MOR Wiki - User contributions [en]

Transfer Function Interpolation

2013-06-17T08:20:15Z

Will: /* Higher dimensional parameter spaces */

[[Category:method]]
[[Category:linear]]
[[Category:parametric]]
[[Category:time invariant]]

'''Transfer function interpolation''' is an approach for parameter-preserving model order reduction which is based on a combination of [[Balanced Truncation|balanced truncation]] (or any other model order reduction method for deterministic linear, time-invariant systems) at certain distinct parameter values (the interpolation points) with interpolation. The approach is quite flexible in allowing the use of numerous interpolation techniques like polynomial, Hermite, rational, sinc, and spline interpolation.

==Description==

The method will exemplarily be introduced for reducing a system given by the transfer function

:<math>
G(s,p) = C(sE - A1- p A2 )^{-1}B.
</math>

The PMOR approach was originally proposed in <ref name="BauB09"> U. Baur and P. Benner, "[http://www.oldenbourg-link.com/doi/abs/10.1524/auto.2009.0787 Modellreduktion für parametrisierte Systeme durch balanciertes Abschneiden und Interpolation (Model Reduction for Parametric Systems Using Balanced Truncation and Interpolation)]",
at-Automatisierungstechnik, vol. 57, no. 8, pp. 411-420, 2009</ref>
using polynomial interpolation. It can simply be extended to a hybrid approach of [[Balanced Truncation|balanced truncation]] applying different kinds of interpolation.

In the following rational interpolation will be employed for the interpolation of the locally reduced-order transfer functions.

It is assumed that for <math>k+1</math> interpolation points <math>p_0, \dots, p_k</math> somehow distributed over the parameter interval,
the underlying non-parametric systems

:<math>

G(s,p_j) = C( sE - A1- p_j A2)^{-1} B, \quad j = 0,\dots k,

</math>

are stable, i.e. all finite eigenvalues of the regular pencils <math>\lambda E - A1 - p_j A2</math> lie in the open left half of the complex plane.

Then, [[Balanced Truncation|balanced truncation]] can be applied to <math>G(s,p_j)</math> leading to reduced-order systems of order <math>r_j</math>

:<math>

\hat G_j(s) := \hat G(s,p_j) = \hat C_j(s \hat E_j - \hat A1_j - \hat A2_j)^{-1}\hat B_j, \quad \textrm{for} \ j=0,\dots,k.

</math>

The reduced-order transfer function over the whole parameter interval is obtained by rational interpolation, e.g. by use of the barycentric formula

:<math>

\hat G_I(s,p) = \frac{\textrm{num}(s,p)}{\textrm{den}(s,p)} =

\frac{\sum\limits_{j=0}^k \frac{u_j}{p - p_j} \hat G_j(s) }{\sum\limits_{j=0}^{k} \frac{u_j}{p - p_j}}

</math>

for real numbers <math>u_j\ne 0</math> and with numerator and denominator of degree at most <math>k</math> <ref>J.-P. Berrut, R. Baltensperger, and H.D. Mittelmann, "[http://link.springer.com/chapter/10.1007%2F3-7643-7356-3_3# Recent developments in barycentric rational interpolation]", in Trends and applications in constructive approximation,
vol. 151 of Internat. Ser. Numer. Math., pp. 27-51. Birkhäuser, Basel, 2005.</ref>.

==Higher dimensional parameter spaces==

For systems including more than one parameter, the problem of finding a reduced-order interpolating function over the whole parameter space

is much more involved. The main reason for that is the exponentially growing number of interpolation points for higher dimensional parameter spaces.

This leads to a high computationally complexity since [[Balanced Truncation|balanced truncation]] has to be applied many times. Furthermore, the order of the reduced-order

system grows exponentially as well.

A strategy for an effective and representative choice of parameter points in higher dimensional parameter spaces

comes through the use of [[wikipedia:Sparse_grid|sparse grids]], see, e.g. <ref>H.-J. Bungartz and M. Griebel, Sparse grids.
Acta Numerica, vol. 13, pp. 147-269, 2004.</ref>, <ref>M. Griebel
Sparse grids and related approximation schemes for higher dimensional problems
In Foundations of Computational Mathematics (FoCM05), Santander, pp. 106-161, 2006.</ref>, <ref>C. Zenger
Sparse grids
In Parallel algorithms for partial differential equations
(Kiel, 1990), vol. 31 of Notes Numer. Fluid Mech., pp. 241-251, 1991.</ref>.

This approach is based on a hierarchical basis and a sparse tensor product construction. Significantly less interpolation points are needed for obtaining a similar accuracy as interpolation in a full grid space.

A coupling of [[Balanced Truncation|balanced truncation]] with piecewise polynomial interpolation using sparse grid points was described in <ref name="BauB09"/>.

==Numerical results==

Numerical results for the [[Anemometer|anemometer benchmark]] can be found in
<ref>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, vol. 17, no. 4, pp. 297-317, 2011.</ref>.

==References==

<references/>

==Contact==

[[User:Baur]]

Transfer Function Interpolation

2013-06-17T08:18:03Z

Will: /* Higher dimensional parameter spaces */

[[Category:method]]
[[Category:linear]]
[[Category:parametric]]
[[Category:time invariant]]

'''Transfer function interpolation''' is an approach for parameter-preserving model order reduction which is based on a combination of [[Balanced Truncation|balanced truncation]] (or any other model order reduction method for deterministic linear, time-invariant systems) at certain distinct parameter values (the interpolation points) with interpolation. The approach is quite flexible in allowing the use of numerous interpolation techniques like polynomial, Hermite, rational, sinc, and spline interpolation.

==Description==

The method will exemplarily be introduced for reducing a system given by the transfer function

:<math>
G(s,p) = C(sE - A1- p A2 )^{-1}B.
</math>

The PMOR approach was originally proposed in <ref name="BauB09"> U. Baur and P. Benner, "[http://www.oldenbourg-link.com/doi/abs/10.1524/auto.2009.0787 Modellreduktion für parametrisierte Systeme durch balanciertes Abschneiden und Interpolation (Model Reduction for Parametric Systems Using Balanced Truncation and Interpolation)]",
at-Automatisierungstechnik, vol. 57, no. 8, pp. 411-420, 2009</ref>
using polynomial interpolation. It can simply be extended to a hybrid approach of [[Balanced Truncation|balanced truncation]] applying different kinds of interpolation.

In the following rational interpolation will be employed for the interpolation of the locally reduced-order transfer functions.

It is assumed that for <math>k+1</math> interpolation points <math>p_0, \dots, p_k</math> somehow distributed over the parameter interval,
the underlying non-parametric systems

:<math>

G(s,p_j) = C( sE - A1- p_j A2)^{-1} B, \quad j = 0,\dots k,

</math>

are stable, i.e. all finite eigenvalues of the regular pencils <math>\lambda E - A1 - p_j A2</math> lie in the open left half of the complex plane.

Then, [[Balanced Truncation|balanced truncation]] can be applied to <math>G(s,p_j)</math> leading to reduced-order systems of order <math>r_j</math>

:<math>

\hat G_j(s) := \hat G(s,p_j) = \hat C_j(s \hat E_j - \hat A1_j - \hat A2_j)^{-1}\hat B_j, \quad \textrm{for} \ j=0,\dots,k.

</math>

The reduced-order transfer function over the whole parameter interval is obtained by rational interpolation, e.g. by use of the barycentric formula

:<math>

\hat G_I(s,p) = \frac{\textrm{num}(s,p)}{\textrm{den}(s,p)} =

\frac{\sum\limits_{j=0}^k \frac{u_j}{p - p_j} \hat G_j(s) }{\sum\limits_{j=0}^{k} \frac{u_j}{p - p_j}}

</math>

for real numbers <math>u_j\ne 0</math> and with numerator and denominator of degree at most <math>k</math> <ref>J.-P. Berrut, R. Baltensperger, and H.D. Mittelmann, "[http://link.springer.com/chapter/10.1007%2F3-7643-7356-3_3# Recent developments in barycentric rational interpolation]", in Trends and applications in constructive approximation,
vol. 151 of Internat. Ser. Numer. Math., pp. 27-51. Birkhäuser, Basel, 2005.</ref>.

==Higher dimensional parameter spaces==

For systems including more than one parameter, the problem of finding a reduced-order interpolating function over the whole parameter space

is much more involved. The main reason for that is the exponentially growing number of interpolation points for higher dimensional parameter spaces.

This leads to a high computationally complexity since [[Balanced Truncation|balanced truncation]] has to be applied many times. Furthermore, the order of the reduced-order

system grows exponentially as well.

A strategy for an effective and representative choice of parameter points in higher dimensional parameter spaces

comes through the use of [[wikipedia:Sparse_grid|sparse grids]], see, e.g. <ref>H.-J. Bungartz and M. Griebel, Sparse grids.
Acta Numerica, vol. 13, pp. 147-269, 2004.</ref>, <ref>M. Griebel
Sparse grids and related approximation schemes for higher dimensional problems
In Foundations of Computational Mathematics (FoCM05), Santander}, pp. 106-161, 2006.</ref>, <ref>C. Zenger
Sparse grids
In Parallel algorithms for partial differential equations
(Kiel, 1990), vol. 31 of Notes Numer. Fluid Mech., pp. 241-251, 1991.</ref>.

This approach is based on a hierarchical basis and a sparse tensor product construction. Significantly less interpolation points are needed for obtaining a similar accuracy as interpolation in a full grid space.

A coupling of [[Balanced Truncation|balanced truncation]] with piecewise polynomial interpolation using sparse grid points was described in <ref name="BauB09"/>.

==Numerical results==

Numerical results for the [[Anemometer|anemometer benchmark]] can be found in
<ref>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, vol. 17, no. 4, pp. 297-317, 2011.</ref>.

==References==

<references/>

==Contact==

[[User:Baur]]

Transfer Function Interpolation

2013-06-17T08:14:56Z

Will: /* Description */

[[Category:method]]
[[Category:linear]]
[[Category:parametric]]
[[Category:time invariant]]

'''Transfer function interpolation''' is an approach for parameter-preserving model order reduction which is based on a combination of [[Balanced Truncation|balanced truncation]] (or any other model order reduction method for deterministic linear, time-invariant systems) at certain distinct parameter values (the interpolation points) with interpolation. The approach is quite flexible in allowing the use of numerous interpolation techniques like polynomial, Hermite, rational, sinc, and spline interpolation.

==Description==

The method will exemplarily be introduced for reducing a system given by the transfer function

:<math>
G(s,p) = C(sE - A1- p A2 )^{-1}B.
</math>

The PMOR approach was originally proposed in <ref name="BauB09"> U. Baur and P. Benner, "[http://www.oldenbourg-link.com/doi/abs/10.1524/auto.2009.0787 Modellreduktion für parametrisierte Systeme durch balanciertes Abschneiden und Interpolation (Model Reduction for Parametric Systems Using Balanced Truncation and Interpolation)]",
at-Automatisierungstechnik, vol. 57, no. 8, pp. 411-420, 2009</ref>
using polynomial interpolation. It can simply be extended to a hybrid approach of [[Balanced Truncation|balanced truncation]] applying different kinds of interpolation.

In the following rational interpolation will be employed for the interpolation of the locally reduced-order transfer functions.

It is assumed that for <math>k+1</math> interpolation points <math>p_0, \dots, p_k</math> somehow distributed over the parameter interval,
the underlying non-parametric systems

:<math>

G(s,p_j) = C( sE - A1- p_j A2)^{-1} B, \quad j = 0,\dots k,

</math>

are stable, i.e. all finite eigenvalues of the regular pencils <math>\lambda E - A1 - p_j A2</math> lie in the open left half of the complex plane.

Then, [[Balanced Truncation|balanced truncation]] can be applied to <math>G(s,p_j)</math> leading to reduced-order systems of order <math>r_j</math>

:<math>

\hat G_j(s) := \hat G(s,p_j) = \hat C_j(s \hat E_j - \hat A1_j - \hat A2_j)^{-1}\hat B_j, \quad \textrm{for} \ j=0,\dots,k.

