-29T17:02:37Z

Mlinaric: Edit categories, edit math

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

==Description: Classical CD Player Model==

This benchmark models the swing arm of a [[wikipedia:CD_Player|CD Player]] holding a lens which can be moved in the horizontal plain.
More details can be found in <ref name="draijer92"/>, <ref name="wortelboer96"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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/CDplayer.zip CDplayer.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{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{120 \times 120}</math>,
<math>B \in \mathbb{R}^{120 \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times 120}</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_cdisc,
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{DraSB92,
author = <nowiki>{W. Draijer and M. Steinbuch and O. Bosgra}</nowiki>,
title = {Adaptive Control of the Radial Servo System of a Compact Disc Player},
journal = {Automatica},
volume = {28},
number = {3},
pages = {455--462},
year = {1992},
doi = {10.1016/0005-1098(92)90171-B}
}

==References==

<references>

<ref name="draijer92"> W. Draijer, M. Steinbuch, O. Bosgra. [https://doi.org/10.1016/0005-1098(92)90171-B Adaptive Control of the Radial Servo System of a Compact Disc Player]. Automatica, 28(3): 455--462, 1992.</ref>

<ref name="wortelboer96"> P. Wortelboer, M. Steinbuch, O. Bosgra. [https://research.tue.nl/en/publications/closed-loop-balanced-reduction-with-cation-to-a-compact-disc-mech Closed-Loop Balanced Reduction with Application to a Compact Disc Mechanism]. Selected Topics in Identification, Modeling and Control, 9: 47--58, 1996.</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>

Butterfly Gyroscope

2023-08-29T16:54:00Z

Mlinaric: Fixes in text, math, bib

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Butterfly1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Butterfly2.jpg|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Butterfly3.jpg|490px|thumb|right|Figure 3]]</figure>

The '''butterfly gyro''' is developed at the [http://www.imego.com Imego Institute] in an ongoing project with [[wikipedia:Saab_Bofors_Dynamics|Saab Bofors Dynamics AB]].
The butterfly is a vibrating micro-mechanical gyro that has sufficient theoretical performance characteristics to make it a promising candidate for use in inertial navigation applications.
The goal of the current project is to develop a micro unit for inertial navigation that can be commercialized in the high-end segment of the rate sensor market.
This project has reached the final stage of a three-year phase where the development and research efforts have ranged from model-based signal processing, via electronics packaging to design and prototype manufacturing of the sensor element.
The project has also included the manufacturing of an [[wikipedia:Application-specific_integrated_circuit|ASIC]], named µSIC, that has been especially designed for the sensor (see Fig. 1).

The gyro chip consists of a three-layer silicon wafer stack, in which the middle layer contains the sensor element.
The sensor consists of two wing pairs that are connected to a common frame by a set of beam elements (see Fig. 2 and Fig. 3);
this is the reason the gyro is called the butterfly.
Since the structure is manufactured using an anisotropic wet-etch process, the connecting beams are slanted.
This makes it possible to keep all electrodes, both for capacitive excitation and detection, confined to one layer beneath the two wing pairs.
The excitation electrodes are the smaller dashed areas shown in Fig. 2.
The detection electrodes correspond to the four larger ones.
By applying DC-biased AC voltages to the four pairs of small electrodes, the wings are forced to vibrate in anti-phase in the wafer plane.
This is the excitation mode.
As the structure rotates about the axis of sensitivity (see Fig. 2), each of the masses will be affected by a Coriolis acceleration.
This acceleration can be represented as an inertial force that is applied at right angles with the external angular velocity and the direction of motion of the mass. The Coriolis force induces an anti-phase motion of the wings out of the wafer plane.
This is the detection mode. The external angular velocity can be related to the amplitude of the detection mode, which is measured via the large electrodes.

When planning for and making decisions on future improvements of the butterfly, it is important to improve the efficiency of the gyro simulations.
Repeated analyses of the sensor structure have to be conducted with respect to a number of important issues.
Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases, and the effect of force-feedback.

The use of model order reduction indeed decreases runtimes for repeated simulations.
Moreover, the reduction technique enables a transformation of the FE representation of the gyro into a state space equivalent formulation.
This will prove helpful in testing the model-based Kalman signal processing algorithms that are being designed for the '''butterfly gyro'''.

The structural model of the gyroscope has been done in [http://www.ansys.com/ ANSYS] using quadratic tetrahedral elements (SOLID187, see Fig. 3).
The model shown is a simplified one with a coarse mesh as it is designed to test the model reduction approaches.
It includes the pure structural mechanics problem only. The load vector is composed of time-varying nodal forces applied at the centers of the excitation electrodes (see Fig. 2).
The amplitude and frequency of each force are equal to <math>0.055 \, \mu \text{N}</math> and <math>2384 \, \text{Hz}</math>, respectively.
The Dirichlet boundary conditions have been applied to all degrees of freedom of the nodes belonging to the top and bottom surfaces of the frame.
The output nodes are listed in Table 2 and correspond to the centers of the detection electrodes (see Fig. 3).
The structural model

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) & = C x(t),
\end{align}
</math>

contains the mass <math>M</math> and stiffness matrices <math>K</math>.
The damping matrix <math>E</math> can be modeled as <math>E = \alpha M + \beta K</math>, where the typical values of <math>\alpha</math> and <math>\beta</math> are <math>0</math> and <math>10^{-6}</math> respectively.
The nature of the damping matrix is in reality more complex (squeeze film damping, thermo-elastic damping, etc.) but this simple approach has been chosen with respect to the model reduction test.
<math>B</math> is the load vector, <math>C</math> is the output matrix.

The statistics for the matrices are shown in Table 1.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: System matrices for the gyroscope.''
|-
!Matrix
!m
!n
!nnz
!Is Symmetric?
|-
|M
|17361
|17361
|178896
|Yes
|-
|K
|17361
|17361
|519260
|Yes
|-
|B
|17361
|1
|8
|No
|-
|C
|12
|17361
|12
|No
|}

The outputs are detailed in Table 2.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2. Outputs for the Butterfly Gyro Model.''
|-
!Index
!Code
!Comment
|-
|1-3
|det1m_Ux, det1m_Uy, det1m_Uz
|Displacements of detection electrode 1, (bottom left large electrode of Fig. 2)
|-
|4-6
|det1p_Ux, det1p_Uy, det1p_Uz
|Displacements of detection electrode 2, (bottom right large electrode of Fig. 2)
|-
|7-9
|det2m_Ux, det2m_Uy, det2m_Uz
|Displacements of detection electrode 3, (top left large electrode of Fig. 2)
|-
|10-12
|det2p_Ux, det2p_Uy, det2p_Uz
|Displacements of detection electrode 4, (top right large electrode of Fig. 2)
|}

The model reduction of the gyroscope model by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] is described in
<ref name="lienemann04"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 35889, see: <ref name="billger05"/>.

==Data==

Download matrices in [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ButterflyGyro-dim1e5-gyro.tar.gz ButterflyGyro-dim1e5-gyro.tar.gz] (7.4 MB)

The matrix name is used as an extension of the matrix file. File <tt>*.C.names</tt> contains a list of output names written consecutively.

==Dimensions==

System structure:
:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) & = B u(t) \\
y(t) & = C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{17361 \times 17361}</math>,
<math>E \in \mathbb{R}^{17361 \times 17361}</math>,
<math>K \in \mathbb{R}^{17361 \times 17361}</math>,
<math>B \in \mathbb{R}^{17361 \times 1}</math>,
<math>C \in \mathbb{R}^{12 \times 17361}</math>.

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

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

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

* For the background on the benchmark:

@INPROCEEDINGS{morBil05,
author = {D. Billger},
title = {The Butterfly Gyro},
booktitle = {Dimension Reduction of Large-Scale Systems},
publisher = {Springer-Verlag, Berlin/Heidelberg, Germany},
year = 2005,
volume = 45,
pages = {349--352},
series = {Lecture Notes in Computational Science and Engineering},
doi = {10.1007/3-540-27909-1_18}
}

==References==

<references>

<ref name="lienemann04"> J. Lienemann, D. Billger, E.B. Rudnyi, A. Greiner, and J.G. Korvink, [http://modelreduction.com/doc/papers/lienemann04MSM.pdf MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices], Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA.</ref>

<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="billger05"> D. Billger, [https://doi.org/10.1007/3-540-27909-1_18 The Butterfly Gyro], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 349--352, 2005.</ref>

</references>

Bone Model

2023-08-29T16:43:20Z

Mlinaric: Edit categories

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop a very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example, to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi-static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low-magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but such an analysis needs to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two hundred thousand to twelve million equations with the goal to research on scalability of model reduction software.
Each model represents a second-order system of the form

:<math>
\begin{align}
M \ddot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t),
\end{align}
</math>

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

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

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math>, and <math>UZ</math> for the first node.

The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

Files (in .mat format) can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M, K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{3 \times n}</math>.

System variants:

<tt>BS01</tt>: <math>n = 127224</math>,
<tt>BS10</tt>: <math>n = 914898</math>,
<tt>B010</tt>: <math>n = 986703</math>,
<tt>B025</tt>: <math>n = 2159661</math>,
<tt>B050</tt>: <math>n = 4934244</math>,
<tt>B120</tt>: <math>n = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_bone,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Bone Model},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Bone_Model}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==References==

<references>

<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="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

All pass system

2023-08-29T16:38:19Z

Mlinaric: Edit categories, replace source tags

[[Category:benchmark]]
[[Category:procedural]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Sparse]]
[[Category:SISO]]

==Description==

This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
which means this system has a single Hankel singular value of multiplicity of the system's order.
The system matrices are constructed based on the scheme:

:<math>
\begin{align}
A &=
\begin{pmatrix}
a_{1,1} & -\alpha_1 \\
\alpha_1 & 0 & -\alpha_2 \\
& \alpha_2 & 0 & \ddots \\
& & \ddots & \ddots & -\alpha_{N-1} \\
& & & \alpha_{N-1} & 0
\end{pmatrix}, \\
B &=
\begin{pmatrix}
b_1 \\
0 \\
\vdots \\
0
\end{pmatrix}, \\
C &=
\begin{pmatrix}
s_1 b_1 & 0 & \cdots & 0
\end{pmatrix}, \\
D &= -s_1 \sigma.
\end{align}
</math>

We choose <math>s_1 \in \{-1,1\}</math> to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
Furthermore, <math>b_1 = 1</math> and <math>\sigma = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.

==Data==

This benchmark is procedural and the state dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:


:<syntaxhighlight lang="matlab">
function [A,B,C,D] = allpass(n)
% allpass (all-pass system)
% by Christian Himpe, 2020
% released under BSD 2-Clause License
%*

A = gallery('tridiag',n,-1,0,1);
A(1,1) = -0.5;
B = sparse(1,1,1,n,1);
C = -B';
D = 1;
end
</syntaxhighlight>

The function call requires one argument; the number of states <math>n</math>.
The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

:<syntaxhighlight lang="matlab">
[A,B,C,D] = allpass(n);
</syntaxhighlight>

An equivalent [https://www.python.org/ Python] code is

:<syntaxhighlight lang="python">
from scipy.sparse import diags, lil_matrix

def allpass(n):
A = diags([-1, 0, 1], offsets=[-1, 0, 1], shape=(n, n), format='lil')
A[0, 0] = -0.5
A = A.tocsc()
B = lil_matrix((n, 1))
B[0, 0] = 1
B = B.tocsc()
C = -B.T
D = 1
return A, B, C, D
</syntaxhighlight>

==Dimensions==

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

System dimensions:

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

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system

@MISC{morwiki_allpass,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {All-Pass System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/All_pass_system}</nowiki>,
year = {2020}
}

==References==

<references>

<ref name="Obe87">R.J. Ober. "[https://doi.org/10.1016/S1474-6670(17)55030-2 Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization]", IFAC Proceedings Volumes, 20(5): 181--185, 1987.</ref>

</references>

Building Model

2023-08-29T16:37:54Z

Mlinaric: Edit categories, replace source tags

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

==Description: Motion Problem in a Building==

This benchmark models the displacement of a multi-story building for example during an Earthquake.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Earthquake Model===

==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/build.zip build.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + K_{so} q(t) &= B_{so} u(t), \\
y(t) &= C_{so} \dot{q}(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
0 & C_{so}
\end{pmatrix}
\end{align}
</math>

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 48
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = B[n2:]
Cso = C[:, n2:]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{48 \times 48}</math>,
<math>B \in \mathbb{R}^{48 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 48}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_v \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{24 \times 24}</math>,
<math>B \in \mathbb{R}^{24 \times 1}</math>,
<math>C_v \in \mathbb{R}^{1 \times 24}</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_build,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Building Model

2023-08-29T15:43:05Z

Mlinaric: Small fix, Python code, second-order form

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

==Description: Motion Problem in a Building==

This benchmark models the displacement of a multi-story building for example during an Earthquake.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Earthquake Model===

==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/build.zip build.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<source lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)
</source>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + K_{so} q(t) &= B_{so} u(t), \\
y(t) &= C_{so} \dot{q}(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
0 & C_{so}
\end{pmatrix}
\end{align}
</math>

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<source lang="python">
n = 48
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = B[n2:]
Cso = C[:, n2:]
</source>

==Dimensions==

===First differential order===

System structure:

:<math>
\begin{align}
\dot{x}(t) &= A x(t) + B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{48 \times 48}</math>,
<math>B \in \mathbb{R}^{48 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 48}</math>.

===Second differential order===

System structure:

:<math>
\begin{align}
\ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C_v \dot{x}(t)
\end{align}
</math>

System dimensions:

<math>E, K \in \mathbb{R}^{24 \times 24}</math>,
<math>B \in \mathbb{R}^{24 \times 1}</math>,
<math>C_v \in \mathbb{R}^{1 \times 24}</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_build,
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{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</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>

Branchline Coupler

2023-08-29T14:48:35Z

Mlinaric: Fixes, extraction, system structure and dimensions

{{preliminary}} 

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==

A '''branchline coupler''' (see Fig. 1) is a microwave semiconductor device, which is simulated by the [http://www.maxwells-equations.com/forms.php#harmonic time-harmonic Maxwell's equation].
A 2-section '''branchline coupler''' consists of four strip line ports, coupled to each other by two transversal bridges.
The energy excited at one port is coupled almost in equal shares to the two opposite ports, when considered as a MIMO-system.
Here, only the SISO case is considered.
The '''branchline coupler''' with <math>0.05 \, \text{mm}</math> thickness is placed on a substrate with <math>0.749 \, \text{mm}</math> thickness and relative permittivity
<math> \epsilon_r = 2.2 </math> and zero-conductivity <math> \sigma = 0 \, \text{S/m} </math>.
The simulation domain is confined to a <math> 23.6 \times 22 \times 7 \, \text{mm}^3 </math> box.
The metallic ground plane of the device is represented by the electric boundary condition. The magnetic boundary
condition is considered for the other sides of the structures. The discrete input port with source impedance <math>50 \, \Omega</math>
imposes <math>1 \, \text{A}</math> current as the input. The voltage along the coupled port at the end of the other side of the coupler is
read as the output.

<figure id="fig:branch">
[[File:BranchlineCoupler.png|frame|<caption>Branchline Coupler Model<ref name="hess13"/></caption>]]
</figure>

Considered parameters are the frequency <math>\omega </math> and the relative permeability <math> \mu_r </math> .

The affine form <math> a(u, v; \omega, \mu_r) = \sum_{q=1}^Q \Theta^q(\omega, \mu_r) a^q(u, v) </math> can be established using <math> Q = 2 </math> affine terms.

The discretized bilinear form is <math> a(u, v; \omega, \mu_r) = \sum_{q=1}^Q \Theta^q(\omega, \mu_r) A^q </math>, with matrices <math> A^q </math>.

The matrices corresponding to the bilinear forms <math> a^q( \cdot , \cdot ) </math> as well as the input and output forms and the H(curl) inner product matrix have been assembled
using the [[wikipedia:Finite_Element_Method|Finite Element Method]], resulting in <math>27679</math> degrees of freedom, after removal of boundary conditions.

The coefficient functions are given by:

:<math> \Theta^1(\omega, \mu_r) = \frac{1}{\mu_r} </math>

:<math> \Theta^2(\omega, \mu_r) = -\omega^2. </math>

The parameter domain of interest is <math> \omega \in [1, 10] \cdot 10^9 \, \text{Hz}</math>, where the factor of <math> 10^9 </math> has already been taken into account
while assembling the matrices, while the material variation occurs between <math> \mu_r \in [0.5, 2.0] </math>. The input functional also has a factor of <math> \omega </math>.

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano project].

==Data==

The files are numbered according to their appearance in the summation and can be found here:

* [[Media:branchline_part1.zip|branchline_part1.zip]]
* [[Media:branchline_part2.zip|branchline_part2.zip]]
* [[Media:branchline_part3.zip|branchline_part3.zip]]

Unzipping these files individually will extract:

* <tt>branchline_coupler_MORwiki_matrices.7z.001</tt>
* <tt>branchline_coupler_MORwiki_matrices.7z.002</tt>
* <tt>branchline_coupler_MORwiki_matrices.7z.003</tt>

Extracting those will then give:

* <tt>BranchlineCoupler_A1.mtx</tt>
* <tt>BranchlineCoupler_A2.mtx</tt>
* <tt>BranchlineCoupler_Input.mtx</tt>
* <tt>BranchlineCoupler_Output.mtx</tt>
* <tt>BranchlineCoupler_L2.mtx</tt>
* <tt>BranchlineCoupler_Hcurl.mtx</tt>

==Dimensions==

System structure:

:<math>
\begin{align}
(\Theta_1(\omega, \mu_r) A_1 + \Theta_2(\omega, \mu_r) A_2) x(\omega, \mu_r) & = b \\
y(\omega, \mu_r) & = c^T x(\omega, \mu_r)
\end{align}
</math>

System dimensions:

<math>A_1, A_2 \in \mathbb{R}^{27679 \times 27679}</math>,
<math>b, c \in \mathbb{R}^{27679 \times 1}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Branchline Coupler'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Branchline_Coupler

@MISC{morwiki_branchcouple,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Branchline Coupler},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Branchline_Coupler}</nowiki>,
year = {2013}
}

* For the background on the benchmark:

@ARTICLE{morHesB13,
author = {M.~W. Hess and P. Benner},
title = {Fast Evaluation of Time-Harmonic {M}axwell's Equations Using the Reduced Basis Method},
journal = {{IEEE} Trans. Microw. Theory Techn.},
year = 2013,
volume = 61,
number = 6,
pages = {2265--2274},
doi = {10.1109/TMTT.2013.2258167}
}

==References==

<references>

<ref name="hess13">M. W. Hess, P. Benner, "[https://doi.org/10.1109/TMTT.2013.2258167 Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, 61(6): 2265--2274, 2013.</ref>

</references>

==Contact==

'' [[User:hessm|Martin Hess]]''

Bone Model

2023-08-29T13:27:50Z

Mlinaric: Typos, N->n

[[Category:benchmark]]
[[Category:Oberwolfach]]

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop a very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example, to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi-static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low-magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but such an analysis needs to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two hundred thousand to twelve million equations with the goal to research on scalability of model reduction software.
Each model represents a second-order system of the form

:<math>
\begin{align}
M \ddot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t),
\end{align}
</math>

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

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

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math>, and <math>UZ</math> for the first node.

The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

Files (in .mat format) can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M, K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{3 \times n}</math>.

System variants:

<tt>BS01</tt>: <math>n = 127224</math>,
<tt>BS10</tt>: <math>n = 914898</math>,
<tt>B010</tt>: <math>n = 986703</math>,
<tt>B025</tt>: <math>n = 2159661</math>,
<tt>B050</tt>: <math>n = 4934244</math>,
<tt>B120</tt>: <math>n = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_bone,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Bone Model},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Bone_Model}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==References==

<references>

<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="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

Batch Chromatography

2023-08-29T01:42:22Z

Mlinaric: Edit math, text fixes

[[Category:benchmark]]
[[Category:PDE]]
[[Category:nonlinear]]
[[Category:Parametric]]

==Description==

<figure id="fig:bach">[[File:Fig_BatchChrom.jpg|490px|thumb|right|<caption>Sketch of a batch chromatographic process for the separation of A and B.</caption>]]</figure>

Preparative liquid chromatography as a crucial separation and purification tool has been widely employed in the food, fine chemical, and pharmaceutical industries. Chromatographic separation at the industry scale can be operated either discontinuously or in a continuous mode.
The continuous case will be discussed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- [[wikipedia:Affinity_chromatography#Batch_and_column_setups|batch chromatography]].
The principle of the batch chromatographic process for the binary separation is shown in Fig. 1.
During the injection period <math>t_{inj}</math>, a mixture of products A and B is injected at the inlet of the column packed with a suitable stationary phase.
With the help of the mobile phase, the feed mixture flows through the column.
Since the to-be-separated solutes exhibit different adsorption affinities to the stationary phase, they move at different velocities
and thus separate from each other when exiting the column.
At the column outlet, component A is collected between <math>t_1</math> and <math>t_2</math>, and component B is collected between <math>t_3</math> and <math>t_4</math>.
Here the positions of <math>t_1</math> and <math>t_4</math> are determined by a minimum concentration threshold that the detector can resolve, and the positions of <math>t_2</math> and <math>t_3</math> are determined by the purity specifications imposed on the products.
After the cycle period <math>t_{cyc}:=t_4-t_1</math>, the injection is repeated.
The feed flow rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables.
By properly choosing them, the process can achieve the desired performance criterion, such as production rate, while respecting the product specifications (e.g., purity, recovery yield).