</math>

The reduced-order transfer function over the whole parameter interval is obtained by rational interpolation, e.g. by use of the barycentric formula

:<math>

\hat G_I(s,p) = \frac{\textrm{num}(s,p)}{\textrm{den}(s,p)} =

\frac{\sum\limits_{j=0}^k \frac{u_j}{p - p_j} \hat G_j(s) }{\sum\limits_{j=0}^{k} \frac{u_j}{p - p_j}}

</math>

for real numbers <math>u_j\ne 0</math> and with numerator and denominator of degree at most <math>k</math> <ref>J.-P. Berrut, R. Baltensperger, and H.D. Mittelmann, "[http://link.springer.com/chapter/10.1007%2F3-7643-7356-3_3# Recent developments in barycentric rational interpolation]", in Trends and applications in constructive approximation,
vol. 151 of Internat. Ser. Numer. Math., pp. 27-51. Birkhäuser, Basel, 2005.</ref>.

==Higher dimensional parameter spaces==

For systems including more than one parameter, the problem of finding a reduced-order interpolating function over the whole parameter space

is much more involved. The main reason for that is the exponentially growing number of interpolation points for higher dimensional parameter spaces.

This leads to a high computationally complexity since [[Balanced Truncation|balanced truncation]] has to be applied many times. Furthermore, the order of the reduced-order

system grows exponentially as well.

A strategy for an effective and representative choice of parameter points in higher dimensional parameter spaces

comes through the use of [[wikipedia:Sparse_grid|sparse grids]], see, e.g. <ref>H.-J. Bungartz and M. Griebel, Sparse grids.
Acta Numerica, vol. 13, pp. 147-269, 2004.</ref>, <ref>M. Griebel
Sparse grids and related approximation schemes for higher dimensional problems
In Foundations of Computational Mathematics (FoCM05), Santander}, pp. 106-161, 2006.</ref>, <ref>C. Zenger
Sparse grids
In Parallel algorithms for partial differential equations
(Kiel, 1990), vol. 31 of Notes Numer. Fluid Mech., pp. 241-251, 1991.</ref>

This approach is based on a hierarchical basis and a sparse tensor product

construction. Significantly less interpolation points are needed for obtaining a similar accuracy as interpolation in a full grid space.

A coupling of [[Balanced Truncation|balanced truncation]] with piecewise polynomial interpolation using sparse grid points was described in <ref name="BauB09"/>.

==Numerical results==

Numerical results for the [[Anemometer|anemometer benchmark]] can be found in
<ref>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, vol. 17, no. 4, pp. 297-317, 2011.</ref>.

==References==

<references/>

==Contact==

[[User:Baur]]

Submission rules

2012-11-24T13:14:19Z

Will: /* 3.3. Matrix format for nonlinear or parametric non-affine systems */

The PMOR benchmark collection contains benchmarks related to model order reduction. Its goal is to supply the dynamical systems in a computer-readable format to researches from different areas. They can then test new algorithms, software, etc.

The collection contains documents that are supposed to be processed by a human being and data files that are supposed to be read automatically by software. As a result, with minor exceptions, the rules concerning documents have a recommendation status, and the rules concerning data files are obligatory.

We start with the description of documents, then we present the publication policy, and finally we describe the data files.

==1. Documents==

The collection consists of documents forming a two-level hierarchy. Top-level documents will be referred to as benchmarks. Each benchmark document may have links to several documents referred to as reports. A benchmark and its reports may be written by different authors.

Each document is written according to conventional scientific practice, that is, it describes the matters in such a way that, at least in principle, anyone could reproduce the results presented. The authors should understand that the document may be read by people from quite different disciplines. Hence, abbreviations should be avoided or at least explained and references to the background ideas should be made.

===1.1. Benchmark===

The goal of a benchmark document is to describe the origin of the dynamic system and its relevance to the application area. It is important to present the mathematical model, the meaning of the inputs and outputs, and the desired behavior from the application viewpoint.

A few points to be addressed:

* The purpose of the model should be explained clearly. (For instance, simulation, iterative system design, feedback control design, ...)
* Why should the model be reduced at all? (For instance, reducing simulation time, reducing implementation effort in observers, controllers...)
* What are the QUALITATIVE requirements to the reduced model? What variables are to be approximated well? Is the step response to be approximated or is it the bode plot? What are typical input signals? (Some systems are driven by a step function and nothing else, others are driven by a wide variety of input signals, others are used in closed loops and can cause instabilities, although being stable themselves.)
* What are the QUANTITATIVE requirements to the reduced model? Best would be if the authors of any individual model can suggest some cost functions (performance indices) to be used for the comparison. These can be in time domain, or in frequency domain (including bandwidth), or both.
* Are there limits of input and state variables known? (application related or generally)? What are the physical limits where the model becomes useless/false? If known a-priori: Out of the technically typical input signals, which one will cause "the most nonlinear" behavior?
* How many parameters are included in the model, are they physical parameters or geometrical parameters.
* Is the system linear or nonlinear? Is it a first or a second order system?
* When the model is used to test, e.g. an algorithm, can some of the parameters in the model be fixed, such that the model includes fewer parameters, which makes the testing easier?
* If possible, the source code which generates the model is encouraged to be provided.

* We stress the importance to describe the software employed, as well as its related options. For example, if the model is generated from ANSYS, the software ANSYS is better to be mentioned. If necessary, please also describe some related options like boundary conditions, initial conditions, and other parameters which need to be assigned when generating the model. Please go to the benchmark webpage [[Silicon nitride membrane]] for a reference.

* If the dynamic system is obtained from partial differential equations, then the information about material properties, geometrical data, initial and boundary conditions should be given. The exception to this rule is the case when the original model comes from industry. In this case, if trade secrets are tied with the information mentioned, it may be kept hidden.

* The authors are encouraged to produce several dynamic models of different dimensions in order to provide an opportunity to apply different software and to research scalability issues. If an author has an interactive page on his/her server to generate benchmarks, a link to this page is welcome.

* The dynamic system may be obtained by means of compound matrices, for example, when the second-order system is converted to first-order. In this case, the document should describe such a transformation but in the data file the original and not the compound matrices should be given. This allows to research other ways of model reduction of the original system.

The above information is in a general sense. Therefore, some items may not be applicable to certain benchmarks. However, it is recommended that the items that apply to the benchmark provided should be fully considered.

===1.2. Report===

A report document may contain:

# The solution of the original benchmark that contains sample outputs for the usual input signals. Plots and numerical values of time and frequency response. Eigenvalues and eigenvectors, singular values, poles, zeros, etc.
# Model reduction and its results as compared to the original system.
# Description of any other related results.
# We stress the importance of describing the software employed as well as its related options.

===1.3. Document Format===

Any document is considered as a Web-page. As such it should have a main page in HTML and all other objects linked to the main page such as pictures and plots (gif, jpeg), additional documents (pdf, html). In particular, a document can have a small introductory part written in HTML and the main part as a linked pdf.

The authors are advised to keep the layout simple.

Scripts included in the Web-page should be avoided, or at least they must not be obligatory to view the page.

Numerical data including the original dynamic system and the simulation results should be given in a special format described in Section 3.

==2. Publishing Method==

A document is submitted to morwiki in the electronic form (see below) as an archive of all the appropriate files (tar.gz or zip). Then it is placed in a special area and enters a reviewing stage for four weeks. An announcement about a new document is send to reviewers chosen by an editorial board. Depending on the comments, the document is published, rejected or sent to authors to make corrections. The decision is taken by an editorial board.

A published document is never changed or deleted. Rather, a new version of the document is submitted again and it is published as a new document if accepted. In this case, the old document receives a status "expired" but it still remains in the public area.

===2.1. Rules for Online Submission===

* Only ZIP or TAR.GZ archives are accepted for the submission.
* The maximum compressed size for these files is 15 Mb.
* The archive should contain at least one HTML file, named “index.html”. This file represents the main document file.
* The archive may only contain files of the following types: *.html, *.htm, *.pdf, *.gif, *.jpg, *.png, *.zip, *.tar.gz
* After the submission, the files are post-processed:
**File types not specified above are deleted.
**Only the body part of every HTML file is kept.
**All the format/style/css information, like “style=..”, “class=..” are removed from the body part.
* If you decide to use PDF documents, use the “index.html” to include links to them.
* There are three states of the submission:
**Submitted: The author and the chief editor receive a notification mail. The submission is only accessible for the chief editor to accept the submission.
**Opened for review: The submission is open for certain users to post their comments and reviews. After that the chief editor can accept the paper.
**Accepted: The submission is open for everybody.

==3. Data files==

All the numerical data for the collection can be considered as a list of matrices, a vector being a m x 1 matrix. As a result, there should be a naming convention for the matrices.

===3.1. Naming convention===

Below <math>x</math> is a state vector, <math>\mu</math> is a vector of parameters in the system, if any.

For the two cases of a linear system of the first and second orders, the naming convention should be written as follows

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

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

We recommend to use for nonlinear models

<math>
\begin{align}
M(\mu) \ddot x + E(\mu) \dot x + K(\mu) x &= B(\mu) u + F(\mu) g(x,u,\mu)\\
y &= C(\mu) x + D(\mu) f(x, u,\mu)
\end{align}
</math>

An author can use another notation in the case when the convention above is not appropriate.

===3.2. Matrix format for linear non-parametric or parametric affine systems===

For the linear non-parametric systems and the parametric affine systems, the matrices of the systems are constant matrices. Such matrices should be written in the Matrix Market format as described at http://math.nist.gov/MatrixMarket/.

A file with a matrix should be named as

problem_name.matrix_name

If there is no file for a matrix, it is assumed to be identity for the E matrix and 0 for the D matrix.

All matrix files for a given problem should be compressed in a single zip or tar.gz archive, there should be a folder in the archive which is named with "problem_name" or with the abbreviated problem name.

===3.3. Matrix format for nonlinear or parametric non-affine systems===

For the parametric non-affine systems, or nonlinear systems, the system matrices are coupled with the parameters and/or the state vectors. In this case, the system matrices should be analytically described if possible. For example, <math>A(1,1)=x_1^2+x_3</math>, <math>A(i,2)=\mu_i^3 x_i, i=1,2,\ldots m</math>, etc., where <math>\mu_i</math> are the parameters, and <math>x_i</math> is the <math>i</math>th entry in the state vector <math>x</math>. If the system matrices are too complicated to be described analytically, or in plain text, it should be allowed to get contact for the source code.

==4. References==

[1] Younès Chahlaoui and Paul Van Dooren. A collection of benchmark examples for model reduction of linear time invariant dynamical systems; SLICOT Working Note 2002-2: February 2002, http://www.win.tue.nl/niconet/NIC2/benchmodred.html.

Editors:

'' [[User:Feng]] ''

'' [[User:Baur]] ''

'' [[User:Grundel]] ''

Submission rules

2012-11-24T13:08:48Z

Will: /* 1.2. Report */

The PMOR benchmark collection contains benchmarks related to model order reduction. Its goal is to supply the dynamical systems in a computer-readable format to researches from different areas. They can then test new algorithms, software, etc.

The collection contains documents that are supposed to be processed by a human being and data files that are supposed to be read automatically by software. As a result, with minor exceptions, the rules concerning documents have a recommendation status, and the rules concerning data files are obligatory.

We start with the description of documents, then we present the publication policy, and finally we describe the data files.

==1. Documents==

The collection consists of documents forming a two-level hierarchy. Top-level documents will be referred to as benchmarks. Each benchmark document may have links to several documents referred to as reports. A benchmark and its reports may be written by different authors.

Each document is written according to conventional scientific practice, that is, it describes the matters in such a way that, at least in principle, anyone could reproduce the results presented. The authors should understand that the document may be read by people from quite different disciplines. Hence, abbreviations should be avoided or at least explained and references to the background ideas should be made.

===1.1. Benchmark===

The goal of a benchmark document is to describe the origin of the dynamic system and its relevance to the application area. It is important to present the mathematical model, the meaning of the inputs and outputs, and the desired behavior from the application viewpoint.