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation.
In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>) in the liquid phase can be written as:

:<math>
\frac{\partial c_i}{\partial t}
+ \frac{1 - \epsilon}{\epsilon} \frac{\partial q_i}{\partial t}
+ u \frac{\partial c_i}{\partial z}
- D_i \frac{\partial^2 c_i}{\partial z^2}
= 0, \qquad
z \in (0, L), \qquad (1)
</math>

where <math>c_i</math> and <math>q_i</math> are the concentrations of solute <math>i</math> in the liquid and solid phases, respectively, <math>u</math> the interstitial liquid velocity, <math>\epsilon</math> the column porosity, <math>t</math> the time coordinate, <math>z</math> the axial coordinate along the column, <math>L</math> the column length, <math>D_i=\frac{uL}{Pe}</math> the axial dispersion coefficient and <math>Pe</math> the [[wikipedia:Péclet_number|Péclet number]].
The adsorption rate is modeled by the LDF approximation:

:<math>
\frac{\partial q_i}{\partial t}
= \kappa_{i} \left(q^{Eq}_i - q_i\right), \qquad
z \in [0, L],
</math>

where <math>\kappa_{i}</math> is the mass-transfer coefficient of component <math>i</math> and <math>q^{Eq}_i</math> is the adsorption equilibrium concentration calculated by the isotherm equation for component <math>i</math>. Here the bi-[[wikipedia:Langmuir_adsorption_model|Langmuir]] isotherm model is used to describe the adsorption equilibrium:

:<math>
q^{Eq}_i
= \frac{H_{i,1}\,c_i}{1 + \sum_{j = A, B}K_{j,1}\,c_j}
+ \frac{H_{i,2}\,c_i}{1 + \sum_{j = A, B}K_{j,2}\,c_j},\;
i = A, B,
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for (1) are specified by the Danckwerts relations:

:<math>
D_i \left.\frac{\partial c_i}{\partial z}\right|_{z = 0}
= u \left(\left.c_i\right|_{z=0} - c^{in}_i\right), \qquad
\left.\frac{\partial c_i}{\partial z}\right|_{z = L}
= 0,
</math>

where <math>c^{in}_i</math> is the concentration of component <math>i</math> at the inlet of the column.
A rectangular injection is assumed for the system and thus

:<math>
c^{in}_i
=
\begin{cases}
c^F_i, & \text{if } t \le t_{inj}, \\
0, & \text{if } t > t_{inj}.
\end{cases}
</math>

Here <math>c^F_i</math> is the feed concentration for component <math>i</math> and <math>t_{inj}</math> is the injection period.
In addition, the column is assumed unloaded initially:

:<math>
c_i(t = 0, z)
= q_i(t = 0,z)
= 0, \quad
z \in [0, L], \;
i = A, B.
</math>

More details about the mathematical modeling for batch chromatography can be found in the literature <ref name="guiochon06"/>.

==Discretization==
In this model, the feed volumetric flow-rate <math>Q</math> and the injection period <math>t_{inj}</math> are considered as the operating parameters, and denoted as the parameter <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we get the full order model (FOM) as follows,

:<math>
\left\{
\begin{aligned}
\mathbf{A} \mathbf{c}_i^{k+1}
&= \mathbf{B} \mathbf{c}_i^{k}
+ d_i^k
- \frac{1 - \epsilon}{\epsilon} \Delta t \mathbf{h}_i^k, \\
\mathbf{q}_i^{k + 1}
&= \mathbf{q}_i^{k}
+ \Delta t \mathbf{h}_i^k,
\end{aligned}
\right.
</math>

where <math>\mathbf{c}_i^k, \mathbf{q}_i^k \in \mathbb{R}^{\mathcal N}, i = A, B</math> are the solution vectors of <math>c_i</math> and <math>q_i</math> at the time instance <math>t = t^k, k = 0, 1, \ldots, K</math>, respectively.
The time step <math>\Delta t </math> is determined by the stability condition.
The equation <math> \mathbf h_i^k = \kappa_{i} (\mathbf q^{Eq}_i - \mathbf q_i^k) </math>, is time- and parameter-dependent, the boldface <math>\mathbf{A,B}</math> are constant matrices. As a result, it is a nonlinear parametric system.

==Generation of ROM==

The reduced order model (ROM) can be obtained by the reduced basis method <ref name="zhang15"/>, which is applicable for nonlinear parametric systems, see
[[Reduced_Basis_PMOR_method|Reduced Basis PMOR method]].
For parametrized time-dependent problems, the reduced basis is often generated by using the POD-Greedy algorithm <ref name="haasdonk08"/>.
Notice that the nonlinear functions <math>\mathbf h_i, i=A, B</math> can be approximated by the empirical interpolation method <ref name="barrault04"/>,
such that the ROM can be obtained efficiently by the strategy of offline-online decomposition.

Assume <math>W_z</math> is the collateral reduced basis (CRB) for the nonlinear operator <math>h_z</math>, and <math>V_{c_z},V_{q_z}</math> are the reduced bases for the field variables <math>c_z</math> and <math>q_z</math>, respectively.
Applying Galerkin projection and empirical operator interpolation, the ROM can be formulated as:

:<math>
\left\{
\begin{aligned}
\hat{A}_{c_z} {a}_{c_z}^{n+1}
&= \hat{B}_{c_z} {a}_{c_z}^{n}
+ d_0^n \hat{d}_{c_z}
- \frac{1 - \epsilon}{\epsilon} \Delta t \hat{H}_{c_z} \beta_z^n, \\
{a}_{q_z}^{n + 1}
&= {a}_{q_z}^{n}
+ \Delta t \hat{H}_{q_z} \beta_z^n,
\end{aligned}
\right.
</math>

where <math> {a}_{c_z}^n, {a}_{q_z}^n \in \mathbb R^N</math> are the solution of the ROM. <math>\hat{A}_{c_z}=V_{c_z}^T A V_{c_z},
\hat{ B}_{c_z}=V_{c_z}^T B V_{c_z},
\hat{d}_{c_z}^{n}=V_{c_z}^T e_1</math>,
<math>\hat{H}_{c_z}:= V_{c_z}^TW_z ,</math>
<math>\hat{H}_{q_z}:= V_{q_z}^TW_z </math> are the reduced matrices, <math>e_1:=(1,0,\cdots,0)</math>.
<math>\beta_z^n \in \mathbb R^M</math> is the coefficients of the CRB <math>W_z</math> for the empirical interpolation.

==Data==
<figure id="fig:cabfr">[[File:cabfrA.png|480px|thumb|right|<caption>Concentrations at the outlet of the column.</caption>]]</figure>

Fig. 2 shows the concentrations at the outlet of the column at a given parameter <math>\mu= (0.1018, 1.3487)</math>, which show that the ROM (<math>N=46, M=151</math>) reproduces the dynamics of the full order model (<math>\mathcal N=1000</math>).

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Batch Chromatography'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Batch_Chromatography

@MISC{morwiki_bone,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Batch Chromatography},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Batch_Chromatography}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{Y. Zhang and L. Feng and S. Li and P. Benner}</nowiki>,
title = {Accelerating {PDE} constrained optimization by the reduced basis method: application to batch chromatography},
journal = {International Journal for Numerical Methods in Engineering},
volume = {104},
number = {11},
pages = {983--1007},
year = {2015},
doi = {10.1002/nme.4950}
}

==References==

<references>

<ref name="guiochon06"> G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, [http://books.google.com/books/about/Fundamentals_of_preparative_and_nonlinea.html?id=UjZRAAAAMAAJ Fundamentals of Preparative and Nonlinear Chromatography], 2nd Edition, Academic Press, 2006.</ref>

<ref name="barrault04"> M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, "[https://doi.org/10.1016/j.crma.2004.08.006 An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations]", C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.</ref>

<ref name="haasdonk08"> B. Haasdonk and M. Ohlberger, "[https://doi.org/10.1051/m2an:2008001 Reduced basis method for finite volume approximations of parameterized linear evolution equations]", Mathematical Modeling and Numerical Analysis, 42 (2008), 277-302.</ref>

<ref name="zhang15"> Y. Zhang, L. Feng, S. Li and P. Benner, "[https://doi.org/10.1002/nme.4950 Accelerating PDE constrained optimization by the reduced basis method: application to batch chromatography]", International Journal for Numerical Methods in Engineering, 104(11): 983--1007, 2015.</ref>

</references>

==Contact==

'' [[User:Zhangy|Yongjin Zhang]]''

'' [[User:Suzhou|Suzhou Li]]''

Artificial Fishtail

2023-08-29T01:28:39Z

Mlinaric: Fixes, N->n

[[Category:benchmark]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:Sparse]]

<figure id="fig:plot1">
[[File:Fishtail.png|4380px|thumb|right|<caption>Schematic 3D-Model-Fishtail</caption>]]
</figure>
<figure id="fig:plot2">
[[File:fishtail_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp</caption>]]
</figure>
<figure id="fig:plot3">
[[File:fishtail_ext_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp_ext</caption>]]
</figure>

==Description==
Today's [[wikipedia:Autonomous_underwater_vehicle|autonomous underwater vehicles]] (AUVs) are a source of noise pollution and inefficiency due to their screw propeller-driven design.
The evolution of fish has, on the other hand, optimized their underwater efficiency and agility over millennia.
The adaption of fish-like drive systems for AUVs is therefore a promising choice.

==Model Description==
This model describes the silicon body of an artificial fishtail supported by a central carbon beam.
The rear part of the fish body without the fins is modeled as a 3D FEM model using linear elasticity.
In the current stage of modeling the tail is rigidly mounted in the front, the states in <math>x</math> represent the displacements of the finite element degrees of freedom.
The fish-like locomotion is enabled by pumping air between two sets of pressure chambers in the left and right halves of the tail.
The single input <math>u</math> of the system is thus the pumping pressure.
The outputs are displacements of certain surface points.
There are two variants of the model.
The first has three outputs representing the displacements of the point of interest, the rear tip of the carbon beam, in the three spatial directions.
For the second variant, six additional points <math>(z_1,z_2,z_3)</math> on the flank are added as outputs, yielding a total of 21 outputs.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! scope="col" style="width: 12ex;" | <math>z_1</math>
! scope="col" style="width: 12ex;" | <math>z_2</math>
! scope="col" style="width: 12ex;" | <math>z_3</math>
|-
| 0.05
| 0.0
| 0.0
|-
| 0.0474526
| 0.0
| 0.0599584
|-
| 0.04032111
| 0.0
| 0.105274
|-
| 0.0326229
| 0.0
| 0.136726
|-
| 0.0250675
| 0.0
| 0.16107
|-
| 0.0168069
| 0.0
| 0.183588
|-
| 0.0
| 0.0
| 0.21
|}

Note that the POI (Point of Interest) is the last row in this table and in Cp_ext in the data files (see below).
The additional outputs show two effects.
On the one hand, for purely input-output-related reduction methods they avoid drastic deviations on the interior states.
On the other hand, they show a smoothing effect for the model's transfer function.

==Origin==
The model was set up and computed at the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Artificial_Fishtail chair of automatic control] at CAU Kiel and first presented in <ref name="SieKM18" />.

==Data==
The model is based on the finite element package [https://www.firedrakeproject.org Firedrake] and uses the material parameters:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|
|<math>\varrho_1</math>
|<math>1.07 \cdot 10^{-3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-style="background-color:#FFFFFF;"
|Hull
|<math>E_1</math>
|<math>0.025 \cdot 10^6</math>
|<math>\frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu_1</math>
|<math>0.48</math>
|
|-
|
|<math>\varrho_2</math>
|<math>1.4 \cdot 10^{3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-
| Beam
|<math> E_2</math>
|<math> 2.96 \cdot 10^{10}</math>
|<math> \frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-
|
|<math>\nu_2</math>
|<math> 0.3</math>
|
|-style="background-color:#FFFFFF;"
| Rayleigh damping
|<math>\alpha_r</math>
|<math> 1.0 \cdot 10^{-4}</math>
|<math>\frac{1}{s}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\beta_r</math>
|<math> 2.0 \cdot 10^{-4}</math>
|<math>\text{s}</math>
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M, E, K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{p \times n}</math>,
with <math>n = 779\,232</math> and <math>m = 1</math>.

The internal damping is modeled as Rayleigh damping <math>E = \alpha_r M + \beta_r K</math> using the coefficients from the table above.

System variants:

* <math>p = 3</math>: <math>C=</math><tt>Cp</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files],
* <math>p = 21</math>: <math>C=</math><tt>Cp_ext</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files].

== Remarks ==
* Physically meaningful inputs are of dimension <math>u(t) = \mathcal{O}(10^3)</math>. As an example, a step signal with around <math>5\,000</math>Pa leads to a horizontal POI displacement of about <math>3</math>cm.
* The interesting operation frequencies are in the range between <math>0</math>Hz and <math>10</math>Hz.
* If required, the finite element mesh behind the model and a CSV file with the output locations are available [https://doi.org/10.5281/zenodo.2565173 separately].
* Warning: the data files are quite large and may exceed the RAM of a typical machine if the user is also running MATLAB.

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

* For the benchmark itself and its data:
@Misc{SieKM19,
author = {Siebelts, D. and Kater, A. and Meurer, T. and Andrej, J.},
title = {Matrices for an Artificial Fishtail},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2019,
doi = {10.5281/zenodo.2558728}
}

* For the background on the benchmark:

@Article{SieKM18,
author = {Siebelts, D. and Kater, A. and Meurer, T.},
title = {Modeling and Motion Planning for an Artificial Fishtail},
journal = {IFAC-PapersOnLine},
volume = 51,
number = 2,
year = 2018,
pages = {319--324},
doi = {10.1016/j.ifacol.2018.03.055},
publisher = {Elsevier {BV}}
}

==References==
<references>
<ref name="SieKM18">D. Siebelts, A. Kater, T. Meurer, [https://doi.org/doi:10.1016/j.ifacol.2018.03.055 Modeling and Motion Planning for an Artificial Fishtail], IFAC-PapersOnLine (9th Vienna International Conference on Mathematical Modelling) 51(2): 319--324, 2018.</ref>
</references>

Anemometer

2023-08-29T01:22:37Z

Mlinaric: Math fixes

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:first differential order]]
[[Category:SISO]]

==Description==

<figure id="fig:plot1">
[[File:Model_Color.pdf|600px|thumb|right|<caption>Schematic 2D-Model-Anemometer</caption>]]
</figure>
<figure id="fig:plot2">

[[file:ContourPlot30.pdf|600px|thumb|right|<caption>Calculated temperature profile for the Anemometer function</caption>]]

</figure>

An '''Anemometer'''<ref name="ernst01" group="a)"/><ref name="benner05" group="a)"/><ref name="moosmann05" group="a)"/><ref name="moosmann07" group="c)"/><ref name="moosmann05b" group="c)"/><ref name="rudnyi06" group="c)"/> (see [[wikipedia:Thermal_mass_flow_meter|thermal mass flow meter]])
is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity Fig. 1.
They are located on a membrane to minimize heat dissipation through the structure.
Without any flow, the heat dissipates symmetrically into the fluid.
This symmetry is disturbed if a flow is applied to the fluid,
which leads to a convection on the temperature field and therefore to a difference between the temperature sensors (see Fig. 2) from which the fluid velocity can be determined.

The physical model can be expressed by the [[wikipedia:Convection–diffusion_equation|convection-diffusion partial differential equation]] <ref name="moosmann04" group="b)"/>:

:<math>
\rho c \frac{\partial T}{\partial t}
= \nabla \cdot (\kappa \nabla T)
- \rho c v \nabla T
+ \dot{q},
</math>

where <math>\rho</math> denotes the mass density, <math>c</math> is the specific heat capacity, <math>\kappa</math> is the thermal conductivity,
<math>v</math> is the fluid velocity, <math>T</math> is the temperature, and <math>\dot q</math> is the heat flow into the system caused by the heater.

The solid model has been generated and meshed in [[wikipedia:ANSYS|ANSYS]].
Triangular [http://www.ansys.stuba.sk/html/elem_55/chapter4/ES4-55.htm PLANE55] elements have been used for the finite element discretization.
The order of the system is <math>n = 29008</math>.

Example with one parameter:

The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation|ODE]] system has the following transfer function
:<math>
G(s, p) = C (s E - A_1 - p (A_2 - A_1))^{-1} B
</math>

with the fluid velocity <math>p(=v)</math> as single parameter.
Here <math>E</math> is the heat capacitance matrix, <math>B</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the [[wikipedia:Finite_Element_Method|FEM]] discretization w.r.t. the spatial variables.
<math>A_i</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection.
Thus, for obtaining pure convection you have to compute <math>A_2 - A_1</math>.

Example with three parameters:

Here, all fluid properties are identified as parameters. Thus, we consider the following transfer function

:<math>
G(s, p_0, p_1, p_2) =
C
(
s \underbrace{(E_s + p_0 E_f)}_{E(p_0)}
- \underbrace{(A_{d,s} + p_1 A_{d,f} + p_2 A_c)}_{A(p_1,p_2)}
)^{-1}
B
</math>

with parameters <math>p_0</math>, <math>p_1</math>, <math>p_2</math> which are combinations of the original fluid parameters <math>\rho</math>, <math>c</math>, <math>\kappa</math>, <math>v</math>: <math>p_0 = \rho c</math>, <math>p_1 = \kappa</math>, and <math>p_2 = \rho c v</math>, see <ref name="baur11" group="c)"/>. So far, we have considered the mass density as fixed, i.e. <math>\rho=1</math>.

==Origin==

* [http://www.imtek.uni-freiburg.de/professuren/simulation/ IMTEK Freiburg, Simulation group], Prof Dr Jan G. Korvink has taken on a position as Director of the Institute of Microstructure Technology (IMT) at the Karlsruhe Institute of Technology (KIT).

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
All matrices (for the one parameter system and for the three parameter case) can be found and uploaded in [[Media:Anemometer.tar.gz|Anemometer.tar.gz]].
The matrix name is used as an extension of the matrix file.
The system matrices have been extracted from ANSYS models by means of [http://simulation.uni-freiburg.de/downloads/mor4fem mor4fem].
For more information about computing the system matrices, the choice of the output, applying the permutation, please look into the [[media:Readme2.pdf|readme file]]. [[File: Readme2.pdf|thumb]]

Example with one parameter:

* <tt>.B</tt>: load vector
* <tt>.E</tt>: heat capacitance matrix
* <tt>.P</tt>: permutation matrix
* <tt>.A</tt>: stiffness matrices (2)

Example with three parameters:

* <tt>.B</tt>: load vector
* <tt>.E</tt>: heat capacitance matrices (2)
* <tt>.A</tt>: stiffness matrices (5)

To test the quality of the reduced order systems, harmonic simulations as well as transient step responses could be computed, see <ref name="baur11" group="c)"/>.

The output matrix <math>C \in \mathbb{R}^{1 \times 29008}</math> is a vector with non-zero elements <math>C_{173} = 1</math> and <math>C_{133} = -1</math>.

==Dimensions==

System structure (1 parameter):

:<math>
\begin{align}
E \dot{x}(t) &= (A_1 + p (A_2 - A_1)) x(t) + B u(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{29008 \times 29008}</math>,
<math>A_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>B \in \mathbb{R}^{29008 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 29008}</math>.

System structure (3 parameter):

:<math>
\begin{align}
(E_1 + p_0 (E_2 - E_1)) \dot{x}(t) &= (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1)) x(t) + B u(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>A_{1,2,3,4,5} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>B \in \mathbb{R}^{29008 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 29008}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_anemom,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Anemometer},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Anemometer}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

==References==

a) About the '''Anemometer'''

<references group="a)">

<ref name="ernst01" group="a)">H. Ernst, "[http://www.freidok.uni-freiburg.de/volltexte/201/ High-Resolution Thermal Measurements in Fluids]," PhD thesis, University of Freiburg, Germany (2001).</ref>

<ref name="benner05" group="a)">P. Benner, V. Mehrmann and D. Sorensen, "[http://dx.doi.org/10.1007/3-540-27909-1 Dimension Reduction of Large-Scale Systems]", Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin/Heidelberg, Germany, 45, 2005.</ref>

<ref name="moosmann05" group="a)">C. Moosmann and A. Greiner, "[http://dx.doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems]", Chapter 16 (pages 341--343) of 2.</ref>

</references>

b) MOR for non-parametrized '''Anemometer'''

<references group="b)">

<ref name="moosmann04" group="b)">C. Moosmann, E. B. Rudnyi, A. Greiner and J. G. Korvink, "[http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf Model Order Reduction for Linear Convective Thermal Flow]",
Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems, THERMINIC2004, 29 Sept - 1 Oct, 2004, Sophia Antipolis, France.</ref>

</references>

c) MOR for parametrized '''Anemometer'''

<references group="c)">

<ref name="baur11" group="c)">U. Baur, P. Benner, A. Greiner, J. G. Korvink, J. Lienemann and C. Moosmann, "[http://dx.doi.org/10.1080/13873954.2011.547658 Parameter preserving model order reduction for MEMS applications]", MCMDS Mathematical and Computer Modeling of Dynamical Systems, 17(4):297--317, 2011.</ref>

<ref name="moosmann07" group="c)">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR - Model Order Reduction for parameterized MEMS applications]", PhD thesis, University of Freiburg, Germany (2007).</ref>

<ref name="moosmann05b" group="c)">C. Moosmann, E. B. Rudnyi, A. Greiner, J. G. Korvink and M. Hornung, "[http://modelreduction.com/doc/papers/moosmann05MSM.pdf Parameter Preserving Model Order Reduction of a Flow Meter]", Technical Proceedings of the 2005 Nanotechnology
Conference and Trade Show, Nanotech 2005, May 8-12, 2005, Anaheim, California, USA, NSTINanotech
2005, vol. 3, p. 684-687.</ref>

<ref name="rudnyi06" group="c)">E. B. Rudnyi, C. Moosmann, A. Greiner, T. Bechtold, J. G. Korvink, "[http://modelreduction.com/doc/papers/rudnyi06mathmod.pdf Parameter Preserving Model Reduction for MEMS System-level Simulation and Design]", Proceedings of MATHMOD 2006, February 8 -
10, 2006, Vienna University of Technology, Austria.</ref>

</references>

==Contact==

[[User:Himpe]]

All pass system

2023-08-29T01:00:53Z

Mlinaric: Fixes, Python code, N->n

[[Category:benchmark]]
[[Category:procedural]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:SISO]]

==Description==

This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
which means this system has a single Hankel singular value of multiplicity of the system's order.
The system matrices are constructed based on the scheme:

:<math>
\begin{align}
A &=
\begin{pmatrix}
a_{1,1} & -\alpha_1 \\
\alpha_1 & 0 & -\alpha_2 \\
& \alpha_2 & 0 & \ddots \\
& & \ddots & \ddots & -\alpha_{N-1} \\
& & & \alpha_{N-1} & 0
\end{pmatrix}, \\
B &=
\begin{pmatrix}
b_1 \\
0 \\
\vdots \\
0
\end{pmatrix}, \\
C &=
\begin{pmatrix}
s_1 b_1 & 0 & \cdots & 0
\end{pmatrix}, \\
D &= -s_1 \sigma.
\end{align}
</math>

We choose <math>s_1 \in \{-1,1\}</math> to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
Furthermore, <math>b_1 = 1</math> and <math>\sigma = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.

==Data==

This benchmark is procedural and the state dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:


:<source lang="matlab">
function [A,B,C,D] = allpass(n)
% allpass (all-pass system)
% by Christian Himpe, 2020
% released under BSD 2-Clause License
%*

A = gallery('tridiag',n,-1,0,1);
A(1,1) = -0.5;
B = sparse(1,1,1,n,1);
C = -B';
D = 1;
end
</source>

The function call requires one argument; the number of states <math>n</math>.
The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

:<source lang="matlab">
[A,B,C,D] = allpass(n);
</source>

An equivalent [https://www.python.org/ Python] code is

:<source lang="python">
from scipy.sparse import diags, lil_matrix

def allpass(n):
A = diags([-1, 0, 1], offsets=[-1, 0, 1], shape=(n, n), format='lil')
A[0, 0] = -0.5
A = A.tocsc()
B = lil_matrix((n, 1))
B[0, 0] = 1
B = B.tocsc()
C = -B.T
D = 1
return A, B, C, D
</source>

==Dimensions==

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

System dimensions:

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

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system

@MISC{morwiki_allpass,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {All-Pass System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/All_pass_system}</nowiki>,
year = {2020}
}

==References==

<references>

<ref name="Obe87">R.J. Ober. "[https://doi.org/10.1016/S1474-6670(17)55030-2 Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization]", IFAC Proceedings Volumes, 20(5): 181--185, 1987.</ref>

</references>

Comparison of Software

2023-07-07T00:16:51Z

Mlinaric: Update for pyMOR 2023.1

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.
For more information on MOR software (especially packages listed in the MORwiki), see <ref name="haasdonk21"/>.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.99 (04.2022)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 1.1 (10.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.2 (Matlab), 1.0 (C, Python, Julia)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause] (Matlab). [https://spdx.org/licenses/GPL-2.0.html GPL-2.0] (C, Python, Julia)
| C, Matlab, Python, Julia
|-
! [[MOR Toolbox]]
| Yes
| (on-going)
| Yes
| (Yes)
| No
| Yes
| Yes
| Yes
|
| 1.1 (Dec. 2020)
| [http://mordigitalsystems.fr/en/ MOR Digital Systems]
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [https://www.mw.tum.de/rt/forschung/modellordnungsreduktion/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.3 (07.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (No)
| Yes
| Yes
|
| 2023.1.0 (07.2023)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

----
See also: [[Further Software]]

References:
<references>

<ref name="haasdonk21">B. Haasdonk "[https://doi.org/10.1515/9783110499001-013 MOR Software]", Model Order Reduction, Volume 3: Applications: 431--460, 2021.</ref>

</references>

PyMOR

2023-07-07T00:14:59Z

Mlinaric: Update for pyMOR 2023.1

[[Category:Software]]
[[Category:Python]]

== Synopsis ==

[https://pymor.org pyMOR] is a [https://opensource.org/licenses/BSD-2-Clause BSD-licensed] software library for building model order reduction applications in the [[:Wikipedia:Python_(programming_language)|Python programming language]].
All algorithms in '''pyMOR''' are formulated in terms of abstract interfaces, allowing generic implementations to work with different backends, from NumPy/SciPy to external partial differential equation solver packages.

== Features ==

Currently, the following model reduction algorithms are provided by '''pyMOR''':
* A generic reduction routine for projection of arbitrary high-dimensional discretizations onto reduced spaces, preserving (possibly nested) affine decompositions of operators and functionals for efficient offline/online decomposition.
* Efficient error estimation for linear affinely decomposed problems.
* Empirical interpolation of arbitrary operators (with efficient evaluation of projected interpolated operators if the operator supports restriction to selected degrees of freedom).
* Parallel adaptive greedy and POD algorithms for reduced space construction.
* Empirical-Interpolation-Greedy and DEIM algorithms for generation of interpolation data for empirical operator interpolation.
* Balanced-based and interpolation-based reduction methods for first-order, second-order, and port-Hamiltonian linear time-invariant systems.
* Model order reduction using artificial neural networks.
* Data-driven model order reduction with Dynamic Mode Decomposition, Loewner matrices, Eigensystem Realization, and parametric AAA.
* [[:Wikipedia:Gram_schmidt|Gram-Schmidt algorithm]] supporting re-orthogonalization for improved numerical accuracy.
* Time-stepping and Newton algorithms, as well as generic iterative linear solvers.
* Low-rank alternating direction implicit (LR ADI) method for large-scale Lyapunov equations and bindings for matric equations solvers in [https://www.slicot.org SLICOT] (via [https://github.com/python-control/Slycot slycot]) and [https://www.mpi-magdeburg.mpg.de/projects/mess Py-M.E.S.S].
* Eigenvalue/pole computation using the implicitly restarted Arnoldi method and the subspace accelerated dominant pole (SAMDP) algorithm.
* Modal truncation for linear time-invariant systems.
* Time-dependent parameters.

All these algorithms are formulated in terms of abstract interfaces for seamless integration with external high-dimensional PDE solvers.
Bindings for the following PDE solver libraries are available:
* [https://www.dealii.org/ deal.II]
* [https://dune-project.org/ DUNE]
* [https://fenicsproject.org/ FEniCS]
* [https://sourceforge.net/projects/ngsolve/ NGSolve]
* [https://scikit-fem.readthedocs.io/ scikit-fem] (experimental)

Pure Python implementations of discretizations using the [https://numpy.org NumPy]/[https://scipy.org SciPy] scientific computing stack are implemented to provide an easy-to-use sandbox for experimentation with new model reduction approaches. '''pyMOR''' offers:
* Structured 1D and 2D grids, as well as an experimental Gmsh-based grid, implementing the same abstract grid interface.
* [[:Wikipedia:Finite_element|Finite element]] and [[:Wikipedia:Finite_volume|finite volume]] operators based on this interface.
* SciPy/[http://crd-legacy.lbl.gov/~xiaoye/SuperLU SuperLU] based iterative and direct solvers for sparse systems.
* [[:Wikipedia:Opengl|OpenGL]], [https://matplotlib.org matplotlib], and [https://github.com/K3D-tools/K3D-jupyter k3d] based visualizations of solutions.

== References ==
* M. Ohlberger, S. Rave, S. Schmidt, S. Zhang. "[https://doi.org/10.1007/978-3-319-05591-6_69 A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries]". Springer Proceedings in Mathematics & Statistics Vol. 78: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Berlin, June 2014.
* R. Milk, S. Rave, F. Schindler. "[https://doi.org/10.1137/15M1026614 pyMOR - Generic Algorithms and Interfaces for Model Order Reduction]". SIAM Journal on Scientific Computing 38(5): S194--S216, 2016.

== Links ==
* Official [https://pymor.org website],
* Development of '''pyMOR''' can be tracked on [https://github.com/pymor/pymor GitHub],
* Online [https://docs.pymor.org documentation].

== Contact ==
For assistance with, and contributions to '''pyMOR''', the developers can be contacted via [https://github.com/pymor/pymor/discussions GitHub discussions].

Comparison of Software

2023-03-30T19:13:38Z

Mlinaric: New pyMOR release

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.
For more information on MOR software (especially packages listed in the MORwiki), see <ref name="haasdonk21"/>.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.99 (04.2022)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 1.1 (10.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.2 (Matlab), 1.0 (C, Python, Julia)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause] (Matlab). [https://spdx.org/licenses/GPL-2.0.html GPL-2.0] (C, Python, Julia)
| C, Matlab, Python, Julia
|-
! [[MOR Toolbox]]
| Yes
| (on-going)
| Yes
| (Yes)
| No
| Yes
| Yes
| Yes
|
| 1.1 (Dec. 2020)
| [http://mordigitalsystems.fr/en/ MOR Digital Systems]
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [https://www.mw.tum.de/rt/forschung/modellordnungsreduktion/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.3 (07.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (No)
| Yes
| Yes
|
| 2022.2.1 (3.2023)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

----
See also: [[Further Software]]

References:
<references>

<ref name="haasdonk21">B. Haasdonk "[https://doi.org/10.1515/9783110499001-013 MOR Software]", Model Order Reduction, Volume 3: Applications: 431--460, 2021.</ref>

</references>

PyMOR

2022-07-21T12:27:15Z

Mlinaric: Improvements and update on pyMOR

[[Category:Software]]
[[Category:Python]]

== Synopsis ==

[https://pymor.org pyMOR] is a [https://opensource.org/licenses/BSD-2-Clause BSD-licensed] software library for building model order reduction applications in the [[:Wikipedia:Python_(programming_language)|Python programming language]].
Implemented algorithms include reduced basis methods for parametric linear and non-linear problems, as well as system-theoretic methods such as balanced truncation and iterative rational Krylov algorithm.
'''pyMOR''' is designed from the ground up for easy integration with external [[List_of_abbreviations#PDE|PDE]] solver packages but also offers Python-based discretizations for getting started easily.

== Features ==

Currently, the following model reduction algorithms are provided by '''pyMOR''':
* A generic reduction routine for projection of arbitrary high-dimensional discretizations onto reduced spaces, preserving (possibly nested) affine decompositions of operators and functionals for efficient offline/online decomposition.
* Efficient error estimation for linear affinely decomposed problems.
* Empirical interpolation of arbitrary operators (with efficient evaluation of projected interpolated operators if the operator supports restriction to selected degrees of freedom).
* Parallel adaptive greedy and [[List_of_abbreviations#POD|POD]] algorithms for reduced space construction.
* Empirical-Interpolation-Greedy and [[List_of_abbreviations#DEIM|DEIM]] algorithms for generation of interpolation data for empirical operator interpolation.
* Balanced-based and interpolation-based reduction methods for first-order and second-order linear time-invariant systems.
* Model order reduction using artificial neural networks.
* Data-driven model order reduction with Dynamic Mode Decomposition
* [[:Wikipedia:Gram_schmidt|Gram-Schmidt algorithm]] supporting re-orthogonalization for improved numerical accuracy.
* Time-stepping and Newton algorithms, as well as generic iterative linear solvers.
* Low-rank alternating direction implicit (LR ADI) method for large-scale Lyapunov equations and bindings for matric equations solvers in [http://slicot.org SLICOT] (via [https://github.com/python-control/Slycot slycot]) and [https://www.mpi-magdeburg.mpg.de/projects/mess Py-M.E.S.S].
* Eigenvalue/pole computation using the implicitly restarted Arnoldi method and the subspace accelerated dominant pole (SAMDP) algorithm.
* Modal truncation for linear time-invariant systems.
* Time-dependent parameters.

All these algorithms are formulated in terms of abstract interfaces for seamless integration with external high-dimensional [[List_of_abbreviations#PDE|PDE]] solvers.
Bindings for the following PDE solver libraries are available:
* [http://www.dealii.org/ deal.II]
* [http://dune-project.org/ DUNE]
* [http://fenicsproject.org/ FEniCS]
* [http://sourceforge.net/projects/ngsolve/ NGSolve]
* [https://scikit-fem.readthedocs.io/ scikit-fem] (experimental)

Pure Python implementations of discretizations using the [https://www.scipy.org NumPy/SciPy] scientific computing stack are implemented to provide an easy-to-use sandbox for experimentation with new model reduction approaches. '''pyMOR''' offers:
* Structured 1D and 2D grids, as well as an experimental Gmsh-based grid, implementing the same abstract grid interface.
* [[:Wikipedia:Finite_element|Finite element]] and [[:Wikipedia:Finite_volume|finite volume]] operators based on this interface.
* SciPy/[http://crd-legacy.lbl.gov/~xiaoye/SuperLU SuperLU] based iterative and direct solvers for sparse systems.
* [[:Wikipedia:Opengl|OpenGL]] and [http://matplotlib.org matplotlib] based visualizations of solutions.

== References ==
* M. Ohlberger, S. Rave, S. Schmidt, S. Zhang. "[http://dx.doi.org/10.1007/978-3-319-05591-6_69 A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries]". Springer Proceedings in Mathematics & Statistics Vol. 78: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Berlin, June 2014.
* R. Milk, S. Rave, F. Schindler. "[https://doi.org/10.1137/15M1026614 pyMOR - Generic Algorithms and Interfaces for Model Order Reduction]". SIAM Journal on Scientific Computing 38(5): S194--S216, 2016.

== Links ==
* Official [https://pymor.org website],
* Development of '''pyMOR''' can be tracked on [https://github.com/pymor/pymor GitHub],
* Online [https://docs.pymor.org documentation].

== Contact ==
For assistance with, and contributions to '''pyMOR''', the developers can be contacted via [https://github.com/pymor/pymor/discussions GitHub discussions].

Comparison of Software

2022-07-21T12:21:06Z

Mlinaric: Update pyMOR version

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.
For more information on MOR software (especially packages listed in the MORwiki), see <ref name="haasdonk21"/>.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.99 (04.2022)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 1.1 (10.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.2 (Matlab), 1.0 (C, Python, Julia)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause] (Matlab). [https://spdx.org/licenses/GPL-2.0.html GPL-2.0] (C, Python, Julia)
| C, Matlab, Python, Julia
|-
! [[MOR Toolbox]]
| Yes
| (on-going)
| Yes
| (Yes)
| No
| Yes
| Yes
| Yes
|
| 1.1 (Dec. 2020)
| [http://mordigitalsystems.fr/en/ MOR Digital Systems]
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [https://www.mw.tum.de/rt/forschung/modellordnungsreduktion/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.3 (07.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (No)
| Yes
| Yes
|
| 2022.1 (07.2022)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

----
See also: [[Further Software]]

References:
<references>

<ref name="haasdonk21">B. Haasdonk "[https://doi.org/10.1515/9783110499001-013 MOR Software]", Model Order Reduction, Volume 3: Applications: 431--460, 2021.</ref>

</references>

User:Mlinaric

2021-09-13T20:41:04Z

Mlinaric:

Petar Mlinarić 
McBryde Room 475 
460 McBryde Hall, Virginia Tech 
225 Stanger Street 
Blacksburg, VA 24061-1026 
United States of America

email: mlinaric@vt.edu

Transmission Lines

2021-07-03T18:23:24Z

Mlinaric: Fix category name

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:DAE index one]]
==Definition==

In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency,
that is, currents with a frequency high enough that their wave nature must be taken into account.
'''Transmission lines''' are used for purposes such as connecting radio transmitters and receivers with their antennas,
distributing cable television signals, and computer network connections.

In many electric circuits, the length of the wires connecting the components can often be ignored.
That is, the voltage on the wire at a given time can be assumed to be the same at all points.
However, when the voltage changes as fast as the signal travels through the wire,
the length becomes important and the wire must be treated as a transmission line, with distributed parameters.
Stated in another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if its length is greater than <math>1/10</math> of the wavelength,
and the interconnect is called "electrically long".
At this length the phase delay and the interference of any reflections on the line (as well as other undesired effects) become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.



==Description==

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods,
such as the [[wikipedia:Partial_element_equivalent_circuit|Partial Element Equivalent Circuit]] (PEEC) method; it stems from the integral equation form of [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The main difference of the PEEC method with other integral-Equation-based techniques, such as the method of moments, resides in the fact that it provides a circuit interpretation of the [[wikipedia:Electric-field_integral_equation|Electric Field Integral Equation]] (EFIE) in terms of partial elements, namely resistances, partial inductances, and coefficients of potential. In the standard approach, volumes and surfaces are discretized into elementary regions, hexahedra, and patches respectively over which the current and charge densities are expanded into a series of basis functions.
Pulse basis functions are usually adopted as expansion and weight functions.
Such choice of pulse basis functions corresponds to assuming constant current density and charge density over the elementary volume (inductive) and surface (capacitive) cells, respectively.
Following the standard Galerkin's testing procedure, topological elements, namely nodes and branches, are generated and electrical lumped elements are identified modeling both the magnetic and electric field coupling.
Conductors are modeled by their ohmic resistance, while dielectrics requires modeling the excess charge due to the dielectric polarization.
Magnetic and electric field coupling are modeled by partial inductances and coefficients of potential, respectively.

The magnetic field coupling between two inductive volume cells <math>\alpha</math> and <math>\beta</math> is described by the partial inductance

:<math> L_{p_{\alpha\beta}}=\frac{\mu}{4\pi}\frac{1}{a_{\alpha}a_{\beta}}\int_{u_{\alpha}}\int_{u_{\beta}}\frac{1}{R_{\alpha\beta}}\,du_{\alpha}\,du_{\beta} </math>

where <math>R_{\alpha\beta}</math> is the distance between any two points in the volumes <math>u_{\alpha}</math> and <math>u_{\beta}</math> with <math>a_{\alpha}</math> and <math>a_{\beta}</math> their cross section. The electric field coupling between two capacitive surface cells <math>\delta</math> and <math>\gamma</math> is modeled by the coefficient of the potential

:<math> P_{\delta\gamma}=\frac{1}{4\pi\epsilon}\frac{1}{S_{\delta}S_{\gamma}}\int_{S_{\delta}}\int_{S_{\gamma}}\frac{1}{R_{\delta\gamma}}\,dS_{\delta}\,dS_{\gamma} </math>

where <math>R_{\delta\gamma}</math> is the distance between any two points on the surfaces <math>\delta</math> and <math>\gamma</math>, while <math>S_{\delta}</math> and <math>S_{\gamma}</math> denote the area of their respective surfaces <ref name="ferranti11"/>.

Generalized Kirchhoff's laws for conductors, when dielectrics are considered, can be rewritten as

:<math> \textbf{P}^{-1}\frac{d\textbf{v}(t)}{dt}-\textbf{A}^T\textbf{i}(t)+\textbf{i}_e(t)=0, </math>

<figure id="fig:peec">[[File:Peec.jpg|400px|frame|<caption>Illustration of PEEC circuit electrical quantities for a conductor elementary cell (Figure from <ref name="ferranti11"/>).</caption>]]</figure>

:<math> -\textbf{A}\textbf{v}(t)-\textbf{L}_p\frac{d\textbf{i}(t)}{dt}-\textbf{v}_d(t)=0, </math>

:<math> \textbf{i}(t)=\textbf{C}_d\frac{d\textbf{v}_d(t)}{dt} </math>

where <math>\textbf{A}</math> is the connectivity matrix, <math>\textbf{v}(t)</math> denotes the node potentials to infinity, <math>\textbf{i}(t)</math> and <math>\textbf{i}_e(t)</math> represent the currents flowing in volume cells and the external currents, respectively, <math>\textbf{v}_d(t)</math> is the excess capacitance voltage drop, which is related to the excess charge by <math>\textbf{v}_d(t)=\textbf{C}_d^{-1}\textbf{q}_d(t)</math>. A selection matrix <math>\textbf{K}</math> is introduced to define the port voltages by selecting node potentials. The same matrix is used to obtain the external currents <math>\textbf{i}_e(t)</math> by the currents <math>\textbf{i}_s(t)</math>, which are of opposite sign with respect to the <math>n_p</math> port currents <math>\textbf{i}_p(t)</math>,

:<math> \textbf{v}_p(t)=\textbf{K}\textbf{v}(t), </math>

:<math> \textbf{i}_e(t)=\textbf{K}^T\textbf{i}_s(t). </math>

An example of PEEC circuit electrical quantities for a conductor elementary cell is illustrated, in the Laplace domain, in <xr id="fig:peec"/>, where the current-controlled voltage sources <math>sL_{p,ij}I_j</math> and the current-controlled current sources <math>I_{cci}</math> model the magnetic and electric coupling, respectively.

Thus, assuming that we are interested in generating an admittance representation having <math>n_p</math> output currents under voltage excitation, and let us denote with <math>n_n</math> the number of nodes, <math>n_i</math> the number of branches where currents flow, <math>n_c</math> the number of branches of conductors, <math>n_d</math> the number of dielectrics, <math>n_d</math> the additional unknowns since dielectrics require the excess capacitance to model the polarization charge, and <math>n_u=n_i+n_d+n_n+n_p</math> the global number of unknowns, and if the [[wikipedia:Modified_nodal_analysis|Modified Nodal Analysis]] (MNA) approach is used, we have:

:::<math> \left[ \begin{array}{cccc} \textbf{P} & \textbf{0}_{n_n,n_i} & \textbf{0}_{n_n,n_d} & \textbf{0}_{n_n,n_p} \\ \textbf{0}_{n_i,n_n} & \textbf{L}_p & \textbf{0}_{n_i,n_d} & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & \textbf{0}_{n_d,n_i} & \textbf{C}_d & \textbf{0}_{n_d,n_p} \\ \textbf{0}_{n_p,n_n} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\frac{d}{dt}\left[ \begin{array}{c}\textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]= </math>

:<math>= - \left[ \begin{array}{cccc}\textbf{0}_{n_n,n_n} & -\textbf{P}\textbf{A}^T & \textbf{0}_{n_n,n_d} & \textbf{P}\textbf{K}^T \\ \textbf{AP} & \textbf{R} & \Phi & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & -\Phi^T & \textbf{0}_{n_d,n_d} & \textbf{0}_{n_d,n_p} \\ -\textbf{K}\textbf{P} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\cdot\left[ \begin{array}{c} \textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]+ \left[ \begin{array}{c}\textbf{0}_{n_n+n_i+n_d,n_p} \\ -\textbf{I}_{n_p,n_p} \end{array}\right] \cdot [ \textbf{v}_p(t) ]. </math>

Here <math>\textbf{0}</math> is a matrix of zeros, <math>\textbf{I}</math> is the identity matrix, both are with appropriate dimensions, and <math>\Phi=\left[ \begin{array}{c} \textbf{0}_{n_c,n_d} \\ \textbf{I}_{n_d,n_d} \end{array}\right]</math>. Then, in a more compact form, the above equation can be written as:

:<math> \left\{ \begin{array}{c} \textbf{C}\frac{d\textbf{x}(t)}{dt}=-\textbf{G}\textbf{x}(t)+\textbf{B}\textbf{u}(t)\\
\textbf{i}_p(t)=\textbf{L}^T\textbf{x}(t) \end{array}\right . \qquad (1)</math>

with <math>\textbf{x}(t)=\left[ \begin{array}{cccc} \textbf{q}(t)\quad\textbf{i}(t)\quad\textbf{v}_d(t)\quad\textbf{i}_s(t) \end{array}\right]^T</math>. Since this is an <math>n_p</math>-port formulation, whereby the only sources are the voltage sources at the <math>n_p</math>-ports nodes, <math>\textbf{B}=\textbf{L}</math> where <math>\textbf{B}\in\mathbb R^{n_u\times n_p}</math> (for more details on this model, refer to <ref name="ferranti11"/>).

==Motivation of MOR==

Since the number of equations produced by 3-D electromagnetic method PEEC is usually very large,
the inclusion of the PEEC model directly into a circuit simulator (like [[wikipedia:SPICE|SPICE]]) is computationally intractable for complex structures,
where the number of circuit elements can be tens of thousands.