A few points to be addressed:

* The purpose of the model should be explained clearly. (For instance, simulation, iterative system design, feedback control design, ...)
* Why should the model be reduced at all? (For instance, reducing simulation time, reducing implementation effort in observers, controllers...)
* What are the QUALITATIVE requirements to the reduced model? What variables are to be approximated well? Is the step response to be approximated or is it the bode plot? What are typical input signals? (Some systems are driven by a step function and nothing else, others are driven by a wide variety of input signals, others are used in closed loops and can cause instabilities, although being stable themselves.)
* What are the QUANTITATIVE requirements to the reduced model? Best would be if the authors of any individual model can suggest some cost functions (performance indices) to be used for the comparison. These can be in time domain, or in frequency domain (including bandwidth), or both.
* Are there limits of input and state variables known? (application related or generally)? What are the physical limits where the model becomes useless/false? If known a-priori: Out of the technically typical input signals, which one will cause "the most nonlinear" behavior?
* How many parameters are included in the model, are they physical parameters or geometrical parameters.
* Is the system linear or nonlinear? Is it a first or a second order system?
* When the model is used to test, e.g. an algorithm, can some of the parameters in the model be fixed, such that the model includes fewer parameters, which makes the testing easier?
* If possible, the source code which generates the model is encouraged to be provided.

* We stress the importance to describe the software employed, as well as its related options. For example, if the model is generated from ANSYS, the software ANSYS is better to be mentioned. If necessary, please also describe some related options like boundary conditions, initial conditions, and other parameters which need to be assigned when generating the model. Please go to the benchmark webpage [[Silicon nitride membrane]] for a reference.

* If the dynamic system is obtained from partial differential equations, then the information about material properties, geometrical data, initial and boundary conditions should be given. The exception to this rule is the case when the original model comes from industry. In this case, if trade secrets are tied with the information mentioned, it may be kept hidden.

* The authors are encouraged to produce several dynamic models of different dimensions in order to provide an opportunity to apply different software and to research scalability issues. If an author has an interactive page on his/her server to generate benchmarks, a link to this page is welcome.

* The dynamic system may be obtained by means of compound matrices, for example, when the second-order system is converted to first-order. In this case, the document should describe such a transformation but in the data file the original and not the compound matrices should be given. This allows to research other ways of model reduction of the original system.

The above information is in a general sense. Therefore, some items may not be applicable to certain benchmarks. However, it is recommended that the items that apply to the benchmark provided should be fully considered.

===1.2. Report===

A report document may contain:

# The solution of the original benchmark that contains sample outputs for the usual input signals. Plots and numerical values of time and frequency response. Eigenvalues and eigenvectors, singular values, poles, zeros, etc.
# Model reduction and its results as compared to the original system.
# Description of any other related results.
# We stress the importance of describing the software employed as well as its related options.

===1.3. Document Format===

Any document is considered as a Web-page. As such it should have a main page in HTML and all other objects linked to the main page such as pictures and plots (gif, jpeg), additional documents (pdf, html). In particular, a document can have a small introductory part written in HTML and the main part as a linked pdf.

The authors are advised to keep the layout simple.

Scripts included in the Web-page should be avoided, or at least they must not be obligatory to view the page.

Numerical data including the original dynamic system and the simulation results should be given in a special format described in Section 3.

==2. Publishing Method==

A document is submitted to morwiki in the electronic form (see below) as an archive of all the appropriate files (tar.gz or zip). Then it is placed in a special area and enters a reviewing stage for four weeks. An announcement about a new document is send to reviewers chosen by an editorial board. Depending on the comments, the document is published, rejected or sent to authors to make corrections. The decision is taken by an editorial board.

A published document is never changed or deleted. Rather, a new version of the document is submitted again and it is published as a new document if accepted. In this case, the old document receives a status "expired" but it still remains in the public area.

===2.1. Rules for Online Submission===

* Only ZIP or TAR.GZ archives are accepted for the submission.
* The maximum compressed size for these files is 15 Mb.
* The archive should contain at least one HTML file, named “index.html”. This file represents the main document file.
* The archive may only contain files of the following types: *.html, *.htm, *.pdf, *.gif, *.jpg, *.png, *.zip, *.tar.gz
* After the submission, the files are post-processed:
**File types not specified above are deleted.
**Only the body part of every HTML file is kept.
**All the format/style/css information, like “style=..”, “class=..” are removed from the body part.
* If you decide to use PDF documents, use the “index.html” to include links to them.
* There are three states of the submission:
**Submitted: The author and the chief editor receive a notification mail. The submission is only accessible for the chief editor to accept the submission.
**Opened for review: The submission is open for certain users to post their comments and reviews. After that the chief editor can accept the paper.
**Accepted: The submission is open for everybody.

==3. Data files==

All the numerical data for the collection can be considered as a list of matrices, a vector being a m x 1 matrix. As a result, there should be a naming convention for the matrices.

===3.1. Naming convention===

Below <math>x</math> is a state vector, <math>\mu</math> is a vector of parameters in the system, if any.

For the two cases of a linear system of the first and second orders, the naming convention should be written as follows

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

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

We recommend to use for nonlinear models

<math>
\begin{align}
M(\mu) \ddot x + E(\mu) \dot x + K(\mu) x &= B(\mu) u + F(\mu) g(x,u,\mu)\\
y &= C(\mu) x + D(\mu) f(x, u,\mu)
\end{align}
</math>

An author can use another notation in the case when the convention above is not appropriate.

===3.2. Matrix format for linear non-parametric or parametric affine systems===

For the linear non-parametric systems and the parametric affine systems, the matrices of the systems are constant matrices. Such matrices should be written in the Matrix Market format as described at http://math.nist.gov/MatrixMarket/.

A file with a matrix should be named as

problem_name.matrix_name

If there is no file for a matrix, it is assumed to be identity for the E matrix and 0 for the D matrix.

All matrix files for a given problem should be compressed in a single zip or tar.gz archive, there should be a folder in the archive which is named with "problem_name" or with the abbreviated problem name.

===3.3. Matrix format for nonlinear or parametric non-affine systems===

For the parametric non-affine systems, or nonlinear systems, the system matrices are coupled with the parameters and/or the state vectors. In this case, the system matrices should be analytically described if possible. For example, <math>A(1,1)=x_1^2+x_3</math>, <math>A(i,2)=\mu_i^3 x_i, i=1,2,\ldots m</math>, etc., where <math>\mu_i</math> are the parameters, and <math>x_i</math> are the <math>i</math>th entry in the state vector <math>x</math>. If the system matrices are too complicated to be described analytically, or in plain text, it should be allowed to get contact for the source code.

==4. References==

[1] Younès Chahlaoui and Paul Van Dooren. A collection of benchmark examples for model reduction of linear time invariant dynamical systems; SLICOT Working Note 2002-2: February 2002, http://www.win.tue.nl/niconet/NIC2/benchmodred.html.

Editors:

'' [[User:Feng]] ''

'' [[User:Baur]] ''

'' [[User:Grundel]] ''

Submission rules

2012-11-24T13:07:31Z

Will: /* 1.1. Benchmark */

The PMOR benchmark collection contains benchmarks related to model order reduction. Its goal is to supply the dynamical systems in a computer-readable format to researches from different areas. They can then test new algorithms, software, etc.

The collection contains documents that are supposed to be processed by a human being and data files that are supposed to be read automatically by software. As a result, with minor exceptions, the rules concerning documents have a recommendation status, and the rules concerning data files are obligatory.

We start with the description of documents, then we present the publication policy, and finally we describe the data files.

==1. Documents==

The collection consists of documents forming a two-level hierarchy. Top-level documents will be referred to as benchmarks. Each benchmark document may have links to several documents referred to as reports. A benchmark and its reports may be written by different authors.

Each document is written according to conventional scientific practice, that is, it describes the matters in such a way that, at least in principle, anyone could reproduce the results presented. The authors should understand that the document may be read by people from quite different disciplines. Hence, abbreviations should be avoided or at least explained and references to the background ideas should be made.

===1.1. Benchmark===

The goal of a benchmark document is to describe the origin of the dynamic system and its relevance to the application area. It is important to present the mathematical model, the meaning of the inputs and outputs, and the desired behavior from the application viewpoint.

A few points to be addressed:

* The purpose of the model should be explained clearly. (For instance, simulation, iterative system design, feedback control design, ...)
* Why should the model be reduced at all? (For instance, reducing simulation time, reducing implementation effort in observers, controllers...)
* What are the QUALITATIVE requirements to the reduced model? What variables are to be approximated well? Is the step response to be approximated or is it the bode plot? What are typical input signals? (Some systems are driven by a step function and nothing else, others are driven by a wide variety of input signals, others are used in closed loops and can cause instabilities, although being stable themselves.)
* What are the QUANTITATIVE requirements to the reduced model? Best would be if the authors of any individual model can suggest some cost functions (performance indices) to be used for the comparison. These can be in time domain, or in frequency domain (including bandwidth), or both.
* Are there limits of input and state variables known? (application related or generally)? What are the physical limits where the model becomes useless/false? If known a-priori: Out of the technically typical input signals, which one will cause "the most nonlinear" behavior?
* How many parameters are included in the model, are they physical parameters or geometrical parameters.
* Is the system linear or nonlinear? Is it a first or a second order system?
* When the model is used to test, e.g. an algorithm, can some of the parameters in the model be fixed, such that the model includes fewer parameters, which makes the testing easier?
* If possible, the source code which generates the model is encouraged to be provided.

* We stress the importance to describe the software employed, as well as its related options. For example, if the model is generated from ANSYS, the software ANSYS is better to be mentioned. If necessary, please also describe some related options like boundary conditions, initial conditions, and other parameters which need to be assigned when generating the model. Please go to the benchmark webpage [[Silicon nitride membrane]] for a reference.

* If the dynamic system is obtained from partial differential equations, then the information about material properties, geometrical data, initial and boundary conditions should be given. The exception to this rule is the case when the original model comes from industry. In this case, if trade secrets are tied with the information mentioned, it may be kept hidden.

* The authors are encouraged to produce several dynamic models of different dimensions in order to provide an opportunity to apply different software and to research scalability issues. If an author has an interactive page on his/her server to generate benchmarks, a link to this page is welcome.

* The dynamic system may be obtained by means of compound matrices, for example, when the second-order system is converted to first-order. In this case, the document should describe such a transformation but in the data file the original and not the compound matrices should be given. This allows to research other ways of model reduction of the original system.

The above information is in a general sense. Therefore, some items may not be applicable to certain benchmarks. However, it is recommended that the items that apply to the benchmark provided should be fully considered.

===1.2. Report===

A report document may contain:

# The solution of the original benchmark that contains sample outputs for the usual input signals. Plots and numerical values of time and frequency response. Eigenvalues and eigenvectors, singular values, poles, zeros, etc.
# Model reduction and its results as compared to the original system.
# Description of any other related results.
# we stress the importance to describe the software employed as well as its related options.

===1.3. Document Format===

Any document is considered as a Web-page. As such it should have a main page in HTML and all other objects linked to the main page such as pictures and plots (gif, jpeg), additional documents (pdf, html). In particular, a document can have a small introductory part written in HTML and the main part as a linked pdf.

The authors are advised to keep the layout simple.

Scripts included in the Web-page should be avoided, or at least they must not be obligatory to view the page.

Numerical data including the original dynamic system and the simulation results should be given in a special format described in Section 3.

==2. Publishing Method==

A document is submitted to morwiki in the electronic form (see below) as an archive of all the appropriate files (tar.gz or zip). Then it is placed in a special area and enters a reviewing stage for four weeks. An announcement about a new document is send to reviewers chosen by an editorial board. Depending on the comments, the document is published, rejected or sent to authors to make corrections. The decision is taken by an editorial board.

A published document is never changed or deleted. Rather, a new version of the document is submitted again and it is published as a new document if accepted. In this case, the old document receives a status "expired" but it still remains in the public area.

===2.1. Rules for Online Submission===

* Only ZIP or TAR.GZ archives are accepted for the submission.
* The maximum compressed size for these files is 15 Mb.
* The archive should contain at least one HTML file, named “index.html”. This file represents the main document file.
* The archive may only contain files of the following types: *.html, *.htm, *.pdf, *.gif, *.jpg, *.png, *.zip, *.tar.gz
* After the submission, the files are post-processed:
**File types not specified above are deleted.
**Only the body part of every HTML file is kept.
**All the format/style/css information, like “style=..”, “class=..” are removed from the body part.
* If you decide to use PDF documents, use the “index.html” to include links to them.
* There are three states of the submission:
**Submitted: The author and the chief editor receive a notification mail. The submission is only accessible for the chief editor to accept the submission.
**Opened for review: The submission is open for certain users to post their comments and reviews. After that the chief editor can accept the paper.
**Accepted: The submission is open for everybody.

==3. Data files==

All the numerical data for the collection can be considered as a list of matrices, a vector being a m x 1 matrix. As a result, there should be a naming convention for the matrices.

===3.1. Naming convention===

Below <math>x</math> is a state vector, <math>\mu</math> is a vector of parameters in the system, if any.

For the two cases of a linear system of the first and second orders, the naming convention should be written as follows

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

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

We recommend to use for nonlinear models

<math>
\begin{align}
M(\mu) \ddot x + E(\mu) \dot x + K(\mu) x &= B(\mu) u + F(\mu) g(x,u,\mu)\\
y &= C(\mu) x + D(\mu) f(x, u,\mu)
\end{align}
</math>

An author can use another notation in the case when the convention above is not appropriate.

===3.2. Matrix format for linear non-parametric or parametric affine systems===

For the linear non-parametric systems and the parametric affine systems, the matrices of the systems are constant matrices. Such matrices should be written in the Matrix Market format as described at http://math.nist.gov/MatrixMarket/.

A file with a matrix should be named as

problem_name.matrix_name

If there is no file for a matrix, it is assumed to be identity for the E matrix and 0 for the D matrix.

All matrix files for a given problem should be compressed in a single zip or tar.gz archive, there should be a folder in the archive which is named with "problem_name" or with the abbreviated problem name.

===3.3. Matrix format for nonlinear or parametric non-affine systems===

For the parametric non-affine systems, or nonlinear systems, the system matrices are coupled with the parameters and/or the state vectors. In this case, the system matrices should be analytically described if possible. For example, <math>A(1,1)=x_1^2+x_3</math>, <math>A(i,2)=\mu_i^3 x_i, i=1,2,\ldots m</math>, etc., where <math>\mu_i</math> are the parameters, and <math>x_i</math> are the <math>i</math>th entry in the state vector <math>x</math>. If the system matrices are too complicated to be described analytically, or in plain text, it should be allowed to get contact for the source code.

==4. References==

[1] Younès Chahlaoui and Paul Van Dooren. A collection of benchmark examples for model reduction of linear time invariant dynamical systems; SLICOT Working Note 2002-2: February 2002, http://www.win.tue.nl/niconet/NIC2/benchmodred.html.

Editors:

'' [[User:Feng]] ''

'' [[User:Baur]] ''

'' [[User:Grundel]] ''

Submission rules

2012-11-24T13:00:34Z

Will: /* 1. Documents */

The PMOR benchmark collection contains benchmarks related to model order reduction. Its goal is to supply the dynamical systems in a computer-readable format to researches from different areas. They can then test new algorithms, software, etc.

The collection contains documents that are supposed to be processed by a human being and data files that are supposed to be read automatically by software. As a result, with minor exceptions, the rules concerning documents have a recommendation status, and the rules concerning data files are obligatory.

We start with the description of documents, then we present the publication policy, and finally we describe the data files.

==1. Documents==

The collection consists of documents forming a two-level hierarchy. Top-level documents will be referred to as benchmarks. Each benchmark document may have links to several documents referred to as reports. A benchmark and its reports may be written by different authors.

Each document is written according to conventional scientific practice, that is, it describes the matters in such a way that, at least in principle, anyone could reproduce the results presented. The authors should understand that the document may be read by people from quite different disciplines. Hence, abbreviations should be avoided or at least explained and references to the background ideas should be made.

===1.1. Benchmark===

The goal of a benchmark document is to describe the origin of the dynamic system and its relevance to the application area. It is important to present the mathematical model, the meaning of the inputs and outputs and the desired behavior from the application viewpoint.

A few points to be addressed:

* The purpose of the model should be explained clearly. (For instance, simulation, iterative system design, feedback control design, ...)
* Why should the model be reduced at all? (For instance, reducing simulation time, reducing implementation effort in observers, controllers...)
* What are the QUALITATIVE requirements to the reduced model? What variables are to be approximated well? Is the step response to be approximated or is it the bode plot? What are typical input signals? (Some systems are driven by a step function and nothing else, others are driven by a wide variety of input signals, others are used in closed loops and can cause instabilities, although being stable themselves)
* What are the QUANTITATIVE requirements to the reduced model? Best would be if the authors of any individual model can suggest some cost functions (performance indices) to be used for the comparison. These can be in time domain, or in frequency domain (including bandwidth), or both.
* Are there limits of input and state variables known? (application related or generally)? What are the physical limits where the model becomes useless/false? If known a-priori: Out of the technically typical input signals, which one will cause "the most nonlinear" behavior?
* How many parameters are included in the model, are they physical parameters or geometrical parameters.
* Is the system linear or nonlinear? Is it a first or second order system?
* When the model is used to test, e.g. an algorithm, can some of the parameters in the model be fixed, such that the model includes fewer parameters, which makes the testing easier?
* If possible, the source code which generates the model is encouraged to be provided.

* We stress the importance to describe the software employed, as well as its related options. For example, if the model is generated from ANSYS, the software ANSYS is better to be mentioned. If necessary, please also describe some related options like boundary conditions, initial conditions, and other parameters need to be assigned when generating the model. Please go to the benchmark webpage [[Silicon nitride membrane]] for a reference.

* If the dynamic system is obtained from partial differential equations, then the information about material properties, geometrical data, initial and boundary conditions should be given. The exception to this rule is the case when the original model came from industry. In this case, if trade secrets are tied with the information mentioned, it may be kept hidden.

* The authors are encouraged to produce several dynamic models of different dimensions in order to provide an opportunity to apply different software and to research scalability issues. If an author has an interactive page on his/her server to generate benchmarks, a link to this page is welcome.

* The dynamic system may be obtained by means of compound matrices, for example, when the second-order system is converted to first-order. In this case, the document should describe such a transformation but in the data file the original and not the compound matrices should be given. In this way, this will allow us to research other ways of model reduction of the original system.

The above information is in a general sense. Therefore, some items may not applicable to certain benchmarks. However, it is recommended that the items apply to the benchmark provided should be fully considered.

===1.2. Report===

A report document may contain:

# The solution of the original benchmark that contains sample outputs for the usual input signals. Plots and numerical values of time and frequency response. Eigenvalues and eigenvectors, singular values, poles, zeros, etc.
# Model reduction and its results as compared to the original system.
# Description of any other related results.
# we stress the importance to describe the software employed as well as its related options.

===1.3. Document Format===

Any document is considered as a Web-page. As such it should have a main page in HTML and all other objects linked to the main page such as pictures and plots (gif, jpeg), additional documents (pdf, html). In particular, a document can have a small introductory part written in HTML and the main part as a linked pdf.

The authors are advised to keep the layout simple.

Scripts included in the Web-page should be avoided, or at least they must not be obligatory to view the page.

Numerical data including the original dynamic system and the simulation results should be given in a special format described in Section 3.

==2. Publishing Method==

A document is submitted to morwiki in the electronic form (see below) as an archive of all the appropriate files (tar.gz or zip). Then it is placed in a special area and enters a reviewing stage for four weeks. An announcement about a new document is send to reviewers chosen by an editorial board. Depending on the comments, the document is published, rejected or sent to authors to make corrections. The decision is taken by an editorial board.

A published document is never changed or deleted. Rather, a new version of the document is submitted again and it is published as a new document if accepted. In this case, the old document receives a status "expired" but it still remains in the public area.

===2.1. Rules for Online Submission===

* Only ZIP or TAR.GZ archives are accepted for the submission.
* The maximum compressed size for these files is 15 Mb.
* The archive should contain at least one HTML file, named “index.html”. This file represents the main document file.
* The archive may only contain files of the following types: *.html, *.htm, *.pdf, *.gif, *.jpg, *.png, *.zip, *.tar.gz
* After the submission, the files are post-processed:
**File types not specified above are deleted.
**Only the body part of every HTML file is kept.
**All the format/style/css information, like “style=..”, “class=..” are removed from the body part.
* If you decide to use PDF documents, use the “index.html” to include links to them.
* There are three states of the submission:
**Submitted: The author and the chief editor receive a notification mail. The submission is only accessible for the chief editor to accept the submission.
**Opened for review: The submission is open for certain users to post their comments and reviews. After that the chief editor can accept the paper.
**Accepted: The submission is open for everybody.

==3. Data files==

All the numerical data for the collection can be considered as a list of matrices, a vector being a m x 1 matrix. As a result, there should be a naming convention for the matrices.

===3.1. Naming convention===

Below <math>x</math> is a state vector, <math>\mu</math> is a vector of parameters in the system, if any.

For the two cases of a linear system of the first and second orders, the naming convention should be written as follows

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

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

We recommend to use for nonlinear models

<math>
\begin{align}
M(\mu) \ddot x + E(\mu) \dot x + K(\mu) x &= B(\mu) u + F(\mu) g(x,u,\mu)\\
y &= C(\mu) x + D(\mu) f(x, u,\mu)
\end{align}
</math>

An author can use another notation in the case when the convention above is not appropriate.

===3.2. Matrix format for linear non-parametric or parametric affine systems===

For the linear non-parametric systems and the parametric affine systems, the matrices of the systems are constant matrices. Such matrices should be written in the Matrix Market format as described at http://math.nist.gov/MatrixMarket/.

A file with a matrix should be named as

problem_name.matrix_name

If there is no file for a matrix, it is assumed to be identity for the E matrix and 0 for the D matrix.

All matrix files for a given problem should be compressed in a single zip or tar.gz archive, there should be a folder in the archive which is named with "problem_name" or with the abbreviated problem name.

===3.3. Matrix format for nonlinear or parametric non-affine systems===

For the parametric non-affine systems, or nonlinear systems, the system matrices are coupled with the parameters and/or the state vectors. In this case, the system matrices should be analytically described if possible. For example, <math>A(1,1)=x_1^2+x_3</math>, <math>A(i,2)=\mu_i^3 x_i, i=1,2,\ldots m</math>, etc., where <math>\mu_i</math> are the parameters, and <math>x_i</math> are the <math>i</math>th entry in the state vector <math>x</math>. If the system matrices are too complicated to be described analytically, or in plain text, it should be allowed to get contact for the source code.

==4. References==

[1] Younès Chahlaoui and Paul Van Dooren. A collection of benchmark examples for model reduction of linear time invariant dynamical systems; SLICOT Working Note 2002-2: February 2002, http://www.win.tue.nl/niconet/NIC2/benchmodred.html.

Editors:

'' [[User:Feng]] ''

'' [[User:Baur]] ''

'' [[User:Grundel]] ''

Branchline Coupler

2012-11-24T12:55:25Z

Will:

[[Category:benchmark]]
[[Category:parametric system]]
[[Category:linear system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:two parameters]]
[[Category:second order system]]

==Model Description==

A branchline coupler is a microwave semiconductor device, which is simulated by the time-harmonic Maxwell's equations.
A 2-section branchline coupler consists of four strip line ports, coupled by two transversal bridges with each other.
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 0.05 mm thickness is placed on a substrate with 0.749 mm thickness and relative permittivity
<math> \epsilon_r = 2.2 </math> and zero-conductivity <math> \sigma = 0 S/m </math>.
The simulation domain is confined to a <math> 23.6 \times 22 \times 7 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 50 ohm
imposes 1 A 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.

[[File:BranchlineCoupler.png]]

==Matrices and Data==

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 Finite Element Method, resulting in 27'679 degrees of freedom, after removal of boundary conditions. The files are numbered according to their
appearance in the summation.

[[File:Matrices_BranchlineCoupler.tar.gz]]

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 [1.0, 10.0] * 10^9 </math> Hz, 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>.

==References==

The models have been developed within the MoreSim4Nano project.