==Data==

All data sets (in a MATLAB formatted data, downloadable in [[Media:TransmissionLines.rar|TransmissionLines.rar]]) in <xr id="tab:peec"/> are referred to as the multiconductor '''transmission lines''' in a MNA form, coming from the PEEC method (then, with dense matrices since they are obtained from the integral formulation of Maxwell's equation).
The LTI descriptor systems have the form of, equation <math>(1)</math>, where <math>C\in\mathbb R^{n\times n}</math> (with <math>C=C^T\ge0</math>), <math>G\in\mathbb R^{n\times n}</math> (with <math>G+G^T\ge0</math>), <math>B\in\mathbb R^{n\times m}</math> and <math>L=B^T</math>, <math>x(t)\in\mathbb R^n</math> is the vector of variables (charges, currents and node potential), the input signal <math>u(t)\in\mathbb R^m</math> are the sources (current or voltage generators depending on what one wants to analyze:
the impedances or the admittances) linked to some node, the output <math>y(t)\in\mathbb R^m</math> are the observation across the node where the sources are inserted. An accurate model of the dynamics of these data sets is generated between 10 kHz and 20 GHz.

<figtable id="tab:peec">
{| border="1"
! Name of the data set !! Matrices !! Dimension !! Number of inputs
|-
| <tt>dsysPEEC-MTLn1600m14</tt> || <math>G</math>, <math>B</math>, <math>C</math> (<math>L=B'</math>) || 1600 || 14
|-
| <tt>dsysPEEC-MTLn2654m30</tt> || dss object (*) || 2654 || 30
|-
| <tt>dsysPEEC-MTLn5248m62</tt> || dss object (*) || 5248 || 62
|}
</figtable>

one can extract the matrices with Matlab command:

[G,B,L,D,C] = dssdata(dssObjectName);

e.g., if one wants to work on one of the last two data sets of this table,
just load it into the Matlab Workspace and type the command aforementioned on the Command Windows;
for the first example, once one loads the data, the Workspace shows directly the matrices.
Note that <math>D = 0</math>.

==Dimensions==

System structure:

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

<tt>dsysPEEC-MTLn1600m14</tt>: <math>N = 1600</math>, <math>M = 14</math>,
<tt>dsysPEEC-MTLn2654m30</tt>: <math>N = 2654</math>, <math>M = 30</math>,
<tt>dsysPEEC-MTLn5248m62</tt>: <math>N = 5248</math>, <math>M = 62</math>

==References==

<references>

<ref name="ferranti11"> F. Ferranti, G. Antonini, T. Dhaene, L. Knockaert, and A. E. Ruehli, "[https://doi.org/10.1109/TCPMT.2010.2101912 Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis]", IEEE Transactions on Components, Packaging and Manufacturing Technology, 1(3): 399--409, 2011.</ref>

</references>

==Contact==

[[User:Deluca]]

[[User:Feng]]

PyMOR

2021-07-03T18:16:52Z

Mlinaric: Add to Python category

[[Category:Software]]
[[Category:Python]]

== Synopsis ==

[https://pymor.org pyMOR] is a [https://opensource.org/licenses/BSD-2-Clause BSD-licensed] software library for building model order reduction applications in the [[:Wikipedia:Python_(programming_language)|Python programming language]].
Implemented algorithms include reduced basis methods for parametric linear and non-linear problems, as well as system-theoretic methods such as balanced truncation and iterative rational Krylov algorithm.
'''pyMOR''' is designed from the ground up for easy integration with external [[List_of_abbreviations#PDE|PDE]] solver packages but also offers Python-based discretizations for getting started easily.

== Features ==

Currently, the following model reduction algorithms are provided by '''pyMOR''':
* A generic reduction routine for projection of arbitrary high-dimensional discretizations onto reduced spaces, preserving (possibly nested) affine decompositions of operators and functionals for efficient offline/online decomposition.
* Efficient error estimation for linear affinely decomposed problems.
* Empirical interpolation of arbitrary operators (with efficient evaluation of projected interpolated operators if the operator supports restriction to selected degrees of freedom).
* Parallel adaptive greedy and [[List_of_abbreviations#POD|POD]] algorithms for reduced space construction.
* Empirical-Interpolation-Greedy and [[List_of_abbreviations#DEIM|DEIM]] algorithms for generation of interpolation data for empirical operator interpolation.
* Balanced-based and interpolation-based reduction methods for first-order and second-order linear time-invariant systems.
* Model order reduction using artificial neural networks.
* [[:Wikipedia:Gram_schmidt|Gram-Schmidt algorithm]] supporting re-orthogonalization for improved numerical accuracy.
* Time-stepping and Newton algorithms, as well as generic iterative linear solvers.
* Low-rank alternating direction implicit (LR ADI) method for large-scale Lyapunov equations and bindings for matric equations solvers in [http://slicot.org SLICOT] (via [https://github.com/python-control/Slycot slycot]) and [https://www.mpi-magdeburg.mpg.de/projects/mess Py-M.E.S.S].
* Eigenvalue/pole computation using the implicitly restarted Arnoldi method and the subspace accelerated dominant pole (SAMDP) algorithm.

All these algorithms are formulated in terms of abstract interfaces for seamless integration with external high-dimensional [[List_of_abbreviations#PDE|PDE]] solvers.
Bindings for the following PDE solver libraries are available:
* [http://www.dealii.org/ deal.II]
* [http://dune-project.org/ DUNE]
* [http://fenicsproject.org/ FEniCS]
* [http://sourceforge.net/projects/ngsolve/ NGSolve]

Pure Python implementations of discretizations using the [https://www.scipy.org NumPy/SciPy] scientific computing stack are implemented to provide an easy to use sandbox for experimentation with new model reduction approaches. '''pyMOR''' offers:
* Structured 1D and 2D grids, as well as an experimental Gmsh-based grid, implementing the same abstract grid interface.
* [[:Wikipedia:Finite_element|Finite element]] and [[:Wikipedia:Finite_volume|finite volume]] operators based on this interface.
* SciPy/[http://crd-legacy.lbl.gov/~xiaoye/SuperLU SuperLU] based iterative and direct solvers for sparse systems.
* Algebraic multigrid solvers through pyAMG bindings.
* [[:Wikipedia:Opengl|OpenGL]] and [http://matplotlib.org matplotlib] based visualizations of solutions.

== References ==
* M. Ohlberger, S. Rave, S. Schmidt, S. Zhang. "[http://dx.doi.org/10.1007/978-3-319-05591-6_69 A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries]". Springer Proceedings in Mathematics & Statistics Vol. 78: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Berlin, June 2014.
* R. Milk, S. Rave, F. Schindler. "[https://doi.org/10.1137/15M1026614 pyMOR - Generic Algorithms and Interfaces for Model Order Reduction]". SIAM Journal on Scientific Computing 38(5): S194--S216, 2016.

== Links ==
* Official [https://pymor.org website],
* Development of '''pyMOR''' can be tracked on [https://github.com/pymor/pymor GitHub],
* Online [https://docs.pymor.org documentation].

== Contact ==
For assistance with, and contributions to '''pyMOR''', the developers can be contacted via [https://github.com/pymor/pymor/discussions GitHub discussions].

MESS

2021-07-03T18:13:53Z

Mlinaric: Fix category name

[[Category:Software]]
[[Category:Linear algebra]]
[[Category:sparse]]
[[Category:Software Sparse Methods]]
[[Category:Octave]]
[[Category:MATLAB]]

[http://www.mpi-magdeburg.mpg.de/projects/mess MESS], the '''M'''atrix '''E'''quations and '''S'''parse '''S'''olvers library, is the successor to the [http://www.netlib.org/lyapack/ Lyapack Toolbox] for MATLAB. It is available as a MATLAB toolbox, as well as, a C-library. It is intended for solving large sparse matrix equations as well as problems from model order reduction and optimal control. The C version provides a large set of auxiliary subroutines for sparse matrix computations and efficient usage of modern multicore workstations.

== Features ==
A list of the main features (some partially finished at the current stage) of both the MATLAB and C versions is:
* Solvers for large and sparse matrix Riccati (algebraic and differential) and Lyapunov (algebraic) equations
* Balanced Truncation based MOR for first and second order state space systems and index 1 DAEs
* <math>\mathcal{H}_2</math>-MOR via the IRKA and TSIA algorithms
* Basic tools for Large sparse linear quadratic optimal control problems

'''The C version moreover provides:'''
* sophisticated Multicore parallelism
* compressed file I/O
* uniform access to linear algebra routines
* specially structured Sylvester equation solvers
* interfaces to [http://www.netlib.org/blas BLAS], [http://www.netlib.org/lapack LAPACK], [http://www.cise.ufl.edu/research/sparse/SuiteSparse/ Suitesparse], [http://www.slicot.org Slicot]

== Contact ==
[[User:Saak| Jens Saak]]

RCL Circuit Equations

2021-07-03T18:12:35Z

Mlinaric: Fix category name

{{preliminary}} 
[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:SISO]]
[[Category:DAE_order_unspecified]]

==Description==
These benchmark 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>{http://modelreduction.org/index.php/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>

Synthetic parametric model

2021-07-03T18:11:26Z

Mlinaric: Add Python code

[[Category:benchmark]]
[[Category:procedural]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:parametric 1 parameter]]
[[Category:first differential order]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==

<figure id="fig1">[[File:synth_poles.png|600px|thumb|right|<caption>System poles for different parameter values.</caption>]]</figure>

On this page you will find a synthetic parametric model with one parameter for which one can easily experiment with different system orders, values of the parameter, as well as different poles and residues (see <xr id="fig1"/>).
Also, the decay of the Hankel singular values can be changed indirectly through the parameter.

===Model===

We consider a dynamical system in the frequency domain given by its pole-residue form of the transfer function

:<math>
\begin{align}
H(s,\varepsilon) & = \sum_{k=1}^{N}\frac{r_{k}}{s-p_{k}}\\
& = \sum_{k=1}^{N}\frac{r_{k}}{s-(\varepsilon a_{k} + jb_{k})},
\end{align}
</math>

with <math>p_{k} = \varepsilon a_{k} + jb_{k}</math> the poles of the system, <math>j</math> the imaginary unit, and <math>r_{k}</math> the residues.
The parameter <math>\varepsilon</math> is used to scale the real part of the system poles.
We can write down the state-space realization of the system's transfer function as

:<math>
\begin{align}
H(s,\varepsilon) = \widehat{C}(sI_{N} - (\varepsilon \widehat{A}_{\varepsilon} + \widehat{A}_{0}))^{-1}\widehat{B},
\end{align}
</math>

with the corresponding system matrices <math>\widehat{A}_{\varepsilon} \in \mathbb{R}^{N \times N}</math>, <math>\widehat{A}_{0} \in \mathbb{C}^{N \times N}</math>, <math>\widehat{B} \in \mathbb{R}^{N}</math>, and <math>\widehat{C}^{T} \in \mathbb{C}^{N}</math> given by

:<math>
\begin{align}
\varepsilon\widehat{A}_{\varepsilon} + \widehat{A}_{0}
& = \varepsilon \begin{bmatrix} a_{1} & & \\ & \ddots & \\ & & a_{N} \end{bmatrix}
+ \begin{bmatrix} jb_{1} & & \\ & \ddots & \\ & & jb_{N} \end{bmatrix},\\
\widehat{B} & = \begin{bmatrix}1, & \ldots, & 1 \end{bmatrix}^{T},\\
\widehat{C} & = \begin{bmatrix}r_{1}, & \ldots, & r_{n} \end{bmatrix}.
\end{align}
</math>

One notices that the system matrices <math>\widehat{A}_{0}</math> and <math>\widehat{C}</math> have complex entries.
For rewriting the system with real matrices, we assume that <math>N</math> is even, <math>N=2m</math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.,

:<math>
\begin{align}
p_{1} & = \varepsilon a_{1} + jb_{1},\\
p_{2} & = \varepsilon a_{1} - jb_{1},\\
& \ldots\\
p_{N-1} & = \varepsilon a_{m} + jb_{m},\\
p_{N} & = \varepsilon a_{m} - jb_{m}.
\end{align}
</math>

Corresponding to the system poles, also the residues are written in complex conjugate pairs

:<math>
\begin{align}
r_{1} & = c_{1} + jd_{1},\\
r_{2} & = c_{1} - jd_{1},\\
& \ldots\\
r_{N-1} & = c_{m} + jd_{m},\\
r_N & = c_{m} - jd_{m}.
\end{align}
</math>

Using this, the realization of the dynamical system can be written with matrices having real entries by

:<math>
\begin{align}
A_{\varepsilon} & = \begin{bmatrix} A_{\varepsilon, 1} & & \\ & \ddots & \\ & & A_{\varepsilon, m} \end{bmatrix}, &
A_{0} & = \begin{bmatrix} A_{0, 1} & & \\ & \ddots & \\ & & A_{0, m} \end{bmatrix}, &
B & = \begin{bmatrix} B_{1} \\ \vdots \\ B_{m} \end{bmatrix}, &
C & = \begin{bmatrix} C_{1}, & \cdots, & C_{m} \end{bmatrix},
\end{align}
</math>

with <math>A_{\varepsilon, k} = \begin{bmatrix} a_{k} & 0 \\ 0 & a_{k} \end{bmatrix}</math>, <math>A_{0, k} = \begin{bmatrix} 0 & b_{k} \\ -b_{k} & 0 \end{bmatrix}</math>, <math>B_{k} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}</math>, <math>C_{k} = \begin{bmatrix} c_{k}, & d_{k} \end{bmatrix}</math>.

<figure id="fig2">[[File:synth_freq_resp.png|600px|thumb|right|<caption>Frequency response of synthetic parametric system for different parameter values.</caption>]]</figure>

===Numerical Values===

<figure id="fig3">[[File:synth_hsv.png|600px|thumb|right|<caption>Hankel singular values of synthetic parametric system for different parameter values.</caption>]]</figure>

We construct a system of order <math>N = 100</math>.
The numerical values for the different variables are

* <math>a_{k}</math> equally spaced in the interval <math>[-10^3, -10]</math>,

* <math>b_{k}</math> equally spaced in the interval <math>[10, 10^3]</math>,

* <math>c_{k} = 1</math>,

* <math>d_{k} = 0</math>,

* <math>\varepsilon \in \left[\frac{1}{50}, 1\right]</math>.

The frequency response of the transfer function <math>H(s,\varepsilon) = C(sI_{N}-(\varepsilon A_{\varepsilon} + A_{0}))^{-1}B</math> is plotted for parameter values <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math> in <xr id="fig2"/>.

Other interesting plots result for small values of the parameter <math>\varepsilon</math>.
For example, for <math>\varepsilon = \frac{1}{100}</math> or <math>\frac{1}{1000}</math>, the peaks in the frequency response become more pronounced, since the poles move closer to the imaginary axis.

For <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math>, we also plotted the decay of the Hankel singular values in <xr id="fig3"/>.
Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

==Data and Scripts==

This benchmark includes one data set. The matrices can be downloaded in the [http://math.nist.gov/MatrixMarket/formats.html MatrixMarket] format:
* [[Media:Synth_matrices.tar.gz|Synth_matrices.tar.gz]] (1.28 kB)
The matrix name is used as an extension of the matrix file.

System data of arbitrary even order <math>N</math> can be generated in MATLAB or Octave by the following script:

<syntaxhighlight lang="octave">
N = 100; % Order of the resulting system.

% Set coefficients.
a = -linspace(1e1, 1e3, N/2).';
b = linspace(1e1, 1e3, N/2).';
c = ones(N/2, 1);
d = zeros(N/2, 1);

% Build 2x2 submatrices.
aa(1:2:N-1, 1) = a;
aa(2:2:N, 1) = a;
bb(1:2:N-1, 1) = b;
bb(2:2:N-2, 1) = 0;

% Set up system matrices.
Ae = spdiags(aa, 0, N, N);
A0 = spdiags([0; bb], 1, N, N) + spdiags(-bb, -1, N, N);
B = 2 * sparse(mod(1:N, 2)).';
C(1:2:N-1) = c.';
C(2:2:N) = d.';
C = sparse(C);
</syntaxhighlight>

or in Python:

<syntaxhighlight lang="python">
import numpy as np
import scipy.sparse as sps

N = 100 # Order of the resulting system.

# Set coefficients.
a = -np.linspace(1e1, 1e3, N//2)
b = np.linspace(1e1, 1e3, N//2)
c = np.ones(N//2)
d = np.zeros(N//2)

# Build 2x2 submatrices.
aa = np.empty(N)
aa[::2] = a
aa[1::2] = a
bb = np.zeros(N)
bb[::2] = b

# Set up system matrices.
Ae = sps.diags(aa, format='csc')
A0 = sps.diags([bb, -bb], [1, -1], (N, N), format='csc')
B = np.zeros((N, 1))
B[::2, :] = 2
C = np.empty((1, N))
C[0, ::2] = c
C[0, 1::2] = d
</syntaxhighlight>

Beside that, the plots in <xr id="fig1"/> and <xr id="fig2"/> can be generated in MATLAB and Octave using the following script:

<syntaxhighlight lang="octave">
% Get residues of the system.
r(1:2:N-1, 1) = c + 1j * d;
r(2:2:N, 1) = c - 1j * d;

ep = [1/50; 1/20; 1/10; 1/5; 1/2; 1]; % Parameter epsilon.
jw = 1j * linspace(0, 1.2e3, 5000).'; % Frequency grid.

% Computations for all given parameter values.
p = zeros(2 * length(a), length(ep));
Hjw = zeros(length(ep), 5000);
for k = 1:length(ep)
p(:, k) = [ep(k) * a + 1j * b; ep(k) * a - 1j * b]; % Poles.
[jww, pp] = meshgrid(jw, p(:, k));
Hjw(k, :) = (r.') * (1 ./ (jww - pp)); % Frequency response.
end

% Plot poles.
figure;
plot(real(p), imag(p), '.', 'MarkerSize', 12);
xlabel('Re(p)');
ylabel('Im(p)');
legend( ...
'\epsilon = 1/50', ...
'\epsilon = 1/20', ...
'\epsilon = 1/10', ...
'\epsilon = 1/5', ...
'\epsilon = 1/2', ...
'\epsilon = 1');

% Plot frequency response.
figure;
loglog(imag(jw), abs(Hjw), 'LineWidth', 2);
axis tight;
xlim([6 1200]);
xlabel('frequency (rad/sec)');
ylabel('magnitude');
legend( ...
'\epsilon = 1/50', ...
'\epsilon = 1/20', ...
'\epsilon = 1/10', ...
'\epsilon = 1/5', ...
'\epsilon = 1/2', ...
'\epsilon = 1');
</syntaxhighlight>

or in Python:

<syntaxhighlight lang="python">
import matplotlib.pyplot as plt

# Get residues of the system.
r = np.empty(N, dtype=complex)
r[::2] = c + 1j * d
r[1::2] = c - 1j * d

ep = [1/50, 1/20, 1/10, 1/5, 1/2, 1] # Parameter epsilon.
jw = 1j * np.geomspace(6, 1.2e3, 5000) # Frequency grid.

# Computations for all given parameter values.
p = np.zeros((len(ep), N), dtype=complex)
Hjw = np.zeros((len(ep), len(jw)), dtype=complex)
for k, epk in enumerate(ep):
# Poles.
p[k, :N//2] = epk * a + 1j * b
p[k, N//2:] = epk * a - 1j * b
# Frequency response.
Hjw[k, :] = (r / (jw[:, np.newaxis] - p[k])).sum(axis=1)

# Plot poles.
fig, ax = plt.subplots()
for k, epk in enumerate(ep):
ax.plot(p[k].real, p[k].imag, '.', label=fr'$\varepsilon$ = {epk}')
ax.autoscale(tight=True)
ax.set_xlabel('Re(p)')
ax.set_ylabel('Im(p)')
ax.legend()

# Plot frequency response.
fig, ax = plt.subplots()
for k, epk in enumerate(ep):
ax.loglog(jw.imag, np.abs(Hjw[k]), label=fr'$\varepsilon$ = {epk}', linewidth=2)
ax.autoscale(tight=True)
ax.set_xlabel('frequency (rad/sec)')
ax.set_ylabel('magnitude')
ax.legend()
</syntaxhighlight>

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= (\varepsilon A_{\varepsilon} + A_{0})x(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A_{\varepsilon} \in \mathbb{R}^{N \times N}</math>,
<math>A_{0} \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>

System variants:

<tt>Synth_matrices</tt>: <math>N = 100</math>,
arbitrary even order <math>N</math> by using the [[#scr1|script]]

==Citation==

To cite this benchmark and its data:
::The MORwiki Community, '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Synthetic_parametric_model

@MISC{morwiki_synth_pmodel,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Synthetic parametric model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Synthetic_parametric_model}</nowiki>,
year = 2005
}

==Contact==

''[[User:Ionita]]''

Synthetic parametric model

2021-07-03T18:06:08Z

Mlinaric: Use syntaxhighlight tag, fix order of plots in code

[[Category:benchmark]]
[[Category:procedural]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:parametric 1 parameter]]
[[Category:first differential order]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==

<figure id="fig1">[[File:synth_poles.png|600px|thumb|right|<caption>System poles for different parameter values.</caption>]]</figure>

On this page you will find a synthetic parametric model with one parameter for which one can easily experiment with different system orders, values of the parameter, as well as different poles and residues (see <xr id="fig1"/>).
Also, the decay of the Hankel singular values can be changed indirectly through the parameter.

===Model===

We consider a dynamical system in the frequency domain given by its pole-residue form of the transfer function

:<math>
\begin{align}
H(s,\varepsilon) & = \sum_{k=1}^{N}\frac{r_{k}}{s-p_{k}}\\
& = \sum_{k=1}^{N}\frac{r_{k}}{s-(\varepsilon a_{k} + jb_{k})},
\end{align}
</math>

with <math>p_{k} = \varepsilon a_{k} + jb_{k}</math> the poles of the system, <math>j</math> the imaginary unit, and <math>r_{k}</math> the residues.
The parameter <math>\varepsilon</math> is used to scale the real part of the system poles.
We can write down the state-space realization of the system's transfer function as

:<math>
\begin{align}
H(s,\varepsilon) = \widehat{C}(sI_{N} - (\varepsilon \widehat{A}_{\varepsilon} + \widehat{A}_{0}))^{-1}\widehat{B},
\end{align}
</math>

with the corresponding system matrices <math>\widehat{A}_{\varepsilon} \in \mathbb{R}^{N \times N}</math>, <math>\widehat{A}_{0} \in \mathbb{C}^{N \times N}</math>, <math>\widehat{B} \in \mathbb{R}^{N}</math>, and <math>\widehat{C}^{T} \in \mathbb{C}^{N}</math> given by

:<math>
\begin{align}
\varepsilon\widehat{A}_{\varepsilon} + \widehat{A}_{0}
& = \varepsilon \begin{bmatrix} a_{1} & & \\ & \ddots & \\ & & a_{N} \end{bmatrix}
+ \begin{bmatrix} jb_{1} & & \\ & \ddots & \\ & & jb_{N} \end{bmatrix},\\
\widehat{B} & = \begin{bmatrix}1, & \ldots, & 1 \end{bmatrix}^{T},\\
\widehat{C} & = \begin{bmatrix}r_{1}, & \ldots, & r_{n} \end{bmatrix}.
\end{align}
</math>

One notices that the system matrices <math>\widehat{A}_{0}</math> and <math>\widehat{C}</math> have complex entries.
For rewriting the system with real matrices, we assume that <math>N</math> is even, <math>N=2m</math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.,

:<math>
\begin{align}
p_{1} & = \varepsilon a_{1} + jb_{1},\\
p_{2} & = \varepsilon a_{1} - jb_{1},\\
& \ldots\\
p_{N-1} & = \varepsilon a_{m} + jb_{m},\\
p_{N} & = \varepsilon a_{m} - jb_{m}.
\end{align}
</math>

Corresponding to the system poles, also the residues are written in complex conjugate pairs

:<math>
\begin{align}
r_{1} & = c_{1} + jd_{1},\\
r_{2} & = c_{1} - jd_{1},\\
& \ldots\\
r_{N-1} & = c_{m} + jd_{m},\\
r_N & = c_{m} - jd_{m}.
\end{align}
</math>

Using this, the realization of the dynamical system can be written with matrices having real entries by

:<math>
\begin{align}
A_{\varepsilon} & = \begin{bmatrix} A_{\varepsilon, 1} & & \\ & \ddots & \\ & & A_{\varepsilon, m} \end{bmatrix}, &
A_{0} & = \begin{bmatrix} A_{0, 1} & & \\ & \ddots & \\ & & A_{0, m} \end{bmatrix}, &
B & = \begin{bmatrix} B_{1} \\ \vdots \\ B_{m} \end{bmatrix}, &
C & = \begin{bmatrix} C_{1}, & \cdots, & C_{m} \end{bmatrix},
\end{align}
</math>

with <math>A_{\varepsilon, k} = \begin{bmatrix} a_{k} & 0 \\ 0 & a_{k} \end{bmatrix}</math>, <math>A_{0, k} = \begin{bmatrix} 0 & b_{k} \\ -b_{k} & 0 \end{bmatrix}</math>, <math>B_{k} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}</math>, <math>C_{k} = \begin{bmatrix} c_{k}, & d_{k} \end{bmatrix}</math>.

<figure id="fig2">[[File:synth_freq_resp.png|600px|thumb|right|<caption>Frequency response of synthetic parametric system for different parameter values.</caption>]]</figure>

===Numerical Values===

<figure id="fig3">[[File:synth_hsv.png|600px|thumb|right|<caption>Hankel singular values of synthetic parametric system for different parameter values.</caption>]]</figure>

We construct a system of order <math>N = 100</math>.
The numerical values for the different variables are

* <math>a_{k}</math> equally spaced in the interval <math>[-10^3, -10]</math>,

* <math>b_{k}</math> equally spaced in the interval <math>[10, 10^3]</math>,

* <math>c_{k} = 1</math>,

* <math>d_{k} = 0</math>,

* <math>\varepsilon \in \left[\frac{1}{50}, 1\right]</math>.

The frequency response of the transfer function <math>H(s,\varepsilon) = C(sI_{N}-(\varepsilon A_{\varepsilon} + A_{0}))^{-1}B</math> is plotted for parameter values <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math> in <xr id="fig2"/>.

Other interesting plots result for small values of the parameter <math>\varepsilon</math>.
For example, for <math>\varepsilon = \frac{1}{100}</math> or <math>\frac{1}{1000}</math>, the peaks in the frequency response become more pronounced, since the poles move closer to the imaginary axis.

For <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math>, we also plotted the decay of the Hankel singular values in <xr id="fig3"/>.
Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

==Data and Scripts==

This benchmark includes one data set. The matrices can be downloaded in the [http://math.nist.gov/MatrixMarket/formats.html MatrixMarket] format:
* [[Media:Synth_matrices.tar.gz|Synth_matrices.tar.gz]] (1.28 kB)
The matrix name is used as an extension of the matrix file.

System data of arbitrary even order <math>N</math> can be generated in MATLAB or Octave by the following script:

<syntaxhighlight lang="octave">
N = 100; % Order of the resulting system.

% Set coefficients.
a = -linspace(1e1, 1e3, N/2).';
b = linspace(1e1, 1e3, N/2).';
c = ones(N/2, 1);
d = zeros(N/2, 1);

% Build 2x2 submatrices.
aa(1:2:N-1, 1) = a;
aa(2:2:N, 1) = a;
bb(1:2:N-1, 1) = b;
bb(2:2:N-2, 1) = 0;

% Set up system matrices.
Ae = spdiags(aa, 0, N, N);
A0 = spdiags([0; bb], 1, N, N) + spdiags(-bb, -1, N, N);
B = 2 * sparse(mod(1:N, 2)).';
C(1:2:N-1) = c.';
C(2:2:N) = d.';
C = sparse(C);
</syntaxhighlight>

Beside that, the plots in <xr id="fig1"/> and <xr id="fig2"/> can be generated in MATLAB and Octave using the following script:

<syntaxhighlight lang="octave">
% Get residues of the system.
r(1:2:N-1, 1) = c + 1j * d;
r(2:2:N, 1) = c - 1j * d;

ep = [1/50; 1/20; 1/10; 1/5; 1/2; 1]; % Parameter epsilon.
jw = 1j * linspace(0, 1.2e3, 5000).'; % Frequency grid.

% Computations for all given parameter values.
p = zeros(2 * length(a), length(ep));
Hjw = zeros(length(ep), 5000);
for k = 1:length(ep)
p(:, k) = [ep(k) * a + 1j * b; ep(k) * a - 1j * b]; % Poles.
[jww, pp] = meshgrid(jw, p(:, k));
Hjw(k, :) = (r.') * (1 ./ (jww - pp)); % Frequency response.
end

% Plot poles.
figure;
plot(real(p), imag(p), '.', 'MarkerSize', 12);
xlabel('Re(p)');
ylabel('Im(p)');
legend( ...
'\epsilon = 1/50', ...
'\epsilon = 1/20', ...
'\epsilon = 1/10', ...
'\epsilon = 1/5', ...
'\epsilon = 1/2', ...
'\epsilon = 1');

% Plot frequency response.
figure;
loglog(imag(jw), abs(Hjw), 'LineWidth', 2);
axis tight;
xlim([6 1200]);
xlabel('frequency (rad/sec)');
ylabel('magnitude');
legend( ...
'\epsilon = 1/50', ...
'\epsilon = 1/20', ...
'\epsilon = 1/10', ...
'\epsilon = 1/5', ...
'\epsilon = 1/2', ...
'\epsilon = 1');
</syntaxhighlight>

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= (\varepsilon A_{\varepsilon} + A_{0})x(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A_{\varepsilon} \in \mathbb{R}^{N \times N}</math>,
<math>A_{0} \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>

System variants:

<tt>Synth_matrices</tt>: <math>N = 100</math>,
arbitrary even order <math>N</math> by using the [[#scr1|script]]

==Citation==

To cite this benchmark and its data:
::The MORwiki Community, '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Synthetic_parametric_model

@MISC{morwiki_synth_pmodel,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Synthetic parametric model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Synthetic_parametric_model}</nowiki>,
year = 2005
}

==Contact==

''[[User:Ionita]]''

PyMOR

2020-12-11T17:06:36Z

Mlinaric: Update and edit links

[[Category:Software]]

== Synopsis ==

[https://pymor.org pyMOR] is a [https://opensource.org/licenses/BSD-2-Clause BSD-licensed] software library for building model order reduction applications in the [[:Wikipedia:Python_(programming_language)|Python programming language]].
Implemented algorithms include reduced basis methods for parametric linear and non-linear problems, as well as system-theoretic methods such as balanced truncation and iterative rational Krylov algorithm.
'''pyMOR''' is designed from the ground up for easy integration with external [[List_of_abbreviations#PDE|PDE]] solver packages but also offers Python-based discretizations for getting started easily.

== Features ==

Currently, the following model reduction algorithms are provided by '''pyMOR''':
* A generic reduction routine for projection of arbitrary high-dimensional discretizations onto reduced spaces, preserving (possibly nested) affine decompositions of operators and functionals for efficient offline/online decomposition.
* Efficient error estimation for linear affinely decomposed problems.
* Empirical interpolation of arbitrary operators (with efficient evaluation of projected interpolated operators if the operator supports restriction to selected degrees of freedom).
* Parallel adaptive greedy and [[List_of_abbreviations#POD|POD]] algorithms for reduced space construction.
* Empirical-Interpolation-Greedy and [[List_of_abbreviations#DEIM|DEIM]] algorithms for generation of interpolation data for empirical operator interpolation.
* Balanced-based and interpolation-based reduction methods for first-order and second-order linear time-invariant systems.
* Model order reduction using artificial neural networks.
* [[:Wikipedia:Gram_schmidt|Gram-Schmidt algorithm]] supporting re-orthogonalization for improved numerical accuracy.
* Time-stepping and Newton algorithms, as well as generic iterative linear solvers.
* Low-rank alternating direction implicit (LR ADI) method for large-scale Lyapunov equations and bindings for matric equations solvers in [http://slicot.org SLICOT] (via [https://github.com/python-control/Slycot slycot]) and [https://www.mpi-magdeburg.mpg.de/projects/mess Py-M.E.S.S].
* Eigenvalue/pole computation using the implicitly restarted Arnoldi method and the subspace accelerated dominant pole (SAMDP) algorithm.

All these algorithms are formulated in terms of abstract interfaces for seamless integration with external high-dimensional [[List_of_abbreviations#PDE|PDE]] solvers.
Bindings for the following PDE solver libraries are available:
* [http://www.dealii.org/ deal.II]
* [http://dune-project.org/ DUNE]
* [http://fenicsproject.org/ FEniCS]
* [http://sourceforge.net/projects/ngsolve/ NGSolve]

Pure Python implementations of discretizations using the [https://www.scipy.org NumPy/SciPy] scientific computing stack are implemented to provide an easy to use sandbox for experimentation with new model reduction approaches. '''pyMOR''' offers:
* Structured 1D and 2D grids, as well as an experimental Gmsh-based grid, implementing the same abstract grid interface.
* [[:Wikipedia:Finite_element|Finite element]] and [[:Wikipedia:Finite_volume|finite volume]] operators based on this interface.
* SciPy/[http://crd-legacy.lbl.gov/~xiaoye/SuperLU SuperLU] based iterative and direct solvers for sparse systems.
* Algebraic multigrid solvers through pyAMG bindings.
* [[:Wikipedia:Opengl|OpenGL]] and [http://matplotlib.org matplotlib] based visualizations of solutions.

== References ==
* M. Ohlberger, S. Rave, S. Schmidt, S. Zhang. "[http://dx.doi.org/10.1007/978-3-319-05591-6_69 A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries]". Springer Proceedings in Mathematics & Statistics Vol. 78: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Berlin, June 2014.
* R. Milk, S. Rave, F. Schindler. "[https://doi.org/10.1137/15M1026614 pyMOR - Generic Algorithms and Interfaces for Model Order Reduction]". SIAM Journal on Scientific Computing 38(5): S194--S216, 2016.

== Links ==
* Official [https://pymor.org website],
* Development of '''pyMOR''' can be tracked on [https://github.com/pymor/pymor GitHub],
* Online [https://docs.pymor.org documentation].

== Contact ==
For assistance with, and contributions to '''pyMOR''', the developers can be contacted via [https://github.com/pymor/pymor/discussions GitHub discussions].

Comparison of Software

2020-12-11T17:03:17Z

Mlinaric: Update pyMOR version

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.8 (05.2020)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 1.1 (10.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.0.1 (Matlab), 1.0 (C,Python,Julia)
| [https://spdx.org/licenses/GPL-2.0.html GPL-2.0]
| C, Matlab, Python, Julia
|-
! [[MORE]]
| Yes
| ?
| Yes
| ?
| ?
| ?
| ?
| ?
|
| ?
| ?
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [https://www.mw.tum.de/rt/forschung/modellordnungsreduktion/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.3 (07.2020)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 2020.2.0 (12.2020)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

Thermal Block

2020-10-27T12:25:28Z

Mlinaric: Fix typos

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:parametric 2-5 parameters]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description==
A parametric semi-discretized heat transfer problem with varying heat transfer coefficients, the parameters, on subdomains. This model is also called the ''cookie baking problem'', and can be viewed as a flattened 2-D version of the ''skyscraper problem'' from high-performance computing.

<figure id="fig1">[[File:ThermalBlockDomain.svg|490px|thumb|right|<caption>The computational domain and boundaries.</caption>]]</figure>
<figure id="fig1">[[File:ThermalBlockTend.png|490px|thumb|right|<caption>A sample heat distribution at time 1.0 for parameter choice [100, 0.01, 0.001, 0.0001].</caption>]]</figure>
<figure id="fig1">[[File:ThermalBlockSigmaMagnitude.png|490px|thumb|right|<caption>Sigma magnitude plot of the single parameter variant.</caption>]]</figure>

===Modeling===
Consider a parameter <math>\mu\in{[10^{-6},10^2]}^4\subset\mathbb{R}^{4}</math> and define the heat conductivity <math>\sigma(\xi; \mu)</math> as <math>1.0</math> when <math>\xi\in\Omega_0</math> and <math>\sigma(\xi; \mu)=\mu_i</math> when <math>\xi\in\Omega_i</math>. The heat distribution is governed by the equation:
:<math>
\partial_t \theta(t, \xi; \mu) + \nabla \cdot (- \sigma(\xi; \mu) \nabla \theta(t, \xi; \mu)) = 0,\text{ for } t\in (0,T), \text{ and } \xi \in \Omega,
</math>
with a heat-inflow condition on the left ([[wikipedia:Neumann_boundary_condition|Neumann boundary]])
:<math>
\sigma(\xi; \mu) \nabla \theta(t, \xi; \mu) \cdot n(\xi) = u(t)\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_{in},
</math>
perfect isolation on the top and bottom ([[wikipedia:Boundary_conditions_in_fluid_dynamics#Wall_boundary_condition|Neumann-zero boundary]])
:<math>
\sigma(\xi; \mu) \nabla \theta(t, \xi; \mu) \cdot n(\xi) = 0\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_N,
</math>
and fixed temperature on the right ([[wikipedia:Dirichlet_boundary_condition|Dirichlet boundary]])
:<math>
\theta(t, \xi; \mu) = 0\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_{D},
</math>
and initial condition
:<math>
\theta(0, \xi; \mu) = 0 \text{ for } \xi \in \Omega.
</math>

===Discretization===
For the discretization, [https://fenicsproject.org/ FEniCS] '''2019.1''' was used on a simplicial grid with first order elements. The mesh is generated from the domain specification using [http://gmsh.info/ gmsh] '''3.0.6''' with '<code>clscale</code>' set to <math>0.1</math>. The Python-based source code for the discretization can be found at [https://doi.org/10.5281/zenodo.3691894 Zenodo].

==Origin==
This benchmark was developed for the [https://imsc.uni-graz.at/modred2019/ MODRED 2019] proceedings<ref name="morRavS20/>.

==Data==
The benchmark includes the basic domain description as a gmsh input file, Python scripts for the matrix assembly, simulation in pyMOR, and visualization as VTK, together with the matrices both as one combined file <code>ABCE.mat</code> or separate matrix market files for all matrices. The sources and the <code>ABCE.mat</code> are available for download at [https://doi.org/10.5281/zenodo.3691894 Zenodo].

Note that the heat transfer coefficients are designed as characteristic functions on the domains, such that the system is only well-posed when all entries in <math>\mu</math> are positive.

==Dimensions==
System structure:
:<math>
\begin{align}
E\dot{x}(t) &= (A_0 + \mu_1 A_1 + \mu_2 A_2 + \mu_3 A_3 + \mu_4 A_4) x(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{N \times N}</math>,
<math>A_0 \in \mathbb{R}^{N \times N}</math>,
<math>A_1 \in \mathbb{R}^{N \times N}</math>,
<math>A_2 \in \mathbb{R}^{N \times N}</math>,
<math>A_3 \in \mathbb{R}^{N \times N}</math>,
<math>A_4 \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{4 \times N}</math>,

where <math>N=7\,488</math> for the system matrices given in <code>ABCE.mat</code>.

==Variants==
Besides the full four parameter setup, the model can be used in variations with other numbers of independent parameters. The following two are recommended in the original work and have been investigated in the literature<ref name="morBenW20c"/>,<ref name="morHim20"/>,<ref name="morBenKS20"/>,<ref name="morMliRS20"/>.

===Single parameter===
The interpretation of the thermal block as the "cookie baking" problem with slight variation in the dough leads to an easy one parameter variant. Here the new single parameter <math>\hat\mu\in [ 10^{-6}, 10^2]</math> is chosen such that <math> \mu = \hat\mu\left[0.2, 0.4, 0.6, 0.8\right]. </math>

===Non-parametric===
The system can be used as a standard LTI state-space model. It is suggested to use <math>\mu = \sqrt{10} [0.2, 0.4, 0.6, 0.8]</math>.

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

* For the benchmark itself and its data:
:: S. Rave and J. Saak, '''Thermal Block'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/Thermal_Block

@MISC{morwiki_thermalblock,
author = <nowiki>{Rave, S. and Saak, J.}</nowiki>,
title = {Thermal Block},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Thermal_Block}</nowiki>,
year = 2020
}

* For the background on the benchmark:
:: S. Rave and J. Saak, [https://arxiv.org/abs/2003.00846 '''A Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction''']. e-prints 2003.00846, arXiv, math.NA (2020).

@TECHREPORT{morRavS20,
author = {Rave, S. and Saak, J.},
title = {A Non-Stationary Thermal-Block Benchmark Model for Parametric
Model Order Reduction},
institution = {arXiv},
type = {e-print},
number = {2003.00846},
note = {math.NA},
year = 2020,
url = <nowiki>{https://arxiv.org/abs/2003.00846}</nowiki>
}

==References==
<references>
<ref name="morBenW20c">P. Benner, S. W. R. Werner, [https://arxiv.org/abs/2002.12682 MORLAB -- the Model Order Reduction LABoratory],
e-print 2002.12682, arXiv, cs.MS (2020).</ref>
<ref name="morHim20">C. Himpe, [https://arxiv.org/abs/2002.12226 Comparing (empirical-Gramian-based) model order reduction algorithms], e-prints 2002.12226, arXiv, math.OC (2020).</ref>
<ref name="morBenKS20">P. Benner, M. Köhler, J. Saak, [https://arxiv.org/abs/2003.02088 Matrix equations, sparse solvers: M-M.E.S.S.-2.0.1 – philosophy, features and application for (parametric) model order reduction], eprints 2003.02088, arXiv, cs.MS (2020).</ref>
<ref name="morRavS20">S. Rave, J. Saak, [https://arxiv.org/abs/2003.00846 An Instationary Thermal-Block Benchmark Model for Parametric Model Order Reduction], e-prints 2003.00846, arXiv, math.NA (2020).</ref>
<ref name="morMliRS20">P. Mlinarić, S. Rave, J. Saak, [https://arxiv.org/abs/2003.05825 Parametric model order reduction using pyMOR], e-prints 2003.05825, arXiv, cs.MS (2020).</ref>
</references>

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

PyMOR

2020-07-23T09:40:18Z

Mlinaric: Update

[[Category:Software]]

== Synopsis ==

[https://pymor.org pyMOR] is a [https://opensource.org/licenses/BSD-2-Clause BSD-licensed] software library for building model order reduction applications in the [[:Wikipedia:Python_(programming_language)|Python programming language]].
Implemented algorithms include reduced basis methods for parametric linear and non-linear problems, as well as system-theoretic methods such as balanced truncation and iterative rational Krylov algorithm.
'''pyMOR''' is designed from the ground up for easy integration with external [[List_of_abbreviations#PDE|PDE]] solver packages but also offers Python-based discretizations for getting started easily.

== Features ==

Currently, the following model reduction algorithms are provided by '''pyMOR''':
* A generic reduction routine for projection of arbitrary high-dimensional discretizations onto reduced spaces, preserving (possibly nested) affine decompositions of operators and functionals for efficient offline/online decomposition.
* Efficient error estimation for linear affinely decomposed problems.
* Empirical interpolation of arbitrary operators (with efficient evaluation of projected interpolated operators if the operator supports restriction to selected degrees of freedom).
* Parallel adaptive greedy and [[List_of_abbreviations#POD|POD]] algorithms for reduced space construction.
* Empirical-Interpolation-Greedy and [[List_of_abbreviations#DEIM|DEIM]] algorithms for generation of interpolation data for empirical operator interpolation.
* Balanced-based and interpolation-based reduction methods for first-order and second-order linear time-invariant systems.
* Model order reduction using artificial neural networks.
* [[:Wikipedia:Gram_schmidt|Gram-Schmidt algorithm]] supporting re-orthogonalization for improved numerical accuracy.
* Time-stepping and Newton algorithms, as well as generic iterative linear solvers.
* Low-rank alternating direction implicit (LR ADI) method for large-scale Lyapunov equations and bindings for matric equations solvers in [http://slicot.org SLICOT] (via [https://github.com/python-control/Slycot slycot]) and [https://www.mpi-magdeburg.mpg.de/projects/mess Py-M.E.S.S].
* Eigenvalue/pole computation using the implicitly restarted Arnoldi method and the subspace accelerated dominant pole (SAMDP) algorithm.

All these algorithms are formulated in terms of abstract interfaces for seamless integration with external high-dimensional [[List_of_abbreviations#PDE|PDE]] solvers.
Bindings for the following PDE solver libraries are available:
* [http://www.dealii.org/ deal.II]
* [http://dune-project.org/ DUNE]
* [http://fenicsproject.org/ FEniCS]
* [http://sourceforge.net/projects/ngsolve/ NGSolve]

Pure Python implementations of discretizations using the [https://www.scipy.org NumPy/SciPy] scientific computing stack are implemented to provide an easy to use sandbox for experimentation with new model reduction approaches. '''pyMOR''' offers:
* Structured 1D and 2D grids, as well as an experimental Gmsh-based grid, implementing the same abstract grid interface.
* [[:Wikipedia:Finite_element|Finite element]] and [[:Wikipedia:Finite_volume|finite volume]] operators based on this interface.
* SciPy/[http://crd-legacy.lbl.gov/~xiaoye/SuperLU SuperLU] based iterative and direct solvers for sparse systems.
* Algebraic multigrid solvers through pyAMG bindings.
* [[:Wikipedia:Opengl|OpenGL]] and [http://matplotlib.org matplotlib] based visualizations of solutions.