[1] www.moresim4nano.org

[2] M. W. Hess, P. Benner, Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method, MPI preprint
http://www.mpi-magdeburg.mpg.de/preprints/2012/MPIMD12-17.pdf

Contact information:

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

Category:Parametric

2012-11-12T09:34:46Z

Will: Blanked the page

Main Page

2012-01-04T11:55:04Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing benchmarks of parametric or non-parametric models and pages explaining applicable (P)MOR methods.

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
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== List of all Categories in the Wiki ==
{{special:categories}}

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{{special:allpages}}

Main Page

2012-01-04T11:54:20Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing benchmarks of parametric or non-parametric models and pages explaining applicable (P)MOR methods (see [[Categories]]).

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:52:57Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing benchmarks (see [[Category:Benchmark]]) of parametric or non-parametric models and pages explaining applicable (P)MOR methods.

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:52:36Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing benchmarks (see [[category:benchmark]]) of parametric or non-parametric models and pages explaining applicable (P)MOR methods.

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:51:53Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing [[category:benchmark benchmarks]] of parametric or non-parametric models and pages explaining applicable (P)MOR methods.

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:51:38Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing [category:benchmark benchmarks] of parametric or non-parametric models and pages explaining applicable (P)MOR methods.

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:49:44Z

Will:

Main Page

2012-01-04T11:48:42Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing [[benchmark]] of parametric or non-parametric models and pages explaining applicable (P)MOR [[method]].

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:44:27Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing [[benchmark]]s of parametric or non-parametric models and pages explaining applicable (P)MOR [[method]]s.

Following the [[submission rules]], one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:42:57Z

Will:

Main Page

2012-01-04T11:39:53Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

MOR Wiki is divided in pages providing benchmarks of parametric or non-parametric models and pages explaining applicable (P)MOR methods.

Following the submission rules, one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:37:08Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas and examples.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

In MOR Wiki, one can find pages which provide benchmarks of parametric or non-parametric models, and some pages which explain applicable (P)MOR methods.

Following the submission rules, one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Main Page

2012-01-04T11:36:28Z

Will:

<big>'''Welcome to MOR Wiki'''</big>

The purpose of MOR Wiki is to bring together experts in the area of model order reduction (MOR), as well as researchers from application areas in an attempt to provide a platform for exchanging ideas.

Modeling and numerical simulation are unavoidable in many application and research areas, e.g. reaction processes, micro-electro-mechanical systems (MEMS) design, and control design. The processes or devices can be modeled by partial differential equations (PDEs). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations (ODEs), or differential algebraic equations (DAEs).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate the large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR has been recognized as very efficient for reducing the simulation time of large-scale systems. Through model order reduction, a small system with reduced number of equations (reduced model) is derived. The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model. As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude. The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can further save much time.

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as Integrated Circuit (IC) design, MEMS design, and chemical engineering. The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate. The reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

In MOR Wiki, one can find pages which provide benchmarks of parametric or non-parametric models, and some pages which explain applicable (P)MOR methods.

Following the submission rules, one can also submit new benchmarks or method pages.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

== List of all Categories in the Wiki ==
{{special:categories}}

== List of all Pages Currently in the Wiki ==
{{special:allpages}}

Category:Time varying

2011-12-12T10:09:10Z

Will:

[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

All benchmark pages describing time varying systems are part of this category.

Category:Time invariant

2011-12-12T10:08:51Z

Will:

Category:Nonzero initial condition

2011-12-12T10:03:05Z

Will:

[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Although reduced basis methods can deal with systems with non-zero initial conditions, some MOR methods like Gramian based MOR ( e.g. balanced truncation ) or moment-matching MOR, cannot be directly applied to systems with non-zero initial conditions. However, systems with non-zero initial conditions are very common, for example, in the filed of chemical engineering, and hence deserve attention.

Category:Time varying

2011-12-08T16:10:48Z

Will:

[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Category:Time invariant

2011-12-08T16:10:36Z

Will:

[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Category:Nonzero initial condition

2011-12-08T16:09:53Z

Will:

[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Microthruster Unit

2011-12-08T16:09:02Z

Will:

[[Category:benchmark]]
[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:three parameters]]

This parametric model is collected in the Oberwolfach Model Reduction Benchmark Collection. The file describing the model is named "Boundary Condition Independent Thermal Model". The parametric model is a 2D-axisymmetric microthruster model with 3 independent parameters. By following this [http://simulation.uni-freiburg.de/downloads/benchmark/Thermal%20Model%20%2838865%29/ link], one can have the detailed description of the parameters and the model. The data of the model can also be downloaded there.

Modified Gyroscope

2011-12-08T16:08:50Z

Will:

[[Category:benchmark]]
[[Category:linear system]]
[[Category:parametric system]]
[[Category:time invariant]]
[[Category:geometric parameters]]
[[Category:two parameters]]
[[Category:second order system]]

==Description of the device==
The device is a MEMS gyroscope based on the butterfly gyroscope [1] developed at the Imego institute in Gothenburg, Sweden (see also: http://simulation.uni-freiburg.de/downloads/benchmark/The Butterfly Gyro (35889)). A gyroscope is a device used to measure angular rates in up to three axes.

The basic working principle of the gyroscope can be described as follows, see also [2]. Without applied external rotation, the paddles vibrate in phase
with the function <math>z(t),</math> see Fig. 1 below. Under the influence of an external rotation about the <math>x</math>-axis (drawn
in red), an additional force due to the Coriolis acceleration acts upon the
paddles. This force leads to an additional small out-of-phase vibration
between two paddles on the same side of the bearing. This
out-of phase vibration is measured as the difference of the z-displacement
of the nodes with the red dots. Thus, measuring the displacement of two
adjacent paddles, the rotation velocity can be ascertained.

==Motivation of MOR==

When planning for and making decisions on future improvements of the Butterfly Gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==Description of the parametrized model==

Two parameters are of special interest for the model. The first one is the
quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by
finite element discretization of the parametrized model (in the form of partial differential equations) for the gyroscope. The details of constructing the parametrized system can be found in [2]. The system is of the following
form:

<math>
M(d)\ddot{x}+D(\theta)\dot{x}+T(d)x=B
y=Cx.
</math>

Here,
<math>M(d)=(M_1+dM_2)\in \mathbb R^{n\times n}</math> is the mass matrix,

<math>D(\theta)=\theta(D_1+dD_2)\in \mathbb R^{n\times n}</math> is the damping matrix,

<math>T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}</math> is the stiffness matrix,

<math>B \in \mathbb R^{n\times m_I}</math> is the load vector, <math>C \in \mathbb R^{m_O\times n}</math> is the output matrix, <math>x \in \mathbb R^{n}</math> is the state vector, and <math>y \in \mathbb R^{m_O}</math> is the output response.

The variables <math>d</math> and <math>\theta</math> are the parameters of the system, where
<math>d</math> is the width of the
bearing and <math>\theta</math> is the rotation velocity along the <math>x</math>-axis.

The interesting output <math>y</math> of the system is <math>\delta</math> which is the
difference of the displacement <math>z(t)</math> between the two red dots on
the same side of the bearing (see Fig.1). The number of degrees of freedom is <math>n=17913</math>.

The interesting range for the parameters are: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>. The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz.

==Data information==

The system matrices <math>M_1, \, M_2, \, D_1, \, D_2, \, T_1, \, T_2</math>, and <math> T_3</math> are in the MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[File: matrices.tgz]]. <math>M_1</math> corresponds to the file Bmass1.dat, <math>M_2</math> corresponds to the file Bmass2.dat., <math>D_1, \, D_2</math> correspond to the files Bdamp1.dat, Bdamp2.dat, respectively. <math>T_1, \, T_2, \, T_3</math> correspond to the files Bstiff1.dat, Bstiff2.dat, Bstiff3.dat, respectively.
The load vector <math>B</math> can be obtained from the file Bload.dat.

==References==

[1] J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "MEMS Compact Modeling Meets Model Order Reduction: Examples of the
Application of Arnoldi Methods to Microsystem Devices," Nanotech, 2004, pp. 303–306.

[2] C. Moosmann, "ParaMOR
Model Order Reduction for parameterized MEMS applications," PhD thesis, Department of Microsystems Engineering,
University of Freiburg, 2007.

[3] B. Salimbahrami, R. Eid, B. Lohmann, "Model Reduction by Second Order
Krylov Subspaces: Extensions, Stability and Proportional Damping,"
IEEE International Symposium on Intelligent Control, 2006, pp. 2997–3002.

Fig.1
[[File:Gyroscope.jpg]]

Scanning Electrochemical Microscopy

2011-12-08T16:08:41Z

Will:

[[Category:benchmark]]
[[Category:parametric system]]
[[Category:linear system]]
[[Category:time varying]]
[[Category:two parameters]]
[[Category:first order system]]
[[Category:nonzero initial condition]]

==Description of the process==

Scanning Electrochemical Microscopy (SECM) has many applications in current problems in the biological field. Quantitative mathematical models have been developed for different operating modes of the SECM. Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border, solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements. After discretization, a high-dimensional system of ordinary differential equations is obtained. Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig.1. The computation domain under the 2D-axisymmetric approximation includes the electrolyte under the electrode. We assume that the concentration does not depend on the rotation angle. A single chemical reaction takes place on the electrode:

<math>Ox+e^-\Leftrightarrow Red, \quad \quad \quad \quad (1) </math>

where <math>Ox</math> and <math>Red</math> are two different species in the reaction.
According to the theory of SECM [2], the species transport in the electrolyte is described by diffusion only. The diffusion partial differential equation is given by the second Fick's law as follows

<math>
\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \quad
\frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,
</math>

where <math>c_1</math> and <math>c_2</math> are the concentration fields of species <math>Ox</math> and <math>Red</math>, respectively. The initial conditions are <math>c_1(0)=c_{1,0}</math> and <math>c_2(0)=c_{2,0}.</math> Conditions at the glass and the bottom of the bath are described by the Neumann boundary conditions of zero flux
<math>\nabla c_1\cdot \vec{n}=0</math> and <math>\nabla c_2\cdot \vec{n}=0</math>. Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions <math>c_1=c_{1,0}</math> and <math>c_2=c_{2,0}</math>. The boundary conditions at the electrode are described by

<math>
\nabla c_1\cdot \vec{n}=j, \,
\nabla c_2\cdot \vec{n}=-j. \quad \quad \quad \quad (2)
</math>

Here <math>j</math> is related to the forward reaction rate <math>k_f</math> and the backward reaction rate <math>k_b</math> through the Buttler-Volmer equation,

<math>
j=k_f \cdot c_1-k_b \cdot c_2.
</math>

The reaction rates <math>k_f</math> and <math>k_b</math> are in the following form,

<math>
k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}, </math>

<math>k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .
</math>

Here, <math>k^0</math> is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction. <math>F</math> is the Faraday-constant, <math>R</math> is the gas constant, <math>T</math> is the temperature, and <math>z</math> is the number of exchanged electrons per reaction. <math>u(t)=v(t)-v^0</math> is the difference between the electrode potential and the reference potential. This difference, to which we refer below as voltage, is changed during the measurement of a voltammogram.

==Description of the model==

The control volume method has been used for the spatial discretization of (1). Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

<math>
E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad
y(t)=C\vec{c},\quad
\vec{c}(0)=\vec{c}_0 \neq 0,
</math>

where E and <math>K(u(t))</math> are system matrices, <math>K(u(t))</math> is a function of voltage that in turn depends on time. The voltage appears in the system matrix due to the boundary conditions (2). The vector <math>\vec{c} \in \mathbb{R}^n</math> is the vector of unknown concentrations, which includes both the <math>Ox</math> and <math>Red</math> species. The vector <math>B</math> is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface.
The matrix <math>K(u(t))</math> has the following form,

<math>
K(u(t))=K_1(u(t))+K_2(u(t)),
</math>

where <math>K_i(u(t))=h_i D_i, \, i=1,2,</math> and <math>h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))</math>. The voltage <math>u(t)</math> is a function of <math>\sigma</math>,

<math>
u(t)=\sigma t-1, \, \text{for } t \leq \frac{2}{ \sigma}, \quad
u(t)=-\sigma t+3, \, \text{for } \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},
</math>

where <math>\sigma</math> can take four different values, <math>\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005</math>. The constant <math>\beta</math> is computed from the parameters <math>\alpha, \, z, \, F, \, R,</math> and <math>T,</math> leading to the value <math>\beta=21.243036728240824</math>.