== References ==
* M. Ohlberger, S. Rave, S. Schmidt, S. Zhang. "[http://dx.doi.org/10.1007/978-3-319-05591-6_69 A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries]". Springer Proceedings in Mathematics & Statistics Vol. 78: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Berlin, June 2014.
* R. Milk, S. Rave, F. Schindler. "[https://doi.org/10.1137/15M1026614 pyMOR - Generic Algorithms and Interfaces for Model Order Reduction]". SIAM Journal on Scientific Computing 38(5): S194--S216, 2016.

== Links ==
* Official website https://pymor.org
* Development of '''pyMOR''' can be tracked on [https://github.com/pymor/pymor GitHub],
* Online [https://docs.pymor.org documentation]

== Contact ==
For assistance with, and contributions to '''pyMOR''', the developers can be contacted via [http://listserv.uni-muenster.de/mailman/listinfo/pymor-dev pymor-dev@listserv.uni-muenster.de]

Comparison of Software

2020-07-23T09:29:12Z

Mlinaric: Fix date

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.8 (05.2020)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.2 (04.2018)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.0.1 (Matlab), 1.0 (C,Python,Julia)
| [https://spdx.org/licenses/GPL-2.0.html GPL-2.0]
| C, Matlab, Python, Julia
|-
! [[MORE]]
| Yes
| ?
| Yes
| ?
| ?
| ?
| ?
| ?
|
| ?
| ?
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [http://www.rt.mw.tum.de/forschung/morlab/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.2.1 (11.2018)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 2020.1.1 (07.2020)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

Comparison of Software

2020-07-23T09:26:05Z

Mlinaric: Update pyMOR version

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.8 (05.2020)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.2 (04.2018)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.0.1 (Matlab), 1.0 (C,Python,Julia)
| [https://spdx.org/licenses/GPL-2.0.html GPL-2.0]
| C, Matlab, Python, Julia
|-
! [[MORE]]
| Yes
| ?
| Yes
| ?
| ?
| ?
| ?
| ?
|
| ?
| ?
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 5.0 (08.2019)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [http://www.rt.mw.tum.de/forschung/morlab/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.2.1 (11.2018)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 2020.1.1 (07.2019)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}

Further Software

2020-04-29T10:34:16Z

Mlinaric: /* Other Model Reduction Software */ Update MOR entry

[[Category:Software]]

Besides the projects in [[:Category:Software|Software]] category, the following model reduction software packages exist:

==Other Model Reduction Software==

* [http://git.io/hapod hapod] - Hierarchical Approximate Proper Orthogonal Decomposition (MATLAB/OCTAVE)
* [http://www.morepas.org/software/kermor/ KerMor] - Kernel Methods for Model Order Reduction (MATLAB)
* [https://github.com/LLNL/libROM libROM] - Model reduction library with an emphasis on large scale parallelism and linear subspace methods (C)
* [https://mordigitalsystems.fr MOR toolbox] - Model reduction toolbox for MATLAB
* [https://modred.readthedocs.io modred] - A Parallelized Model Reduction Library (Python)
* [http://homepages.rpi.edu/~hahnj/Model_Reduction/ Nonlinear Model Reduction Routines] (MATLAB)
* [https://github.com/Pressio/pressio pressio] - Projection-based model reduction for nonlinear dynamical systems (C++)
* [http://www.rt.mw.tum.de/forschung/morlab/software/psssmor/ psssMOR] - Parametric Sparse State-Space and Model Order Reduction Toolbox (MATLAB)
* [https://github.com/CurtinIC/pyROM pyROM] - Reduced Order Modelling Framework for Python
* [https://redbkit.github.io/redbKIT/ redbKIT] - a MATLAB(R) library for reduced-order modeling of parametrized PDEs
* [http://simplifytoolbox.tumblr.com/ SiMpLIfy] - Structured ModeL reductIon Toolbox for MATLAB
* [http://slicot.org/matlab-toolboxes/model-reduction SLICOT] - Model and Controller Reduction Toolbox (MATLAB)
* [https://zenodo.org/record/2553902 SOMDDPA] - Second-Order Modally-Damped Dominant Pole Algorithm (MATLAB)
* [http://sumo.intec.ugent.be/SUMO SUMO] - SUrrogate MOdeling (SUMO) Toolbox (MATLAB)
* [https://github.com/jeffrey-hokanson/sysmor SYSMOR] - System-Theoretic Model Order Reduction (Python)

==Discontinued Model Reduction Software==

*[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.580.2500&rep=rep1&type=pdf A MATLAB Toolbox for Teaching Model Order Reduction Techniques]
* [https://doi.org/10.1109/37.24832 EXPRED] - Expert system for optimization of model reduction techniques (M1)
* [https://web.archive.org/web/20080920171815/http://www.imtek.uni-freiburg.de/simulation/mor4ansys/ mor4ansys] - Model Order Reduction for ANSYS (now [https://www.cadfem.de/produkte/cadfem-ansys-extensions/model-reduction-inside-ansys.html ANSYS extension])
* [https://web.archive.org/web/20080727002739/http://scowl.ge.uiuc.edu/~ssivakum/research.html MRedTool] - Model Reduction of Multi-dimensional and Uncertain systems (MATLAB)
* [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.453.5492&rep=rep1&type=pdf NLMR] - MATLAB toolbox for exploring projection-based nonlinear model reduction
* [https://doi.org/10.1109/ICICIC.2007.436 OPTIMRED] - A Software Application for Computation of Optimal Reduced Model (MATLAB)
* [https://web.archive.org/web/20130620070226/http://www3.uji.es/~quintana/plic/plicmr/ PLiCMR] - Parallel Library in Control: Model Reduction (integrated into PSLICOT)
* [https://web.archive.org/web/20151124062504/http://augustine.mit.edu:80/methodology/methodology_rbMIT_System.htm rbMIT] - reduced basis MIT (MATLAB)
* [https://www.robotic.de/94 RASP-MODRED] - Regelungstechnische Analyse und Synthese Programme Model Reduction (integrated into SLICOT)
* [https://web.archive.org/web/20151115030532/http://bnbond.com/software/smores/ SMORES] - A Matlab tool for Simulation and Model Order Reduction of Electrical Systems
* [https://web.archive.org/web/20130618203457/http://www3.uji.es/~quintana/plic/spared/index.html SpaRed] - A Parallel Library for Model Reduction (C)

MOR Wiki:Current events

2020-04-29T10:30:23Z

Mlinaric: /* Upcoming Workhops and Conferences */ Add SAMM20

[[Category:Miscellaneous]]

== MOR Wiki Users Meetings ==

* The first general assembly of MOR Wiki users took place on December 10th - 2013, at the [http://mpim.iwww.mpg.de/research/groups/csc MPI in Magdeburg] (Germany).

* The second MOR Wiki meeting took place on April 23rd - 2015, at the [http://mpim.iwww.mpg.de/research/groups/csc MPI in Magdeburg] (Germany).

* The third MOR Wiki meeting took place on January 11th - 2017, during the [http://www.mpi-magdeburg.mpg.de/csc/events/modred2017 ModRed in Odense] (Denmark).

== Upcoming Workhops and Conferences ==

* [https://icerm.brown.edu/programs/sp-s20/ Model and dimension reduction in uncertain and dynamic systems], January 27th - May 1st, 2020; Providence (USA)
** [https://icerm.brown.edu/programs/sp-s20/w1/ Mathematics of Reduced Order Models], February 17th - 21st
** [https://icerm.brown.edu/programs/sp-s20/w2/ Algorithms for Dimension and Complexity Reduction], March 23rd - 27th ('''VIRTUAL''')
** [https://icerm.brown.edu/programs/sp-s20/w3/ Computational Statistics and Data-Driven Models], April 20th - 24th ('''VIRTUAL''')

* [https://www.wccm-eccomas2020.org 14th WWCM and ECCOMAS Congress 2020], July 19th - 24th, 2020; Paris (France)
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a93.pdf MS93: Reduced Order and Surrogate Modeling for Uncertainty Analysis in Structural Mechanics]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a139.pdf MS139: Advances in Intrusive and Non-Intrusive Order Reduction Techniques for Flow Analysis, Control and Optimization]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a152.pdf MS152: Model Order Reduction Methods for Parametrized Mechanical Systems]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a170.pdf MS170: Numerical Model Reduction and Data-Driven Surrogates for Multi-Physics Applications]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a179.pdf MS179: Numerical Techniques for the Simulations and Model Reduction of Complex Physical Systems]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a204.pdf MS204: Analysing Parameterised Reduced Order Models]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a371.pdf MS371: Model Order Reduction for Vibroacoustic Problems]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a415.pdf MS415: Model Order Reduction for Nonlinear (Time, Space, Parameter) Multiscale Problems and Applications]"
** "[https://www.wccm-eccomas2020.org/admin/Files/FileAbstract/a458.pdf MS458: Coupled Multiphysics Problems and Reduced Order Methods Applied to Compute Digital Twin Models in Industrial Applications]"

* [https://www.mpi-magdeburg.mpg.de/csc/events/samm20 SAMM20 - Learning Models from Data: Model Reduction, System Identification and Machine Learning], July 27th - 31st, 2020 ('''VIRTUAL''')

* [https://www.mfo.de/occasion/2045a/www_view Banach Center – Oberwolfach Graduate Seminar: Model Reduction and Approximation: Projection-, Tensor- and Data-based Methods], November 1st - 7th, 2020; Bedlewo (Poland)

* [https://ims.nus.edu.sg/events/2021/wred/ International Workshop on Reduced Order Methods], May 17th - 21st, 2021; Singapore (Singapore)

* [https://conferences.cirm-math.fr/2021-calendar.html Data Assimilation and Model Reduction in High-Dimensional Problems], July 17th - August 27th, 2021; Luminy (France)

* MoRePaS 2021 - Model Reduction of Parametrized Systems V, November 8th - 12th, 2021; Austin (USA)

== Past Events ==

=== 2020 ===

* [https://siam-uq20.ma.tum.de SIAM Conference on Uncertainty Quantification (UQ20)], March 24th - 27th, 2020; Munich (Germany) ('''CANCELLED''')
** "[https://www.events.tum.de/frontend/index.php?page_id=3712&v=List&do=15&day=490&ses=1625# CT20: ROM and Surrogate Models]"
** "[https://www.events.tum.de/frontend/index.php?page_id=3712&v=List&do=15&day=489&ses=1723# MS311: Dynamical low rank and reduced basis methods for random or parametric time dependent problems (Part I of II) ]"
** "[https://www.events.tum.de/frontend/index.php?page_id=3712&v=List&do=15&day=489&ses=1724# MS312: Dynamical low rank and reduced basis methods for random or parametric time dependent problems (Part II of II)]"
** "[https://www.events.tum.de/frontend/index.php?page_id=3712&v=List&do=15&day=490&ses=1717# MS331: Recent Advances in Reduced-Order Models for Many Query and Time-Critical Problems]"
** "[https://www.events.tum.de/frontend/index.php?page_id=3712&v=List&do=15&day=491&ses=1774# MS351: Reduced order methods for uncertainty quantification in CFD parametric problems]"

=== 2019 ===

* [https://mortech2019.sciencesconf.org/ MORTech - 5th International Workshop on Reduced Basis, POD or PGD-Based Model Reduction Technique], November 20th - 22nd, 2019; Paris (France)

* [https://www.tudelft.nl/en/events/2019/dcse/dcse-fall-school-november-4-8-2019-on-reduced-order-modeling-and-uncertainty-quantification/ DCSE Fall School November 4-8, 2019, on "Reduced-Order Modeling and Uncertainty Quantification"] November 4th - 8th, 2019; Delft (Netherlands)

* [https://school.pymor.org/ pyMOR School 2019], October 7th - 11th, 2019; Magdeburg (Germany)

* [https://www.enumath2019.eu ENUMATH 2019], September 30th - October 4th, 2019; Egmond aan Zee (Netherlands)
** "[https://www.enumath2019.eu/program/show_slot/30 MS2: Recent Advances in Model Order Reduction]"
** "[https://www.enumath2019.eu/program/show_slot/76 MS14: Reduced Order Models for parametric PDEs: special focus on time-dependent phenomena and time-harmonic wave problems (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/87 MS14: Reduced Order Models for parametric PDEs: special focus on time-dependent phenomena and time-harmonic wave problems (Part 2)]"
** "[https://www.enumath2019.eu/program/show_slot/113 MS18: Model order reduction in optimisation, control, and data assimilation (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/124 MS18: Model order reduction in optimisation, control, and data assimilation (Part 2)]"
** "[https://www.enumath2019.eu/program/show_slot/6 MS26: New challenges and opportunities for model order reduction (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/17 MS26: New challenges and opportunities for model order reduction (Part 2)]"
** "[https://www.enumath2019.eu/program/show_slot/7 MS29: Low-rank modelling in uncertainty quantification (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/18 MS29: Low-rank modelling in uncertainty quantification (Part 2)]"
** "[https://www.enumath2019.eu/program/show_slot/8 MS31: Numerical methods for identification and model reduction of nonlinear systems (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/19 MS31: Numerical methods for identification and model reduction of nonlinear systems (Part 2)]"
** "[https://www.enumath2019.eu/program/show_slot/151 MS38: Modeling of reduced order submanifolds in non-linear spaces (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/162 MS38: Modeling of reduced order submanifolds in non-linear spaces (Part 2)]"
** "[https://www.enumath2019.eu/program/show_slot/121 Computing and Model Order Reduction (Part 1)]"
** "[https://www.enumath2019.eu/program/show_slot/132 Computing and Model Order Reduction (Part 2)]"

* [https://www.win.tue.nl/~hbansal/morss.html Model Order Reduction Summer School - MORSS], September 23rd - 27th, 2019; Eindhoven (Netherlands)

* [https://congress.cimne.com/complas2019/frontal/default.asp International Conference on Computational Plasticity. Fundamentals and Applications.], September 3rd - 5th, 2019; Barcelona (Spain)
** "[https://congress.cimne.com/complas2019/frontal/ProgramPrint.asp?id=WeM1 Model Order Reduction with Emphasis on Non-linear and Multi-scale Problems I]"
** "[https://congress.cimne.com/complas2019/frontal/ProgramPrint.asp?id=WeE1 Model Order Reduction with Emphasis on Non-linear and Multi-scale Problems II]"

* [https://imsc.uni-graz.at/modred2019 4th Workshop on Model Reduction of Complex Dynamical Systems - MODRED], August 28th - 30th, 2019; Graz (Austria)

* [http://15.usnccm.org US National Congress on Computational Mechanics (USNCCM)], July 28th - 1st August, 2019; Austin (USA)
** "[http://15.usnccm.org/208 Data Assimilation in Model Order Reduction Techniques for Computational Mechanics]"
** "[http://15.usnccm.org/1001 Model Order Reduction for Computational Continuum Mechanics]"

* [https://iciam2019.org International Congress on Industrial and Applied Mathematics], July 15th - 19th, 2019; Valencia (Spain)
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FT-2-4%201 MS FT-2-4 1: New trends in dimensionality reduction of parametrized and stochastic PDEs - Part 1 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FT-2-4%202 MS FT-2-4 2: New trends in dimensionality reduction of parametrized and stochastic PDEs - Part 2 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-3-3%202 MS GH-3-3 2: Model order reduction methods and their broad applications in engineering - Part 1 of 3]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-3-3%203 MS GH-3-3 3: Model order reduction methods and their broad applications in engineering - Part 2 of 3]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-3-3%204 MS GH-3-3 4: Model order reduction methods and their broad applications in engineering - Part 3 of 3]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20A1-1-1%205 MS A1-1-1 5: Advances in reduced order methods for parameter-dependent problems - Part 1 of 1]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FE-1-3%205 MS FE-1-3 5: Network based model reduction in large-scale simulations, imaging and data-science - Part 1 of 3]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FE-1-3%206 MS FE-1-3 6: Network based model reduction in large-scale simulations, imaging and data-science - Part 2 of 3]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FE-1-3%207 MS FE-1-3 7: Network based model reduction in large-scale simulations, imaging and data-science - Part 3 of 3]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-3-3%205 MS GH-3-3 5: Model Reduction and Coupled Problems in Industry Applications - Part 1 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-3-3%206 MS GH-3-3 6: Model Reduction and Coupled Problems in Industry Applications - Part 2 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FE-1-G%207 MS FE-1-G 7: Model-reduction, randomization, and other techniques for large-scale inversion and UQ - Part 1 of 1]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FT-2-4%207 MS FT-2-4 7: Reduced Order Modeling for Parametric CFD Problems - Part 1 of 4]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FT-2-4%208 MS FT-2-4 8: Reduced Order Modeling for Parametric CFD Problems - Part 2 of 4]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FT-2-4%209 MS FT-2-4 9: Reduced Order Modeling for Parametric CFD Problems - Part 3 of 4]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FT-2-4%2010 MS FT-2-4 10: Reduced Order Modeling for Parametric CFD Problems - Part 4 of 4]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-0-1%207 MS GH-0-1 7: Reduced-order modeling and data-driven estimation in waves and fluids - Part 1 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20GH-0-1%208 MS GH-0-1 8: Reduced-order modeling and data-driven estimation in waves and fluids - Part 2 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FE-1-G%209 MS FE-1-G 9: Recent Advancements in Model Reduction for Stochastic and Nonlinear Systems - Part 1 of 2]"
** "[https://iciam2019.com/programa/sesiones.html?codSes=MS%20FE-1-G%2010 MS FE-1-G 10: Recent Advancements in Model Reduction for Stochastic and Nonlinear Systems - Part 2 of 2]"
* [https://indico.sissa.it/event/34/ Summer School on Reduced Order Methods in Computational Fluid Dynamics], July 8th - 12th, 2019; Trieste (Italy)
* [https://2019.compdyn.org 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering - COMPDYN], June 24th - 26th, 2019; Crete (Greece)
** "MS 5: Surrogate and Reduced-Order Modeling for Stochastic Simulation of Physical Systems"
** "MS 6: Uncertainty Computations with Reduced Order Models and Low-Rank Representations"
** "MS 27: Advances in model reduction techniques in computational structural dynamics"
* [http://modelreduction.net/workshops/7th-international-workshop/ 7th International Workshop on Model Reduction in Reactive Flow (IWMRRF)], June 18th - 21st, 2019; Trondheim (Norway)
* [https://isc.tamu.edu/events/Spring2019/ Spring 2019 Model Reduction Workshop] April 29th, 2019; Houston (USA)
* [https://www.dropbox.com/s/yqqnyxufa36tfdf/EECI-IGSC-2019-Model%20Reduction.pdf?dl=0 Model Reduction for Linear and Nonlinear Systems], April 29th - May 3rd, 2019; London (England)
* [http://www.math.sissa.it/course/phd-course/reduced-order-methods-computational-mechanics Reduced Order Methods for Computational Mechanics], April 15th - 18th, 2019; Trieste (Italy)
* [https://cmsa.fas.harvard.edu/machine-learning/ Machine Learning for Multiscale Model Reduction Workshop], March 27th - 29th, 2019; Cambridge (USA)
* [https://www.siam.org/Conferences/CM/Main/cse19 2019 SIAM Conference on Computational Science and Engineering], February 25th - March 1st, 2019; Spokane (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65932 MS10: Model Reduction, Adaptivity, and High Dimensionality in Uncertainty Quantification - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65933 MS44: Model Reduction, Adaptivity, and High Dimensionality in Uncertainty Quantification - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65806 MS14: Emerging Trends for Structure Preserving Model Order Reduction - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65807 MS49: Emerging Trends for Structure Preserving Model Order Reduction - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65793 MS25: Discovering and Exploiting Low-dimensional Structures in Computational Models - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65794 MS60: Discovering and Exploiting Low-dimensional Structures in Computational Models - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66052 MS93: Nonlinear Reduced Order Modeling of Realistic Engineering Fluid Flows]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65894 MS112: Recent Advances in Model Reduction and Uncertainty Quantification - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65895 MS145: Recent Advances in Model Reduction and Uncertainty Quantification - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65632 MS119: Model Reduction for Problems with Strong Convection, Sharp Gradients, and Discontinuities - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65633 MS153: Model Reduction for Problems with Strong Convection, Sharp Gradients, and Discontinuities - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65903 MS131: Homogenization and Reduced Order Modelling for Wave Equations - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65904 MS165: Homogenization and Reduced Order Modelling for Wave Equations - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66036 MS188: Reduced Order Modeling for Parametric CFD Problems- Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66037 MS221: Reduced Order Modeling for Parametric CFD Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65744 MS191: New Challenges and Opportunities for Model Order Reduction - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65745 MS225: New Challenges and Opportunities for Model Order Reduction - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65757 MS243: Reduced Order Models for Fluids: Achievements and Open Problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65758 MS276: Reduced Order Models for Fluids: Achievements and Open Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65898 MS267: Model Reduction and Reduced-order Modeling of Dynamical Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65899 MS301: Model Reduction and Reduced-order Modeling of Dynamical Systems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66160 MS308: Data-driven and Mathematical Model Reductions for Combustion System Simulation and Design]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66143 MS316: Rational Approximation and its Applications]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65578 MS343: Data-augmented Reduced-order Modeling: Operator Learning and Closure/error Modeling - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65579 MS376: Data-augmented Reduced-order Modeling: Operator Learning and Closure/error Modeling - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65662 MS352: Recent Developments in Model Order Reduction Methods - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=65663 MS384: Recent Developments in Model Order Reduction Methods - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66087 MS361: Structure-exploiting Techniques for Approximation, Inference and Control of Complex Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=66088 MS393: Structure-exploiting Techniques for Approximation, Inference and Control of Complex Systems - Part II of II]"

=== 2018 ===

* [http://meetings.aps.org/Meeting/DFD18/Content/3571 71st Annual Meeting of the APS Division of Fluid Dynamics], November 18th - 20th, 2018; Atlanta (USA)
** "[http://meetings.aps.org/Meeting/DFD18/Session/A01 Session A01: Nonlinear Dynamics: Model Reduction I]"
** "[http://meetings.aps.org/Meeting/DFD18/Session/F15 Session F15: Flow Control: Coherent Structures and Reduced Order Modeling]"
** "[http://meetings.aps.org/Meeting/DFD18/Session/Q01 Session Q01: Nonlinear Dynamics: Model Reduction II]"

* [https://www.math.uni-hamburg.de/morss2018/ Model Order Reduction Summer School], September 24th - 28th, 2018; Hamburg (Germany)

* [https://alop.uni-trier.de/eucco2018 5th European Conference on Computational Optimization - EUCCO 2018], September 10th - 12th; Trier (Germany)
** "Focus Session: Model order reduction and low-rank approximation for nonlinear problems"

* Minisymposium at [https://sim.mathematik.uni-halle.de/numdiff/Numdiff15/home NumDiff 15], September 3rd - 7th, 2018; Halle (Germany)
** "Model order reduction for dynamical systems"

* [http://597.euromech.org/ Colloquium 597 Reduced Order Modeling in Mechanics of Materials], August 28th - 31st, 2018; Bad Herrenalb (Germany)

* Minisymposia at "[http://www.wccm2018.org/ 13th World Congress in Computational Mechanics]", July 22nd - 27th, 2018; New York (USA)
** "[http://www.wccm2018.org/MS_715 MS715: Reduced Order Methods for Parametric CFD Problems]"
** "[http://www.wccm2018.org/MS_1802 MS1802: Model Reduction, Big Data and Dynamic Data-Driven Systems]"
** "[http://www.wccm2018.org/MS_1804 MS1804: Machine Learning and Reduced-order Models for Complex Systems]"

* Minisymposia at "[http://ecmi.bolyai.hu/ The 20th European Conference on Mathematics for Industry]", June 18th - 22nd, 2018; Budapest (Hungary)
** "MS35: Reduced Order Modelling for Industrial and Scientific Applications"

* Sessions at the [http://www.ecc18.eu/ European Control Conference 2018] June 12th - 15th, 2018; Limassol (Cyprus)
** [https://controls.papercept.net/conferences/conferences/ECC18/program/ECC18_ContentListWeb_4.html#fra3 Model Reduction and Control in Large-Scale Networks]
** [https://controls.papercept.net/conferences/conferences/ECC18/program/ECC18_ContentListWeb_4.html#frb3 Model and Order Reduction I]
** [https://controls.papercept.net/conferences/conferences/ECC18/program/ECC18_ContentListWeb_4.html#frc3 Model and Order Reduction II]

* Minisymposia at "[http://www.eccm-ecfd2018.org ECCM-ECFD 2018]", June 11th - 15th, 2018; Glasgow (Scotland)
** "[http://www.eccm-ecfd2018.org/admin/Files/FileAbstract/a48.pdf MS48: Model reduction, big data and dynamic data-driven systems]"
** "[http://www.eccm-ecfd2018.org/admin/Files/FileAbstract/a126.pdf MS126: Reduced order modeling with error control and adaptivity]"
** "[http://www.eccm-ecfd2018.org/admin/Files/FileAbstract/a130.pdf MS130: Advances in reduced basis techniques for flow problems in analysis, control and optimization]"
** "[http://www.eccm-ecfd2018.org/admin/Files/FileAbstract/a149.pdf MS149: Model order reduction for multiscale problems in geo-engineering]"
** "[http://www.eccm-ecfd2018.org/admin/Files/FileAbstract/a170.pdf MS170: Reduced order modeling for uncertainty quantification in subsurface flow problems]"
** "[http://www.eccm-ecfd2018.org/admin/Files/FileAbstract/a192.pdf MS192: Developments in reduced-order modelling of the cardiovascular system -- methods and applications.]"

* "[http://www.itm.uni-stuttgart.de/iutam2018/ IUTAM Symposium on Model Order Reduction for Coupled Systems (MORCOS18)]" May 22nd - 25th, 2018; Stuttgart (Germany)

* "[https://sites.google.com/view/rom4cvs Workshop on Reduced Models for the Cardiovascular Systems]" April 26th - 27th, 2018; Atlanta (USA)

* Sessions at "[http://www.siam.org/meetings/uq18/ SIAM Conference on Uncertainty Quantification 2018]" April 16th - 20th, 2018; Garden Grove (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63738 MS5 - Model reduction and fast sampling methods for Bayesian inference - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63739 MS18 - Model reduction and fast sampling methods for Bayesian inference - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63731 MS54 - Dimension reduction in Bayesian inference - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63732 MS67 - Dimension reduction in Bayesian inference - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63733 MS80 - Dimension reduction in Bayesian inference - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63754 MS39 - recent advances in model reduction and data-enabled modeling - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63755 MS53 - recent advances in model reduction and data-enabled modeling - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63756 MS66 - recent advances in model reduction and data-enabled modeling - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63821 MS75 - Reduced Order Modeling for Uncertainty Quantification Targeting Exascale Computing Applications]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63791 MS79 - Reduced-order modeling techniques for large-scale UQ problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63792 MS93 - Reduced-order modeling techniques for large-scale UQ problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63817 MS106 - Advances in Reduced Order Modeling for Uncertainty Quantification - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=63818 MS119 - Advances in Reduced Order Modeling for Uncertainty Quantification - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=64390 CP2 - Reduced-order Modeling and Dynamical Systems I]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=64404 CP13 - Reduced-order Modeling and Dynamical Systems II]"
* "[http://morepas2018.sciencesconf.org MoRePaS 2018 - Model Reduction of Parametrized Systems IV]" April 9th - 13th, 2018; Nantes (France)
* "[http://www.newton.ac.uk/event/unqw03 Reducing dimensions and cost for UQ in complex systems]" March 5th - 9th, 2018; Cambridge (UK)
* Minisymposium at [http://www.mathmod.at/ 9th MathMod] February 21st - 23rd, 2018; Vienna (Austria)
** "Model Reduction"
* "[https://sisu.ut.ee/eu_mornet/node/17747 EU-MORNET Workshop]" February 8th - 9th, 2018; Tartu (Estonia)
* "[http://www.jade-hs.de/unsere-hochschule/fachbereiche/ingenieurwissenschaften/forschung-und-praxis/modellierung-und-simulation-mechatronischer-systeme/february-2nd-2018-eu-mornet-workshop-mor-4-mechatronics/ MOR 4 MECHATRONICS]" February 2nd, 2018; Wilhemshaven (Germany

=== 2017 ===

* [http://math.lbl.gov/~mjzahr/wcrw2017/ West Coast ROM Workshop], November 17th, 2017; Berkeley (USA)
* "[http://www.imus.us.es/IWROM4/ MORTech 2017]" November 8th - 10th; Seville (Spain)
* "[http://www.math.rug.nl/gcsc/morschool.html Groningen Autumn School on Model Order Reduction]" October 30th - November 1st, 2017; Groningen (Netherlands)
* Minisymposia at [http://www.uib.no/en/enumath2017 ENUMATH 2017] September 25th - 29th, 2017; Voss (Norway)
** "[http://www.uib.no/en/enumath2017/98144/minisymposia MS10 - Reduced order models for time-dependent problems]"
** "[http://www.uib.no/en/enumath2017/98144/minisymposia MS28 - Model reduction methods for simulation and (optimal) control]"
* "[http://scale-freeback.eu/grenoble-workshop-2017/ Modelling Reduction Tools for Large Scale Complex Networks]" September 21st - 22nd, 2017; Grenoble (France)
* "[http://www.rbss2017.de Reduced Basis (and Friends) Summer School 2017]" September 19th - 22nd, 2017; Goslar (Germany)
* "[http://www.gipsa-lab.fr/summerschool/auto2017/ Approximation of Large-Scale Dynamical Models]" September 11th - 15th, 2017; Grenoble (France)
* "[http://www.rt.mw.tum.de/workshops-seminare/2-mor-doktoranden-workshop/ 2. MOR Doktoranden-Workshop]" September 6th - 8th, 2017; Munich (Germany)
* "[http://www.maths.dur.ac.uk/lms/107/index.html EPSRC Durham Symposium Model Order Reduction]" August 7th - 17th, 2017; Durham (UK)
* "[http://modelreduction.net/workshops/6th-international-workshop 6th International Workshop on Model Reduction in Reacting Flows]" July 11th - 14th, 2017; Princeton (USA)
* "[http://www.siam.org/meetings/dr17/ SIAM Workshop on Parameter Space Dimension Reduction (DR17)]" July 9th - 10th, 2017; Pittsburgh (USA)
* "[https://www.math.vt.edu/GFD_conference2017/ Conference on Classical and Geophysical Fluid Dynamics: Modeling, Reduction and Simulation]" June 26th - 28th, 2017; Blacksburg (USA)
* Session at [http://acc2017.a2c2.org/ American Control Conference] May 24th - 26th, 2017; Seattle (USA)
** "[https://css.paperplaza.net/conferences/scripts/rtf/2017ACC_ContentListWeb_3.html#thc05 Reduced Order Modeling]"
* "[http://eumornetlux.weebly.com/ 2nd Exploratory Workshop on Applications of Model Order Reduction Methods in Industrial Research and Development]" March 10th, 2017; Luxembourg (Luxembourg)
* "[http://www.eu-mor.net/model-reduction-course-hydra/ Model Reduction Course HYDRA]" March 6th - 9th, 2017; Eindhoven (Netherlands)
* Minisymposia at [http://www.siam.org/meetings/cse17/ SIAM CSE'17] February 27th, 2017; Atlanta (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60778 MS8 - Smooth, Reduced, Sparse -- Exploiting Structures for Surrogate Modeling in CSE - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60779 MS37 - Smooth, Reduced, Sparse -- Exploiting Structures for Surrogate Modeling in CSE- Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60819 MS66 - Smooth, Reduced, Sparse -- Exploiting Structures for Surrogate Modeling in CSE - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60968 MS75 - Model Reduction Software: Nonlinear Problems and Data-Driven Solutions]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61076 MS106 - Reduced Order Models for Fluids: Achievements and Open Problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61077 MS133 - Reduced Order Models for Fluids: Achievements and Open Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61078 MS161 - Reduced Order Models for Fluids: Achievements and Open Problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61079 MS188 - Reduced Order Models for Fluids: Achievements and Open Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60854 MS107 - Model Order Reduction: Perspectives from Junior Researchers - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61077 MS134 - Model Order Reduction: Perspectives from Junior Researchers - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61070 MS145 - Reduced Order Modeling Techniques in Large Scale & Data-Driven PDE Problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61071 MS172 - Reduced Order Modeling Techniques in Large Scale & Data-Driven PDE Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60909 MS162 - Model Reduction for Optimal Control Problems: Perspectives from Junior Researchers - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=60910 MS189 - Model Reduction for Optimal Control Problems: Perspectives from Junior Researchers - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=61275 MS313 - Model and Solution Reduction Methods in Computational Mechanics: Challenges and Perspectives]"
* "[http://www.hda2017.unsw.edu.au 7th Workshop on High-Dimensional Approximation]" February 13th - 17th, 2017; Sydney (Australia)
* "[http://matperso.mines-paristech.fr/Personnel/david.ryckelynck Doctoral Workshop on Model Reduction in Nonlinear Mechanics]" February 13th - 17th, 2017; Paris (France)
* "[http://www.mpi-magdeburg.mpg.de/csc/events/modred2017 3rd Workshop on Model Reduction of Complex Dynamical Systems]" January 11th - 13th, 2017; Odense (Denmark)

=== 2016 ===

* "[http://www.komso.org/events/challenge-workshops/reduced-order-modeling-simulation-and-optimization-powerful-algorithms Reduced-Order Modeling for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing]" November 17th - 18th; Renningen (Germany)
* Workshop "[http://www.ihp.fr/en/CEB/T3-2016/workshop2 Recent developments in numerical methods for model reduction]" November 7th - 10th, 2016; Paris (France)
* "[http://www.mpi-magdeburg.mpg.de/3104866/RBSS2016 Reduced Basis Summer School 2016]" October 4th - 7th, 2016; Hedersleben (Germany)
* Workshop on "[http://www.eventbrite.it/e/reduced-order-modelling-and-multi-physics-coupling-for-reactor-applications-registration-24511634960 Reduced Order Modelling and Multiphysics Coupling for Rector Application]" September 30th, 2016; Milano (Italy)
* "[http://alop.uni-trier.de/event/alop-workshop-reduced-order-models-in-optimization/ ALOP Workshop: Reduced Order Models in Optimization] September 26th - 28th, 2016; Trier (Germany)
* "[http://www.mathos.unios.hr/index.php/443 Workshop on Model Reduction Methods and Optimization]" September 20th - 21st, 2016; Opatija (Croatia)
* "[http://www.mechbau.uni-stuttgart.de/EMMA/worm2016 3rd International Workshop on Order-Reduction Methods for Mechanics of Materials]" August 29th - 31st, 2016; Bad Herrenalb (Germany)
* Sessions at [http://www.siam.org/meetings/an16/ SIAM Annual Meeting 2016] 11th - 15, 2016; Boston (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23289 MS32 - Reduced Order Modeling Techniques in Uncertainty Quantification - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23290 MS46 - Reduced Order Modeling Techniques in Uncertainty Quantification - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23600 MS70 - Model Reduction and Krylov-Subspace Methods]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23394 MS71 - Model Reduction for Inverse Problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23395 MS87 - Model Reduction for Inverse Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23453 MS103 - Model Reduction for Wavefield Simulations - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23454 MS118 - Model Reduction for Wavefield Simulations - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23099 MS104 - Model Reduction of Parametrized PDEs in Continuum Mechanics - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23100 MS119 - Model Reduction of Parametrized PDEs in Continuum Mechanics - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=23146 MS137 - Model Reduction of Parametrized PDEs: Application to Optimization and Uncertainty Quantification]"
* Minisymposium "[http://ilas2016.cs.kuleuven.be/minisymposium.php?sid=41 Data-Driven Model Reduction]" at ILAS July 11th - 15th, 2016; Leuven (Belgium)
* Sessions at [http://sites.google.com/a/umn.edu/mtns-2016/ 22nd International Symposium on Mathematical Theory of Networks and Systems], July 11th - 15th, 2016; Minnesota (USA)
** "[http://controls.papercept.net/conferences/conferences/MTNS16/program/MTNS16_ContentListWeb_1.html#tua06 Model Reduction I]"
** "[http://controls.papercept.net/conferences/conferences/MTNS16/program/MTNS16_ContentListWeb_1.html#tub06 Model Reduction II]"
* Session at [http://ecc16.eu European Control conference 2016], June 29th - July 1st, 2016; Aalborg (Denmark)
** "[http://controls.papercept.net/conferences/conferences/ECC16/program/ECC16_ContentListWeb_4.html#fra6 Model Reduction]"
* Seminar "[http://www.aices.rwth-aachen.de/news-events/eu-regional-school/eu-regional-school-2016/courses/heinkenschloss-seminar Model Reduction in PDE Constrained Optimization]" at EU Regional School, June 14th - 15th, 2016; Aachen (Germany)
* Minisymposium "Reduced Basis, POD and PGD Model Order Reduction Techniques" at [http://www.eccomas2016.org ECCOMAS Congress 2016] June 5th - 10th, 2016; Crete Island (Greece)
* Workshop on "[http://www.eu-mor.net/workshop-model-order-reduction-control-inverse-problems-morcip-2016/ Model Order Reduction for Control and Inverse Problems (MORCIP)]" May 19th - 20th, 2016; Lausanne (Switzerland)
* Spring School "[http://www.ercoftac.org/events/ercoftac_montestigliano_spring_school_2016/ Reduced-order models for non-linear dynamics in fluid flows]" May 17th - 23rd, 2016; Siena (Italy)
* Sessions at [http://www.siam.org/meetings/uq16/ SIAM Conference on Uncertainty Quantification] April 5th - 8th, 2016; Lausanne (Switzerland)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=21926 MS13 - Reduced Order Modelling for UQ PDEs Problems: Optimization, Control, Data Assimilation - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=21927 MS28 - Reduced Order Modelling for UQ PDEs Problems: Optimization, Control, Data Assimilation - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=22194 MS89 - Reduced-order Modeling in Uncertainty Quantification - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=22195 MS104 - Reduced-order Modeling in Uncertainty Quantification - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=22196 MS119 - Reduced-order Modeling in Uncertainty Quantification - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=22362 MS149 - Model Reduction in Stochastic Dynamical Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=22363 MS164 - Model Reduction in Stochastic Dynamical Systems - Part II of II]"
* Workshop "[http://www.ians.uni-stuttgart.de/agh/misc/events/morml2016/index.html Data-Driven Model Order Reduction and Machine Learning (MORML 2016)]" March 30th - April 1st, 2016; University of Stuttgart (Germany)
* Symposium "[http://www.humboldt-foundation.de/web/gafos-2016-sessions.html Model Reduction for Complex Systems]" at [http://www.humboldt-foundation.de/web/gafos-2016.html 20th German-American Symposium] March 10th - 13th, 2016; Potsdam (Germany)
* "Doctoral Workshop on Model Reduction in nonlinear dynamics of fluids and structures" January 25th - 29th, 2016; Paris (France)

=== 2015 ===

* West Coast ROM Workshop, November 19th, 2015; Livermore (USA)
* [http://sites.google.com/site/mor4mems2015/ MOR 4 MEMS] November 17th - 18th, 2015; Karlsruhe (Germany). [http://www2.mpi-magdeburg.mpg.de/mpcsc/events/MOR4MEMS2015/ Here] you find the presentation slides.
* [http://rom2015.sciencesconf.org Reduced Basis, POD and PGD Model Reduction Techniques] November 4th - 6th, 2015; Cachan (France)
* [http://eumornetlux.weebly.com/ Exploratory Workshop on Applications of Model Order Reduction Methods in Industrial Research and Development] November 6th, 2015; Luxembourg (Luxembourg)
* [http://indico.sissa.it/event/4/ MoRePaS III] October 13th - 16th, 2015; Trieste (Italy)
* [http://www.mathos.unios.hr/index.php/351 3rd International School on Model Reduction for Dynamical Control Systems] October 5th - 10th, 2015; Dubrovnik (Croatia)
* Minisymposium "[http://enumath2015.iam.metu.edu.tr/minisymposia20.html Local and adaptive model reduction for partial differential equations]" at [http://enumath2015.iam.metu.edu.tr ENuMath] September 14th - 18th, 2015; Ankara (Turkey)
* Minisymposium "[http://scicade2015.math.uni-potsdam.de/scicade2015/minisymposiadetails.html#MS34 Parametric Model Order Reduction: Challenges and Solutions]" at [http://scicade2015.math.uni-potsdam.de SciCADE] September 14th - 18th, 2015; Potsdam (Germany)
* [http://www.math.uni-konstanz.de/numerik/pod/rbss_2015/ Reduced Basis Summer School 2015] September 14th - 18th, 2015; Konstanz (Germany)
* [http://www.cs.cas.cz/more2015/index.php Workshop on MOdel REduction] September 6th - 10th, 2015; Pilsen (Czech Republic)
* Session at [http://www.iciam2015.cn 8th International Congresson Industrial and Applied Mathematics] August 10th - 14th, 2015; Beijing (China)
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-Mo-D-32.88 MS-Mo-D-32 - Reduced-order modeling in uncertainty quantification and computational fluiddynamics - Part I of III]"
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-Mo-E-32.aD MS-Mo-E-32 - Reduced-order modeling in uncertainty quantification and computational fluiddynamics - Part II of III]"
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-We-E-03.cC MS-We-E-03 - Reduced-order modeling in uncertainty quantification and computational fluiddynamics - Part III of III]"
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-We-E-55.XZ MS-We-E-55 - New advances in model order reduction: methods, algorithms, and applications - Part I of II]"
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-Th-BC-55.A6 MS-Th-BC-55 - New advances in model order reduction: methods, algorithms, and applications - Part II of II]"
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-Tu-E-14.1F MS-Tu-E-14 - Optimality in reduced order modeling and inversion - Part I of II]"
** "[http://www.iciam2015.cn/ICIAM%202015-program.pdf#MS-We-D-14.l1 MS-We-D-14 - Optimality in reduced order modeling and inversion - Part II of II]"
* Reduced Order Methods Session at [http://www.yic.rwth-aachen.de/ YIC GACM ACCCES] July 20th - 23rd; Aachen (Germany)
* Session at [https://desreg.jku.at/ecc15/ European Control Conference 2015] July 15th -17th, 2015; Linz (Austria)
** "[http://controls.papercept.net/conferences/conferences/ECC15/program/ECC15_ContentListWeb_2.html#wea11 Model Reduction I]"
** "[http://controls.papercept.net/conferences/conferences/ECC15/program/ECC15_ContentListWeb_2.html#wec11 Model Reduction II]"
* [http://modelreduction.net/workshops/5th-annual 5th International Workshop on Model Reduction in Reacting Flows] June 28th - July 1st; Spreewald (Germany)
* [http://www3.math.tu-berlin.de/numerik/MoRTransPhen/ Model Reduction for Transport Dominated Phenomena] May 19th - 20th; Berlin (Germany)
* International Symposium: [http://www.tum-ias.de/bigdata2015/program.html Big Data and Predictive Computational Modelling] May 18th - 21st, 2015; München (Germany)
* Minisymposia at [http://www.siam.org/meetings/cse15 SIAM Conference on Computational Science and Engineering] March 14th - 18th, 2015; Salt Lake City (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20081 MS4 - Adaptive Model Order Reduction - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20082 MS30 - Adaptive Model Order Reduction - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20083 MS55 - Adaptive Model Order Reduction - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20296 MS91 - Parametric Model Reduction and Inverse Problems - Part I of IV]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20297 MS116 - Parametric Model Reduction and Inverse Problems - Part II of IV]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20300 MS143 - Parametric Model Reduction and Inverse Problems - Part III of IV]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=20301 MS169 - Parametric Model Reduction and Inverse Problems - Part IV of IV]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19988 MS148 - Reduced-order Models for PDE-constrained Optimization Problems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19989 MS174 - Reduced-order Models for PDE-constrained Optimization Problems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19907 MS187 - Recent Advances in Model Reduction - Part I of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19908 MS211 - Recent Advances in Model Reduction - Part II of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19960 MS236 - Recent Advances in Model Reduction - Part III of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19961 MS260 - Recent Advances in Model Reduction - Part IV of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19962 MS285 - Recent Advances in Model Reduction - Part V of V]"
* Session at [http://www.mathmod.at Vienna Conference on Mathematical Modelling (MathMod)] February 18th - 20th, 2015; Vienna (Austria)
** "Model Reduction"
* [http://congress.cimne.com/mor/frontal/objectives.asp Short Course on Model Order Reduction] January 26th - 29th, 2015; Barcelona (Spain)

=== 2014 ===
* [http://www.mfo.de/occasion/1448b/www_view Oberwolfach Seminar on Projection Based Model Reduction] November 23rd - 29th, 2014; Oberwolfach (Germany)
* [http://eu-mor.net EU-MORNET] Kick-Off Meeting September 18th - 19th; Eindhoven (Netherlands)
* [http://www2.le.ac.uk/departments/mathematics/extranet/conferences/model-reduction-across-disciplines Model reduction across disciplines] August 19th - 22nd, 2014; Leicester (United Kingdom)
* [http://wwwmath.uni-muenster.de/rbss2014 Reduced Basis Summer School 2014] August 18th - 22nd, 2014; Muenster (Germany)
* Bay Area ROM Workshop, August 7th, 2014; Livermore (USA)