Although the system is a time-varying system, it can be considered as a parametrized system with two parameters <math>h_1</math> and <math>h_2</math>.

==Data information==

The data of the system matrices <math>E, \ D_1, \ D_2, \ A, \ B, C</math> as well as the initial state <math>\vec{c}_0=x_0</math> are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[File: matrices_SECM.tgz]]. The interesting output of the model is the current which is computed by <math>I(t)=C(5,:)\vec{c}</math> in MATLAB notation. The interesting plot of the output is called the cyclic voltammogram, which is the plot of the current changing with the voltage <math>u(t)</math>.

Fig.1
[[Image:Fig.1.JPG|thumb|left|300px|]]

==References==

[1] L. Feng, D. Koziol, E. B. Rudnyi, and J. G. Korvink, "Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms," Sensor Letters, Vol. 4, 1-10, 2006, pp.1-10.

[2] M. V. Mirkin, "Theory in scanning electrochemical microscopy," A. J. Bard and M. V. Mirkin, Eds. (2001). New York, John Wiley & Sons. pp. 145 – 199.

Gas Sensor

2011-12-08T16:08:32Z

Will:

[[Category:benchmark]]
[[Category:parametric system]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:physical parameters]]
[[Category:four parameters]]
[[Category:time invariant]]

This is an extension of the non-parametrized model of Gas sensor in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)) to a parametrized model.

==Description of the device==

There is a large demand for gas sensing devices in various domains. They are
desired in e. g. safety applications where combustible or toxic gases are present or
in comfort applications, such as climate controls of buildings and vehicles where
good air quality is required. Additionally, gas monitoring is needed in process
control and laboratory analytics. All of these applications demand cheap, small
and user-friendly gas sensing devices which show high sensitivity, selectivity and
stability with respect to a given application.

A micromachined gas sensor is not only a challenge with respect to thermal
design but also with respect to mechanical design. Only by choosing the right
mechanical design a large intrinsic or thermal-induced membrane stress leading
to membrane deformation/ breaking of the membrane can be avoided. It is further
necessary to build a chemometrics calibration model which correlates the set of
sensor resistance measurements to the sensed gas concentration. Prior to fabrication,
a thermal simulation is performed to determine the heating efficiency and
temperature homogeneity of the gas sensitive regions. As the device is connected to circuitry for heating power
control and sensing resistor readout, a system-level simulation is also needed.
Hence, a compact thermal model must be generated. (The text above is taken from [1].)

==Description of the model==

The heat transfer within a hotplate is described through the governing heat transfer equation [2]

<math>
\nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad
Q=j^2R(T), \quad (1)
</math>

where <math>\kappa(r)</math> is the thermal conductivity in <math>W/(m*K)</math> at the position <math>r, \, c_p</math> is the
specific heat capacity in <math>J /(kg*K), \, \rho(r)</math> is the mass density in
<math>kg /m^3,</math> and <math>T(r,t)</math> is the temperature distribution. We assume a
homogeneous heat generation rate over a lumped resistor:

<math>
Q = \frac{u^2(t)}{R(T)}
</math>

with unit <math>W/m^3</math>.
We use the initial condition <math> T_0 = 273K </math>, and the
Dirichlet boundary condition <math> T = 273 K </math> at the bottom of
the computational domain. The convection boundary condition at the top of the membrane is

<math>
q=h(T-T_{air}),
</math>

where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W/(m^2 K)</math>.

Assuming <math>T_{air}=0</math>, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,

<math>
(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
y=C^T \cdot T.
</math>

Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>, which depends linearly on the temperature. Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=1.469 \times 10^{-3}</math>. The model was created and meshed in ANSYS. It contains a constant load vector corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.

The input function <math>u(t)</math> is a step function with the value <math>1</math>, which disappears at the time <math>0.02s</math>. This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero. However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>. This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off. If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc...
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input. (It is also called a weak nonlinear system.)

==Data information==

The system matrices are in MatrixMarket format (http://math.nist.gov/MatrixMarket/) and can be downloaded here [[File: Matrices_gassensor.tgz]]. The files named by *.<math>A_i, \, i=0,1,2</math> correspond to the system matrices <math>A_i, \, i=0,1,2</math>, respectively. The files named by <math>*.E_i, \, i=0,1,2</math> correspond to <math>E_i, \, i=0,1</math>. The file named by <math>*.B</math> corresponds to the load vector <math>B</math> and the file named by <math>*.C</math> corresponds to the output matrix <math>C</math>.

==References==
[1] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering,
University of Freiburg, 2005.

[2] T. Bechtold, D. Hohfel, E. B. Rudnyi and M. Guenther, "Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction," J. Micromech. Microeng. 20(2010) 045030 (13pp).

Synthetic parametric model

2011-12-08T16:08:09Z

Will:

[[Category:benchmark]]
[[Category:parametric system]]
[[Category:linear system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:one parameter]]
[[Category:first order system]]
[[Category:synthetic model]]

== Introduction ==
On this page you will find a synthetic parametric model for which one can easily experiment with different system orders <math> n </math>, values of the parameter <math> \varepsilon </math>, as well as different poles and residues.

Also, the decay of the Hankel singular values can be changed indirectly through the parameter <math> \varepsilon </math>.

== Model description ==

The parameter <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> H(s,\varepsilon) = \sum_{i=1}^{n}\frac{r_i}{s-p_i} = \sum_{i=1}^{n}\frac{r_i}{s-(\varepsilon a_i+jb_i)} </math>

we can write down the state-space realization

:<math> H(s,\varepsilon) = \widehat{C}\Big(sI-\varepsilon \widehat{A}_\varepsilon - \widehat{A}_0\Big)^{-1}\widehat{B}+D</math>

with system matrices defined as

:<math>\varepsilon \widehat{A}_\varepsilon + \widehat{A}_0 = \varepsilon \left[\begin{array}{ccc} a_1 & & \\ & \ddots & \\ & & a_n\end{array}\right] +\left[\begin{array}{ccc} jb_1 & & \\ & \ddots & \\ & & jb_n\end{array}\right] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

:<math> p_1 = \varepsilon a_1+jb_1, p_2 = \varepsilon a_1-jb_1, \ldots, p_{n-1} = \varepsilon a_k+jb_k, p_n = \varepsilon a_k-jb_k, </math>

and the residues also form complex conjugate pairs

:<math> r_1 = c_1+jd_1, r_2 = c_1-jd_1, \ldots, r_{n-1} = c_k+jd_k, r_n = c_k-jd_k. </math>

Then a realization with matrices having real entries is given by

:<math> A_\varepsilon = \left[\begin{array}{ccc} A_{\varepsilon,1} & & \\ & \ddots & \\ & & A_{\varepsilon,k}\end{array}\right], \quad A_0 = \left[\begin{array}{ccc} A_{0,1} & & \\ & \ddots & \\ & & A_{0,k}\end{array}\right], \quad B = \left[\begin{array}{c} B_1 \\ \vdots \\ B_k\end{array}\right], \quad C = \left[\begin{array}{ccc} C_1 & \cdots & C_k\end{array}\right], \quad D = 0,</math>

with <math> A_{\varepsilon,i} = \left[\begin{array}{cc} a_i& 0 \\ 0 & a_i \end{array}\right] </math>,
<math> A_{0,i} = \left[\begin{array}{cc} 0& b_i \\ -b_i & 0 \end{array}\right] </math>,
<math> B_{i} = \left[\begin{array}{c} 2 \\ 0 \end{array}\right] </math>,
<math> C_{i} = \left[\begin{array}{cc} c_i& d_i\end{array}\right] </math>.

== Numerical values ==

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

* <math>a_i </math> equally spaced in <math> [-10^3, -10]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB, the system matrices are easily formed as follows:

<source lang="matlab">
n = 100;
a = -linspace(1e1,1e3,n/2).'; b = linspace(1e1,1e3,n/2).';
c = ones(n/2,1); d = zeros(n/2,1);
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;
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);
</source>

The above system matrices <math>A_\varepsilon, A_0, B, C</math> are also available in [http://math.nist.gov/MatrixMarket/formats.html MatrixMarket] format [[Media:Synth_matrices.tar.gz|Synth_matrices.tar.gz]].

== Plots ==

We plot the frequency response <math>H(s,\varepsilon) = C\big(sI-\varepsilon A_\varepsilon - A_0\big)^{-1}B</math> and poles for parameter values <math>\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] </math>.

:[[Image:synth_freq_resp.png|frame|border|left|Frequency response of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]
[[Image:synth_poles.png|frame|center|border|Poles of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]

In MATLAB, the plots are generated using the following commands:

<source lang="matlab">
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
for j = 1:length(ep)
p(:,j) = [ep(j)*a+1j*b; ep(j)*a-1j*b]; % poles
[jww,pp] = meshgrid(jw,p(:,j));
Hjw(j,:) = (r.')*(1./(jww-pp)); % freq. resp.
end
figure, loglog(imag(jw),abs(Hjw),'LineWidth',2)
axis tight, xlim([6 1200])
xlabel('frequency (rad/sec)')
ylabel('magnitude')
title('Frequency response for different \epsilon')
figure, plot(real(p),imag(p),'.')
title('Poles for different \epsilon')
</source>

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

Next, for <math>\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] </math>, we also plot the decay of the Hankel singular values. Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

[[Image:synth_hsv.png|frame|border|center|Hankel singular values of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]

'' [[User:Ionita|Ionita]] 14:38, 29 November 2011 (UTC) ''

Anemometer

2011-12-08T16:07:28Z

Will:

[[Category:benchmark]]
[[Category:parametric system]]
[[Category:linear system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:four parameters]]
[[Category:first order system]]

==Description==

An anemometer 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. 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.1 below) from which the fluid
velocity can be determined.

The physical model can be expressed by the
convection-diffusion partial differential equation [4]:

<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,
<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 ANSYS.
Triangular PLANE55 elements have been used for the finite element discretization. The order of the system is <math>n = 29008</math>.

Example with 1 parameter:

The <math>n</math> dimensional ODE system has the following transfer function
<math>
G(p) = c((sE - A_{v0}- p(A_{v1} - A_{v0}))^{-1}b)
</math>

with the fluid velocity <math>p(=v) \in [0, 1]</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 FEM discretization w.r.t. the spatial variables. <math>A_i</math> are the stiffness matrices with <math>i=v0</math> for pure diffusion and <math>i=v1</math> for diffusion and convection. Thus, for obtaining pure convection you have to compute <math>A_{v1} - A_{v0}</math>.

Example with 3 parameters:

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

<math>
G(p_0,p_1,p_2) = c((s \underbrace{(M_s + p_0 M_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 \in [0, 1], \, p_1\in [0.1, 2], \, p_2 \in [1, 2]</math> which are combinations of the original fluid parameters <math>\rho, \, c, \, \kappa, \, \vec v: \quad p_0 = \rho c, \, p_1=\kappa,</math> and <math>p_2 =\rho c v,</math> see [5].

==Provenance==

IMTEK Freiburg, group of Jan Korvink.

==Data information==

Matrices are in the Matrix Market format(http://math.nist.gov/MatrixMarket/). Data for the example with one parameter: [[File: matrices_1d.tar.gz]], and data for the example with three parameters: [[File: matrices_3d.tgz]]. The matrix name is used as an extension of the matrix file.
The system matrices have been extracted from ANSYS models by means of mor4fem.
For more information about permutation, computing the matrices, output, look into readme file [[File: readme1.pdf]].

Example with 1 parameter:

*.B: load vector
*.E: damping matrix
*.C: permutation matrix
*.A: stiffness matrices
<math>\_v0</math>: diffusion
<math>\_v1</math>: diffusion and convection

Example with 3 parameters:

*.B: load vector
*.E: damping matrices (2)
*.A: stiffness matrices (5)

==Related paper==

a) About the anemometer

[1] H. Ernst, "High-Resolution Thermal Measurements in Fluids," PhD thesis, University of Freiburg, Germany (2001).

[2] P. Benner, V. Mehrmann and D. Sorensen, "Dimension Reduction of Large-Scale Systems," Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin/Heidelberg, Germany, 45, 2005.

[3] C. Moosmann and A. Greiner, "Convective Thermal Flow Problems," Chapter 16 (pages 341--343) of [2].

b) MOR for non-parametrized anemometer

[4] C. Moosmann, E. B. Rudnyi, A. Greiner and J. G. Korvink, "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.

c) MOR for parametrized anemometer

[5] U. Baur, P. Benner, A. Greiner, J. G. Korvink, J. Lienemann and C. Moosmann, "Parameter preserving model order reduction for MEMS applications," MCMDS Mathematical and Computer Modeling of Dynamical Systems, 17(4):297--317, 2011.

[6] C. Moosmann, "ParaMOR - Model Order Reduction for parameterized MEMS applications," PhD thesis, University of Freiburg, Germany (2007).

[7] C. Moosmann, E. B. Rudnyi, A. Greiner, J. G. Korvink and M. Hornung, "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.

[8] E. B. Rudnyi, C. Moosmann, A. Greiner, T. Bechtold, J. G. Korvink, "Parameter Preserving
Model Reduction for MEMS System-level Simulation and Design," Proceedings of MATHMOD 2006, February 8 -
10, 2006, Vienna University of Technology, Austria.

Fig. 1 2d-model-anemometer

Schematics: [[File:Model_Color.pdf]]

Calculated temperature profile for anemometer function: [[File:ContourPlot30.pdf]]

Moment-matching PMOR method

2011-12-08T15:42:27Z

Will:

[[Category:method]]
[[Category:parametric method]]

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized
systems, where the parameters of the system play an important role
in practical applications such as Integrated Circuit (IC) design,
MEMS design, Chemical engineering etc.. The parameters could be the variables describing
geometrical measurements, material properties, the damping of the
system or the component flow-rate. The reduced models are constructed such that all the
parameters can be preserved with acceptable accuracy.
Usually the time of simulating the reduced model is much shorter
than directly simulating the original large system.

The method introduced here is described in [1] and [2], and applies to a linear parametrized system,
which has the following form in the frequency domain:

<math>
(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad
y=L^{\mathrm{T}}x, \quad \quad \quad \quad (1)
</math>

where <math>s_1, s_2, \ldots, s_{p}</math> are the parameters of the system. They can be any scalar functions of some source parameters, like <math>s_1=e^t</math>, where <math>t</math> is time, or combinations of several physical parameters like <math>s_1=\rho v</math>, where <math>\rho</math> and <math>v</math> are two physical parameters. <math>x(t)\in \mathbb{R}^n</math> is the state vector, <math>u \in \mathbb{R}^{d_I}</math> and <math>y \in
\mathbb{R}^{d_O}</math> are the inputs and outputs of the
system, respectively.

To obtain the reduced model in (2), a
projection matrix <math>V</math> which is independent of all the parameters has
to be computed.

<math>V^T(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)Vx=V^TBu(s_1,\ldots,s_p), </math>

<math>y=L^{\mathrm{T}}Vx. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad(2)
</math>

The matrix <math>V</math> is derived by orthogonalizing a number of moment
matrices of the system in (1), see [1] or [2].

By defining <math>B_M=\tilde{E}^{-1}B, M_i=-\tilde{E}^{-1}E_i,i=1,2,\ldots,p</math> and

<math>
\tilde{E}=E_0+s_1^0E_1+s_2^0E_2+\cdots+s_p^0E_p,
</math>

we can expand <math>x</math> in (1) at <math>s_1, s_2, \ldots, s_p</math> around a set of
expansion points <math>p_0=[s_1^0,s_2^0,\cdots,s_p^0]</math> as below,

<math>
x=[I-(\sigma_1M_1+\ldots +\sigma_pM_p)]^{-1}B_Mu(s_1,\ldots,s_p)
=\sum\limits_{i=0}^{\infty}(\sigma_1M_1+\ldots+\sigma_pM_p)^iB_Mu(s_1,\ldots,s_p).
</math>

Here <math>\sigma_i=s_i-s_i^0, i=1,2,\ldots,p</math>. We call the coefficients
in the above series expansion moment matrices of the parametrized
system, i.e. <math>B_M, M_1B_M, \ldots, M_pB_M, M_1^2B_M, (M_1M_2+M_2M_1)B_M, \ldots, (M_1M_p+M_pM_1)B_M, M_p^2B_M, M_1^3B_M, \ldots</math>. The corresponding moments are those moment
matrices multiplied by <math>L^{\mathrm{T}}</math> from the left. The matrix <math>V</math> can be
generated by first explicitly computing some of the moment matrices
and then orthogonalizing them as suggested in [1].
The resulting <math>V</math> is desired to expand the subspace:

<math>
\mathop{\mathrm{range}}\{V\}=\mathop{\mathrm{span}}\{B_M, \ M_1B_M,\ldots, M_pB_M,\ M_1^2B_M, \, (M_1M_2+M_2M_1)B_M, \ldots, (M_1M_p+M_pM_1)B_M,
M_p^2B_M, M_1^3B_M,\ldots, M_1^rB_M, \ldots,M_p^rB_M \}. \quad \quad \quad \quad (3)
</math>

However, <math>V</math> does not really span the whole subspace, because the
latterly computed vectors in the subspace become linearly dependent
due to numerical instability. Therefore, with this matrix <math>V</math> one
cannot get an accurate reduced model which matches all the moments
included in the subspace.

Instead of directly computing the moment matrices in (3), a
numerically robust method is proposed in [2] ( the
detailed algorithm is described in [3] ), which combines
the recursion in (5) with the modified Gram-Schmidt
process to implicitly compute the moment matrices. The computed <math>V</math>
is actually an orthonormal basis of the subspace as below,

<math>
\mathop{\mathrm{range}}\{V\}=\mathop{\mathrm{span}}\{R_0, R_1,\ldots, R_r \}. \quad \quad \quad \quad (4)
</math>

<math> R_0 </math>=[ <math>B_M </math> ],

<math>R_1=[M_1R_0,\ldots, M_pR_0], </math>

<math>R_2=[M_1R_1,\ldots, M_pR_1], \quad \quad \quad \quad (5) </math>

<math> \vdots,</math>

<math>R_r=[M_1R_{r-1},\ldots, M_pR_{r-1}]</math>

<math> \vdots.</math>

Due to the numerical stability properties of
the repeated modified Gram-Schmidt process employed in
[2] and [3], the reduced model derived from <math>V</math>
in (4) is computed in a numerically stable and accurate way.

==References==

[1] L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J.~White. "A
multiparameter moment-matching model-reduction approach for
generating geometrically parameterized interconnect performance
models," IEEE Trans. Comput.-Aided Des. Integr.
Circuits Syst, 22(5): 678--693, 2004.

[2] L. Feng and P. Benner, "A Robust Algorithm for Parametric Model
Order Reduction," In Proc. Applied Mathematics and
Mechanics (ICIAM 2007)}, 7(1): 10215.01--02, 2007.

[3] L.~Feng and P.~Benner, "A robust algorithm for parametric model
order reduction based on implicit moment matching," submitted.

Category:Benchmark

2011-12-08T15:41:05Z

Will: Created page with 'All pages describing benchmarks for model order reduction are part of this category.'

All pages describing benchmarks for model order reduction are part of this category.

Category:Method

2011-12-08T15:40:08Z

Will: Created page with 'All pages describing a model order reduction method are part of this category.'

All pages describing a model order reduction method are part of this category.

Bilinear PMOR method

2011-12-08T15:37:47Z

Will:

[[Category:method]]
[[Category:non-linear method]]
[[Category:parametric method]]

The model reduction method we present here is applicable to linear parameter-varying (LPV) systems of the form

<center>
<math>
\dot{x}(t)=Ax(t) + \sum_{i=1}^d p_i(t)A_i x(t)+B_0u_0(t),\quad y(t)=Cx(t),
</math>
</center>

where
<math>
A,A_i \in \mathbb R^{n\times n}, B_0 \in \mathbb R^{n\times m}
</math>
and
<math> C \in \mathbb R^{p\times n}.
</math>

The main idea is that the structure of the above type of systems is quite similar to so-called bilinear control systems. Although belonging to the class of nonlinear control systems, the latter exhibit many features of linear time-invariant systems. In more detail, a bilinear control system is given as follows

<center>
<math>
\dot{x}(t)=Ax(t) + \sum_{i=1}^m N_i x(t) u_i(t) + B u(t),
</math>
</center>

where
<math>
A,N_i \in \mathbb R^{n\times n}, B \in \mathbb R^{n\times m}
</math>
and
<math> C \in \mathbb R^{p\times n}.
</math>

As one can see, there seems to be a close connection between LPV and bilinear systems which may be advantageous. To be more precise, following [2], we can interpret LPV systems as special bilinear system by simply setting

<math>
\tilde{A}=A,\;\; \tilde{N}_i=0,\;\;i=1,\dots,m,\;\; \tilde{N}_i=A_i,\;\; i=m+1,\dots,m+d, \;\; \tilde{B}=\begin{bmatrix} B_0 & 0\end{bmatrix},\;\; \tilde{C}=C, \;\; \tilde{u}=\begin{bmatrix} u_0 \\ p_1(t) \\ \vdots \\ p_d(t)\end{bmatrix} .
</math>

It is important to note that in this way the parameter dependency has been hidden in the structure of a bilinear system and if we use a bilinear model reduction method, we automatically end up with a structure-preserving model reduction method for LPV systems as well. Due to the above mentioned similarity of bilinear and linear control systems, one can generalize some useful and well-known linear concepts. For example, it is possible to extend the method of balanced truncation for bilinear systems. Here, we have to solve generalized Lyapunov equations of the form

<center>
<math>
AX+XA^T +\sum_{i=1}^m N_i X N_i^T + BB^T = 0, \qquad A^TY+YA +\sum_{i=1}^m N_i^T Y N_i + C^TC = 0.
</math>
</center>

For a detailed analysis of this approach, we refer to e.g. [3]. Another possibility is to aim at a model reduction method which is optimal with respect to a certain system norm. In [1], the authors investigate a possible extension of the linear <math>H_2</math>-norm and propose an algorithm which extends the well-known iterative rational Krylov algorithm (IRKA) to the bilinear setting specified above.

Both methods have been tested on several LPV control systems and seem to be a reasonable alternative to existing parametric model reduction methods.

==References==

[1] P. Benner and T. Breiten, Interpolation-based H2-model reduction of bilinear control systems, 2011, Preprint MPIMD/11-02.

[2] P. Benner and T. Breiten, On H2-model reduction of linear parameter-varying
systems, In Proceedings in Applied Mathematics and Mechanics. Wiley InterScience,
2011.

[3] P. Benner and T. Damm, Lyapunov Equations, Energy Functionals, and Model Order Reduction
of Bilinear and Stochastic Systems, SIAM J. Cont. Optim., 49 (2011), pp. 686-711.

Gas Sensor

2011-12-08T15:32:07Z

Will:

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:physical parameters]]
[[Category:four parameters]]
[[Category:time invariant]]

This is an extension of the non-parametrized model of Gas sensor in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)) to a parametrized model.

==Description of the device==

There is a large demand for gas sensing devices in various domains. They are
desired in e. g. safety applications where combustible or toxic gases are present or
in comfort applications, such as climate controls of buildings and vehicles where
good air quality is required. Additionally, gas monitoring is needed in process
control and laboratory analytics. All of these applications demand cheap, small
and user-friendly gas sensing devices which show high sensitivity, selectivity and
stability with respect to a given application.

A micromachined gas sensor is not only a challenge with respect to thermal
design but also with respect to mechanical design. Only by choosing the right
mechanical design a large intrinsic or thermal-induced membrane stress leading
to membrane deformation/ breaking of the membrane can be avoided. It is further
necessary to build a chemometrics calibration model which correlates the set of
sensor resistance measurements to the sensed gas concentration. Prior to fabrication,
a thermal simulation is performed to determine the heating efficiency and
temperature homogeneity of the gas sensitive regions. As the device is connected to circuitry for heating power
control and sensing resistor readout, a system-level simulation is also needed.
Hence, a compact thermal model must be generated. (The text above is taken from [1].)