* Minisymposia at [http://www.wccm-eccm-ecfd2014.org IACM-ECCOMAS 2014] July 20th - 25th, 2014; Barcelona (Spain)
** "[http://congress.cimne.com/iacm-eccomas2014/admin/Files/FileAbstract/a239.pdf MS239: Multibody System Dynamics and Modal Reduction]"
** "[http://congress.cimne.com/iacm-eccomas2014/admin/Files/FileAbstract/a15.pdf MS015: Reduced Basis, POD and PGD Model Reduction Techniques]"
** "[http://congress.cimne.com/iacm-eccomas2014/admin/Files/FileAbstract/a141.pdf MS141: Reduced Order Models in Vibroacoustics]"
** "[http://congress.cimne.com/iacm-eccomas2014/admin/Files/FileAbstract/a69.pdf MS069: Advanced Reduced-order Modeling Strategies for Parametrized PDEs and Applications]"

* Session at [http://www.siam.org/meetings/an14/ SIAM Annual Meeting 2014], July 7th - 11th; Chicago (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=19618 CP1 - Reduced Order Models]"
* Sessions at the [http://ecc14.eu European Control Conference 2014] June 24th- 27th, 2014; Strasbourg (France)
** "[https://controls.papercept.net/conferences/conferences/ECC14/program/ECC14_ContentListWeb_3.html#tha2 Model and Controller Reduction I]"
** "[https://controls.papercept.net/conferences/conferences/ECC14/program/ECC14_ContentListWeb_4.html#frc2 Model and Controller Reduction II]"
* Session at [http://acc2014.a2c2.org/ American Control Conference] June 4th - 6th, 2014; Portland (USA)
** "[https://css.paperplaza.net/conferences/conferences/2014ACC/program/2014ACC_ContentListWeb_1.html#wea04 Reduced Order Modeling]"
* Sessions at [http://siam.org/meetings/uq14/ SIAM Conference on Uncertainty Quantification] March 31st - April 3rd; Savannah (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=18137 MS10 - Model-Reduction Techniques for Quantifying and Controlling Uncertainty]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=18298 MT3 - Reduced Order Methods for Modelling and Computational Reduction in UQ Problems]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=18664 CP11 - Reduced-order Modeling]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=18192 MS67 - Surrogate and Reduced Order Modeling for Statistical Inversion and Prediction - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=18193 MS77 - Surrogate and Reduced Order Modeling for Statistical Inversion and Prediction - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=17858 MS87 - Uncertainty Quantification via Dimension Reduction: Deterministic and Stochastic Approaches]"
* [http://jahrestagung.gamm-ev.de/index.php/scientific-program/minisymposia Parametric Model Reduction of Dynamical Systems] Minisymposium at the [http://jahrestagung.gamm-ev.de/ GAMM annual meeting 2014] March 10th - 14th, 2014; Erlangen (Germany)

===2013===
* [http://www2.mpi-magdeburg.mpg.de/mpcsc/events/ModRed/2013/ ModRed 2013] December 11th - 13th, 2013; Magdeburg (Germany)
* [http://www.mathos.unios.hr/locschool2013/ DAAD International School on Linear Optimal Control of Dynamic Systems] September 23rd - 28th; Osijek (Croatia)
* [http://www.ma.tum.de/IGDK1754/SummerSchool2013 Summerschool "Reduced Basis Methods - Fundamentals and Applications"] September 16th - 19th; Munich (Germany)
* Reduced Basis Summer School 2013 August, 2013; Aachen (Germany)
* [http://enumath2013.epfl.ch Enumath] Minisymposium "Reduced order modelling for the simulation of complex systems" August 26th - 30th, 2013, Lausanne (Switzerland)
* Sessions at the [http://www.ecc13.ethz.ch European control Conference 2013] July 17th - 19th, 2013; Zurich (Switzerland)
** "[https://controls.papercept.net/conferences/conferences/ECC13/program/ECC13_ContentListWeb_4.html#fra12 Reduced Order Systems]"
** "[https://controls.papercept.net/conferences/conferences/ECC13/program/ECC13_ContentListWeb_4.html#frc12 Model Reduction]"
* [http://modelreduction.net/workshops/4th-annual 4th International Workshop on Model Reduction in Reacting Flows] June 19th - 21st, 2013, San Francisco (USA)
* Sessions at [http://acc2013.a2c2.org American Control Conference] June 17th - 19th, 2013; Washington D.C. (USA)
** "[https://css.paperplaza.net/conferences/conferences/2013ACC/program/2013ACC_ContentListWeb_3.html#web05 Reduced-Order Modeling]"
* [http://modredcirm2013.uni-muenster.de/ CIRM workshop Model Reduction and Approximation for Complex Systems] June 10th - 14th, 2013; Marseille (France)
* [http://www.tapir.caltech.edu/~rom-gr/ Reduced Order Modelling in General Relativity] June 6th - 7th, 2013; Pasadena (USA)
* Sessions at the [http://www.siam.org/meetings/cse13 SIAM Conference on Computational Science and Engineering] February 25th - March 1st; Boston (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15551 MS20 - Structure-preserving Model Order Reduction of Large-scale Dynamical Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15552 MS40 - Structure-preserving Model Order Reduction of Large-scale Dynamical Systems - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15728 MS14 - Model Reduction and Surrogate Modeling Advances in Porous Media Flow Simulation and Optimization - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15729 MS34 - Model Reduction and Surrogate Modeling Advances in Porous Media Flow Simulation and Optimization - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15547 MS104 - Data-Driven Model Reduction - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15548 MS138 - Data-Driven Model Reduction - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15549 MS194 - Data-Driven Model Reduction - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15676 MS144 - Nonlinear Model Reduction of Complex Flows: Modeling, Analysis, and Computations - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15677 MS182 - Nonlinear Model Reduction of Complex Flows: Modeling, Analysis, and Computations - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15639 MS45 - Data Driven and Nonliner Model Reduction - Parts I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15640 MS68 - Data Driven and Nonliner Model Reduction - Parts II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15641 MS116 - Data Driven and Nonliner Model Reduction - Parts III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15422 MS95 - Reduced Order Modelling for Complex Systems in CFD - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15423 MS164 - Reduced Order Modelling for Complex Systems in CFD - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15424 MS244 - Reduced Order Modelling for Complex Systems in CFD - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15758 MS219 - Model Order Reduction: Recent Advances and Challenges - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15759 MS257 - Model Order Reduction: Recent Advances and Challenges - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15760 MS272 - Model Order Reduction: Recent Advances and Challenges - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=15952 CP12 - Model Reduction and Data-Driven Approaches]"

===2012===
* [http://www.morepas.org/workshop2012/index.html MoRePaS 2] October 2nd - 5th, 2012; Günzburg (Germany)
* [http://www.mathematik.uni-stuttgart.de/fak8/ians/lehrstuhl/agh/misc/events/rbm_workshop_2012.html Reduced Basis Summer School 2012] August 28th - 31st; Stuttgart (Germany)
* [http://www.math.uni-hamburg.de/moa/ Workshop on Adaptivity and Model Order Reduction in PDE Constrained Optimization] July 23rd - 27th; Hamburg (Germany)
* Sessions at [http://acc2012.a2c2.org/ American Control Conference] June 27th - 29th, 2012; Montreal (Canada)
** "[https://css.paperplaza.net/conferences/conferences/2012ACC/program/2012ACC_ContentListWeb_1.html#wec20 Reduced Order Modeling]"
* [https://www.cecam.org/workshop-2-681.html Reduced Basis, POD and Reduced Order Methods for model and computational reduction: towards real-time computing and visualization?] May 14 - 16th; Lausanne (Switzerland)
* [https://www2.mpi-magdeburg.mpg.de/mpcsc/news/program_tegernsee.pdf Workshop on Nonlinear MOR] May 6th - 9th; Tegernsee (Germany)
* Minisymposia at [http://siam.org/meetings/uq12/ SIAM Conference on Uncertaity Quantification] April 2nd - 6th; Raleigh (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=14305 MS58 - Model Reduction for Nonlinear Dynamical Systems]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=14520 MS80 - Reduced Order Modeling for High Dimensional Nonlinear Models]"

===2011===
* [http://www.math.uni-konstanz.de/numerik/pod/workshop Advances in POD and RB Model-Order Reduction] November 21st, 2011; Konstanz (Germany)
* [http://www.uni-ulm.de/mawi/mawi-numerik/aktuelles/summer-school-rbm.html Reduced Basis Summer School 2011] October 25th - 28th; Ulm (Germany)
* [http://www2.mpi-magdeburg.mpg.de/mpcsc/events/trogir/ Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems] October 10th - 15th; Trogir (Croatia)
* [http://www.math.uni-bremen.de/zetem/cms/detail.php?template=gamm_parse_title&person=gamm11/program GAMM Workshop Applied and Numerical Linear Algebra with Special Emphasis on Model Reduction] September 21st - 22nd; Bremen (Germany)
* Sessions at [http://iciam.org/event/iciam-2011-%E2%80%93-vancouver 7th International Congress on Industrial and Applied Mathematics] July 18th - 22nd; Vancouver (Canada)
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11230 MS38 - Optimal Parameter Sampling in Model Reduction - Part I of III]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11231 MS87 - Optimal Parameter Sampling in Model Reduction - Part II of III]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11232 MS136 - Optimal Parameter Sampling in Model Reduction - Part III of III]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11296 MS148 - Structure-Preserving Model Reduction - Part I of III]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11297 MS199 - Structure-Preserving Model Reduction - Part II of III]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11298 MS256 - Structure-Preserving Model Reduction - Part III of III]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11452 MS340 - Dynamical Systems Approaches to Model Reduction -- Part I of II]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=12051 MS388 - Dynamical Systems Approaches to Model Reduction -- Part II of II]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11877 MS363 - Reduced Basis Methods and their Applications - Part I of IV]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11878 MS416 - Reduced Basis Methods and their Applications - Part II of IV]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11879 MS462 - Reduced Basis Methods and their Applications - Part III of IV]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11880 MS511 - Reduced Basis Methods and their Applications - Part IV of IV]"
** "[https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=13350 CP36 - Reduced Order Models]"
* [http://www.modelreduction.net/iwmrrf2011/site/ 3rd International Workshop on Model Reduction in Reacting Flows] April 27th - 29th; Corfu (Greece)
* Session at the [http://siam.org/meetings/cse11 SIAM Conference on Computational Science and Engineering] February 28th - March 4th; Reno (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10982 MS18 - Model Reduction of Nonlinear and Parametrized Systems - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10983 MS26 - Model Reduction of Nonlinear and Parametrized Systems - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10984 MS40 - Model Reduction of Nonlinear and Parametrized Systems - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=11497 MS39 - Model Order Reduction Using Graph Theory and Numerical Linear Algebra]"
* [http://www.wias-berlin.de/workshops/MOR2011/ Workshop on Model Order Reduction in Optimization and Control with PDEs] January 26th - 28th; Berlin (Germany)

===2010===
* [http://www.ians.uni-stuttgart.de/MoRePaS/events/Ulm10/index.html Workshop on RB Methods] December 7th - 8th, 2010; Ulm (Germany)
* [http://www3.math.tu-berlin.de/modred2010/ ModRed 2010] December 2nd - 4th, 2010; Berlin (Germany)
* Sessions at [http://www.siam.org/meetings/an10/ SIAM Annual Meeting 2010], July 12th - 16th, 2010; Pittsburgh (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10125 MS2 - Advances in Model Reduction for Large-Scale Problems - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10126 MS38 - Advances in Model Reduction for Large-Scale Problems - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10127 MS51 - Advances in Model Reduction for Large-Scale Problems - Part III of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=10202 MS33 - Reduced Order Models for Dynamical Systems]"
* Minisymposium "[http://www.eccomas-cfd2010.org/minisymposia.php Model Order Reduction of Complex Systems in CFD]" at [http://www.eccomas-cfd2010.org ECCOMAS CFD]; June 14th - 17th, 2010; Lisbon (Portugal)

===2009===
* Sessions at [https://archive.siam.org/meetings/la09/index.php SIAM Conference on Applied Linear Algebra] October 26 th 29th, 2009; Monterey Bay-Seaside (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=8992 MS14 - Model Order Reduction for Dynamical Systems - Part I of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=8993 MS20 - Model Order Reduction for Dynamical Systems - Part II of III]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=8994 MS32 - Model Order Reduction for Dynamical Systems - Part III of III]"
* [http://www.win.tue.nl/casa/meetings/special/mor09/ Autumn School on Future Developments in Model Order Reduction] September 21st - 25th, 2009; Terschelling (Netherlands)
* [http://www.uni-muenster.de/CeNoS/ocs/index.php/MRP/MRP09/ MoRePaS] September 16th - 18th, 2009; Münster (Germany)
* Session at [http://www.siam.org/meetings/an09/ SIAM Annual Meeting 2009] July 6th - 10th, 2009; Denver (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=8863 CP20 - Model Reduction]"
* [http://modelreduction.net/iwmrrf2009/ 2nd International Workshop on Model Reduction in Reacting Flows] March 30th - April 1st, 2009 Notre Dame (USA)
* [http://www.cl.eps.manchester.ac.uk/medialand/maths/archived-events/workshops/www.mims.manchester.ac.uk/events/workshops/HSMoR09/ CICADA Workshop on Hybrid Systems and Model Reduction] March 19th - 20th, 2009 Manchester (England)
* Sessions at the [http://siam.org/meetings/cse09 SIAM Conference on Computational Science and Engineering] March 2nd - 6th, 2009; Miami (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=8306 CP12 - Model Reduction]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=7918 MS106 - Model Reduction - Part I of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=7919 MS119 - Model Reduction - Part II of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=7920 MS127 - Model Reduction - Part III of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=7921 MS141 - Model Reduction - Part IV of V]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=7922 MS151 - Model Reduction - Part V of V]"

===2008===

* [https://www.math.uni-hamburg.de/spag/zms/syrene/ Model Reduction for Circuit Simulation] October 30th - 31st, 2008; Hamburg (Germany)
* Session at [http://acc2008.a2c2.org/ American Control Conference] June 11th - 13th, 2008; Seattle (USA)
** "[https://css.paperplaza.net/conferences/conferences/2008ACC/program/2008ACC_ContentListWeb_3.html#fra11 Model Reduction]"

===2007===
* [http://www.win.tue.nl/casa/meetings/special/mor07/ Symposium on Recent Advances in Model Order Reduction] November 23rd, 2007; Eindhoven (Netherlands)
* [http://modelreduction.net/ModelReductionWorkshop/Home%20Page.html 1st International Workshop on Model Reduction in Reacting Flows] September 3rd - 5th, 2007; Rome (Italy)
* Sessions at [http://iciam.org/event/iciam-2007-%E2%80%93-zurich 6th International Congress for Industrial and Applied Mathematics] July 16th - 20th, 2007; Zurich (Switzerland)
** "[http://iciamold.jsiam.org/iciam2007/ICIAM07-ProgramBook.pdf#IC/MP/002/S/346 Model reduction: theory, methodology and software #1]"
** "[http://iciamold.jsiam.org/iciam2007/ICIAM07-ProgramBook.pdf#IC/MP/002/S/346-2 Model reduction: theory, methodology and software #2]"
** "[http://iciamold.jsiam.org/iciam2007/iciam07_programbook.pdf#IC/MP/020/S/347 Model reduction: structured and higher-order systems #1]"
** "[http://iciamold.jsiam.org/iciam2007/iciam07_programbook.pdf#IC/MP/020/S/347-2 Model reduction: structured and higher-order systems #2]"
** "[http://iciamold.jsiam.org/iciam2007/iciam07_programbook.pdf#IC/MP/020/P/348 Model reduction in circuit simulation #1]"
** "[http://iciamold.jsiam.org/iciam2007/ICIAM07-ProgramBook.pdf#IC/MP/020/P/348-2 Model reduction in circuit simulation #2]"
* Session at [http://acc2007.a2c2.org/ American Control Conference] July 11th - 13th, 2007; New York (USA)
** "[https://css.paperplaza.net/conferences/conferences/2007ACC/program/2007ACC_ContentListWeb_1.html#web07 Reduced Order Modeling]"
* Sessions at [https://archive.siam.org/meetings/ct07/index.php SIAM Conference on Control and its Application] June 29th - July 1st, 2007; San Francisco (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=6338 MS16 - Model Reduction for Control and Dynamical Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=6339 MS21 - Model Reduction for Control and Dynamical Systems - Part II of II]"
* Sessions at [http://siam.org/meetings/cse07 SIAM Conference on Computational Science and Engineering] February 19th - 23rd, 2007; Costa Mesa (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=5955 MS36 - Model Order Reduction and Automated Behavioural Modeling - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=6029 MS48 - Model Order Reduction and Automated Behavioural Modeling - Part II of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=6137 CP12 - Reduced Order Modelling]"

===2006===
* Session "Model Reduction" at [http://www.eurosime.org/b06.htm EuroSimE 2006] April 24th - 26th, 2006; Como (Italy)

* Minisymposia at [http://www.siam.org/meetings/an06/index.php SIAM Annual Meeting 2006] July 10th - 14th, 2006; Boston (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=5357 MS13 - Model Reduction of Dynamical Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=5358 MS31 - Model Reduction of Dynamical Systems - Part II of II]"

===2005===
* [http://www.lorentzcenter.nl/lc/web/2005/160/info.php3?wsid=160. Model order reduction, coupled problems and optimization] September 19th - 23rd, 2005; Leiden (Netherlands)

* Minisymposia at [http://www.siam.org/meetings/an05/index.htm SIAM Annual Conference 2005] July 11th - 15th, 2005; New Orleans (USA)
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=4372 MS43 - Model Reduction for Large Scale Dynamical Systems - Part I of II]"
** "[http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=4373 MS55 - Model Reduction for Large Scale Dynamical Systems - Part II of II]"

===2004===
* [https://www.birs.ca/events/2004/5-day-workshops/04w5513 Model Reduction Problems and Matrix Methods] April 3rd - 8th; Banff (Canada)

===2003===
* [http://web.archive.org/web/20070612131401/http://www.math.tu-berlin.de/numerik/mt/NumMat/Meetings/0310_MFO/ Dimensional Reduction of Large-Scale Systems] October 19th - 25th, 2003 Oberwolfach (Germany)

===2001===
* [https://research.tue.nl/en/activities/macsi-workshop-on-model-reduction MACSI workshop on Model Reduction] October 8th - 9th, 2001; Eindhoven (Netherlands)
* Minisymposium at [http://user.it.uu.se/~icosahom/ International Conference On Spectral and High Order Methods] June 11th - 15th, 2001; Uppsala (Sweden)
** "Reduced-Basis Methods for Partial Differential Equations"

===2000===
* Session at the [https://archive.siam.org/meetings/cse00/ First SIAM Conference on Computational Science and Engineering] September 21st - 24th, 2000; Washington D.C. (USA)
** "[https://archive.siam.org/meetings/cse00/cp25.htm CP25 - Model Reduction, Validation and Data Assimilation]"

===1999===
* Minisymposia at the [http://iciam.org/event/iciam-1999-%E2%80%93-edinburgh 4th International Council for Industrial and Applied Mathematics Congress] July 5th - 9th, 1999; Edinburgh (Scotland)
** "[http://www.macs.hw.ac.uk/archive/iciam99/PrintedProgramme/BookOfAbstracts.pdf#page=171 MSP-187 Reduced-Order Modeling of Large-Scale Systems and Applications in Industry]"
** "[http://www.macs.hw.ac.uk/archive/iciam99/PrintedProgramme/BookOfAbstracts.pdf#page=187 MSP-205,206 Methods of Dimension Reduction]"

Comparison of Software

2020-03-13T01:06:58Z

Mlinaric: Unify license style

[[Category:software]]

The following table provides a '''Comparison of Software''' for the model reduction software projects listed in the MORwiki.

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! Linear
! Nonlinear
! First Order
! Second Order
! Parametric
! DAE
! Dense
! Sparse
!
! Latest Version
! License
! Language
|-
! [[DPA|DPA]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| ?
| ?
| Matlab
|-
! [[Emgr|emgr]]
| Yes
| Yes
| Yes
| Yes
| Yes
| (Yes)
| Yes
| (Yes)
|
| 5.7 (02.2019)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[EZyRB|EZyRB]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.2 (04.2018)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[ITHACA-FV]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 2.1 (02.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| C++
|-
! [[ITHACA-SEM]]
| (Yes)
| (Yes)
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| (02.2019)
| [https://spdx.org/licenses/MIT.html MIT]
| C++
|-
! [[MESS]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| (Yes)
| Yes
|
| 2.0.1 (Matlab), 1.0 (C,Python,Julia)
| [https://spdx.org/licenses/GPL-2.0.html GPL-2.0]
| C, Matlab, Python, Julia
|-
! [[MORE]]
| Yes
| ?
| Yes
| ?
| ?
| ?
| ?
| ?
|
| ?
| ?
| Matlab
|-
! [[MOREMBS]]
| Yes
| No
| Yes
| Yes
| (Yes)
| Yes
| Yes
| Yes
|
| ?
| ?
| C++, Matlab
|-
! [[MORLAB]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| (Yes)
|
| 4.0 (12.2018)
| [https://spdx.org/licenses/AGPL-3.0.html AGPL-3.0]
| Matlab
|-
! [[MORPACK]]
| Yes
| No
| Yes
| Yes
| No
| Yes
| Yes
| Yes
|
| v3.0.099 (07.2015)
| ?
| Matlab
|-
! [http://www.rt.mw.tum.de/forschung/morlab/software/psssmor/ psssMOR]
| Yes
| No
| Yes
| No
| Yes
| (No)
| Yes
| Yes
|
| 1.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[PyDMD|PyDMD]]
| ?
| ?
| ?
| ?
| ?
| ?
| ?
| ?
|
| 0.2.1 (11.2018)
| [https://spdx.org/licenses/MIT.html MIT]
| Python
|-
! [[PyMOR|pyMOR]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 2019.2 (12.2019)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Python
|-
! [[sssMOR]]
| Yes
| No
| Yes
| (No)
| (Yes)
| (No)
| Yes
| Yes
|
| 2.00 (09.2017)
| [https://spdx.org/licenses/BSD-2-Clause.html BSD-2-Clause]
| Matlab
|-
! [[RBmatlab]]
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|
| 1.16.09 (09.2016)
| [https://spdx.org/licenses/AFL-3.0.html AFL-3.0]
| Matlab
|-
! [[RBniCS]]
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|
| 0.1.0 (06.2019)
| [https://spdx.org/licenses/LGPL-3.0.html LGPL-3.0]
| Python
|}