==Description of the model==

The heat transfer within a hotplate is described through the governing heat transfer equation [2]

<math>
\nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad
Q=j^2R(T), \quad (1)
</math>

where <math>\kappa(r)</math> is the thermal conductivity in <math>W/(m*K)</math> at the position <math>r, \, c_p</math> is the
specific heat capacity in <math>J /(kg*K), \, \rho(r)</math> is the mass density in
<math>kg /m^3,</math> and <math>T(r,t)</math> is the temperature distribution. We assume a
homogeneous heat generation rate over a lumped resistor:

<math>
Q = \frac{u^2(t)}{R(T)}
</math>

with unit <math>W/m^3</math>.
We use the initial condition <math> T_0 = 273K </math>, and the
Dirichlet boundary condition <math> T = 273 K </math> at the bottom of
the computational domain. The convection boundary condition at the top of the membrane is

<math>
q=h(T-T_{air}),
</math>

where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W/(m^2 K)</math>.

Assuming <math>T_{air}=0</math>, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,

<math>
(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
y=C^T \cdot T.
</math>

Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>, which depends linearly on the temperature. Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=1.469 \times 10^{-3}</math>. The model was created and meshed in ANSYS. It contains a constant load vector corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.

The input function <math>u(t)</math> is a step function with the value <math>1</math>, which disappears at the time <math>0.02s</math>. This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero. However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>. This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off. If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc...
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input. (It is also called a weak nonlinear system.)

==Data information==

The system matrices are in MatrixMarket format (http://math.nist.gov/MatrixMarket/) and can be downloaded here [[File: Matrices_gassensor.tgz]]. The files named by *.<math>A_i, \, i=0,1,2</math> correspond to the system matrices <math>A_i, \, i=0,1,2</math>, respectively. The files named by <math>*.E_i, \, i=0,1,2</math> correspond to <math>E_i, \, i=0,1</math>. The file named by <math>*.B</math> corresponds to the load vector <math>B</math> and the file named by <math>*.C</math> corresponds to the output matrix <math>C</math>.

==References==
[1] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering,
University of Freiburg, 2005.

[2] T. Bechtold, D. Hohfel, E. B. Rudnyi and M. Guenther, "Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction," J. Micromech. Microeng. 20(2010) 045030 (13pp).

Gas Sensor

2011-12-08T15:31:57Z

Will:

[[Category:benchmark, linear, time invariant, four physical parameters, first order system]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:physical parameters]]
[[Category:four parameters]]
[[Category:time invariant]]

This is an extension of the non-parametrized model of Gas sensor in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)) to a parametrized model.

==Description of the device==

There is a large demand for gas sensing devices in various domains. They are
desired in e. g. safety applications where combustible or toxic gases are present or
in comfort applications, such as climate controls of buildings and vehicles where
good air quality is required. Additionally, gas monitoring is needed in process
control and laboratory analytics. All of these applications demand cheap, small
and user-friendly gas sensing devices which show high sensitivity, selectivity and
stability with respect to a given application.

A micromachined gas sensor is not only a challenge with respect to thermal
design but also with respect to mechanical design. Only by choosing the right
mechanical design a large intrinsic or thermal-induced membrane stress leading
to membrane deformation/ breaking of the membrane can be avoided. It is further
necessary to build a chemometrics calibration model which correlates the set of
sensor resistance measurements to the sensed gas concentration. Prior to fabrication,
a thermal simulation is performed to determine the heating efficiency and
temperature homogeneity of the gas sensitive regions. As the device is connected to circuitry for heating power
control and sensing resistor readout, a system-level simulation is also needed.
Hence, a compact thermal model must be generated. (The text above is taken from [1].)

==Description of the model==

The heat transfer within a hotplate is described through the governing heat transfer equation [2]

<math>
\nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad
Q=j^2R(T), \quad (1)
</math>

where <math>\kappa(r)</math> is the thermal conductivity in <math>W/(m*K)</math> at the position <math>r, \, c_p</math> is the
specific heat capacity in <math>J /(kg*K), \, \rho(r)</math> is the mass density in
<math>kg /m^3,</math> and <math>T(r,t)</math> is the temperature distribution. We assume a
homogeneous heat generation rate over a lumped resistor:

<math>
Q = \frac{u^2(t)}{R(T)}
</math>

with unit <math>W/m^3</math>.
We use the initial condition <math> T_0 = 273K </math>, and the
Dirichlet boundary condition <math> T = 273 K </math> at the bottom of
the computational domain. The convection boundary condition at the top of the membrane is

<math>
q=h(T-T_{air}),
</math>

where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W/(m^2 K)</math>.

Assuming <math>T_{air}=0</math>, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,

<math>
(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
y=C^T \cdot T.
</math>

Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>, which depends linearly on the temperature. Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=1.469 \times 10^{-3}</math>. The model was created and meshed in ANSYS. It contains a constant load vector corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.

The input function <math>u(t)</math> is a step function with the value <math>1</math>, which disappears at the time <math>0.02s</math>. This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero. However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>. This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off. If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc...
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input. (It is also called a weak nonlinear system.)

==Data information==

The system matrices are in MatrixMarket format (http://math.nist.gov/MatrixMarket/) and can be downloaded here [[File: Matrices_gassensor.tgz]]. The files named by *.<math>A_i, \, i=0,1,2</math> correspond to the system matrices <math>A_i, \, i=0,1,2</math>, respectively. The files named by <math>*.E_i, \, i=0,1,2</math> correspond to <math>E_i, \, i=0,1</math>. The file named by <math>*.B</math> corresponds to the load vector <math>B</math> and the file named by <math>*.C</math> corresponds to the output matrix <math>C</math>.

==References==
[1] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering,
University of Freiburg, 2005.

[2] T. Bechtold, D. Hohfel, E. B. Rudnyi and M. Guenther, "Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction," J. Micromech. Microeng. 20(2010) 045030 (13pp).

Category:Nonzero initial condition

2011-12-08T15:29:56Z

Will: Created page with 'Category:parametric Category:linear system Category:first order system Category:second order system Category:non-linear system'

[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Category:Time varying

2011-12-08T15:26:52Z

Will: Created page with 'Category:parametric Category:linear system Category:first order system Category:second order system Category:non-linear system'

[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Scanning Electrochemical Microscopy

2011-12-08T15:26:19Z

Will:

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:time varying]]
[[Category:two parameters]]
[[Category:first order system]]
[[Category:nonzero initial condition]]

==Description of the process==

Scanning Electrochemical Microscopy (SECM) has many applications in current problems in the biological field. Quantitative mathematical models have been developed for different operating modes of the SECM. Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border, solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements. After discretization, a high-dimensional system of ordinary differential equations is obtained. Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig.1. The computation domain under the 2D-axisymmetric approximation includes the electrolyte under the electrode. We assume that the concentration does not depend on the rotation angle. A single chemical reaction takes place on the electrode:

<math>Ox+e^-\Leftrightarrow Red, \quad \quad \quad \quad (1) </math>

where <math>Ox</math> and <math>Red</math> are two different species in the reaction.
According to the theory of SECM [2], the species transport in the electrolyte is described by diffusion only. The diffusion partial differential equation is given by the second Fick's law as follows

<math>
\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \quad
\frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,
</math>

where <math>c_1</math> and <math>c_2</math> are the concentration fields of species <math>Ox</math> and <math>Red</math>, respectively. The initial conditions are <math>c_1(0)=c_{1,0}</math> and <math>c_2(0)=c_{2,0}.</math> Conditions at the glass and the bottom of the bath are described by the Neumann boundary conditions of zero flux
<math>\nabla c_1\cdot \vec{n}=0</math> and <math>\nabla c_2\cdot \vec{n}=0</math>. Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions <math>c_1=c_{1,0}</math> and <math>c_2=c_{2,0}</math>. The boundary conditions at the electrode are described by

<math>
\nabla c_1\cdot \vec{n}=j, \,
\nabla c_2\cdot \vec{n}=-j. \quad \quad \quad \quad (2)
</math>

Here <math>j</math> is related to the forward reaction rate <math>k_f</math> and the backward reaction rate <math>k_b</math> through the Buttler-Volmer equation,

<math>
j=k_f \cdot c_1-k_b \cdot c_2.
</math>

The reaction rates <math>k_f</math> and <math>k_b</math> are in the following form,

<math>
k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}, </math>

<math>k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .
</math>

Here, <math>k^0</math> is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction. <math>F</math> is the Faraday-constant, <math>R</math> is the gas constant, <math>T</math> is the temperature, and <math>z</math> is the number of exchanged electrons per reaction. <math>u(t)=v(t)-v^0</math> is the difference between the electrode potential and the reference potential. This difference, to which we refer below as voltage, is changed during the measurement of a voltammogram.

==Description of the model==

The control volume method has been used for the spatial discretization of (1). Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

<math>
E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad
y(t)=C\vec{c},\quad
\vec{c}(0)=\vec{c}_0 \neq 0,
</math>

where E and <math>K(u(t))</math> are system matrices, <math>K(u(t))</math> is a function of voltage that in turn depends on time. The voltage appears in the system matrix due to the boundary conditions (2). The vector <math>\vec{c} \in \mathbb{R}^n</math> is the vector of unknown concentrations, which includes both the <math>Ox</math> and <math>Red</math> species. The vector <math>B</math> is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface.
The matrix <math>K(u(t))</math> has the following form,

<math>
K(u(t))=K_1(u(t))+K_2(u(t)),
</math>

where <math>K_i(u(t))=h_i D_i, \, i=1,2,</math> and <math>h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))</math>. The voltage <math>u(t)</math> is a function of <math>\sigma</math>,

<math>
u(t)=\sigma t-1, \, \text{for } t \leq \frac{2}{ \sigma}, \quad
u(t)=-\sigma t+3, \, \text{for } \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},
</math>

where <math>\sigma</math> can take four different values, <math>\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005</math>. The constant <math>\beta</math> is computed from the parameters <math>\alpha, \, z, \, F, \, R,</math> and <math>T,</math> leading to the value <math>\beta=21.243036728240824</math>.

Although the system is a time-varying system, it can be considered as a parametrized system with two parameters <math>h_1</math> and <math>h_2</math>.

==Data information==

The data of the system matrices <math>E, \ D_1, \ D_2, \ A, \ B, C</math> as well as the initial state <math>\vec{c}_0=x_0</math> are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[File: matrices_SECM.tgz]]. The interesting output of the model is the current which is computed by <math>I(t)=C(5,:)\vec{c}</math> in MATLAB notation. The interesting plot of the output is called the cyclic voltammogram, which is the plot of the current changing with the voltage <math>u(t)</math>.

Fig.1
[[Image:Fig.1.JPG|thumb|left|300px|]]

==References==

[1] L. Feng, D. Koziol, E. B. Rudnyi, and J. G. Korvink, "Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms," Sensor Letters, Vol. 4, 1-10, 2006, pp.1-10.

[2] M. V. Mirkin, "Theory in scanning electrochemical microscopy," A. J. Bard and M. V. Mirkin, Eds. (2001). New York, John Wiley & Sons. pp. 145 – 199.

Modified Gyroscope

2011-12-08T15:24:04Z

Will:

[[Category:benchmark]]
[[Category:linear system]]
[[Category:parametric]]
[[Category:time invariant]]
[[Category:geometric parameters]]
[[Category:two parameters]]
[[Category:second order system]]

==Description of the device==
The device is a MEMS gyroscope based on the butterfly gyroscope [1] developed at the Imego institute in Gothenburg, Sweden (see also: http://simulation.uni-freiburg.de/downloads/benchmark/The Butterfly Gyro (35889)). A gyroscope is a device used to measure angular rates in up to three axes.

The basic working principle of the gyroscope can be described as follows, see also [2]. Without applied external rotation, the paddles vibrate in phase
with the function <math>z(t),</math> see Fig. 1 below. Under the influence of an external rotation about the <math>x</math>-axis (drawn
in red), an additional force due to the Coriolis acceleration acts upon the
paddles. This force leads to an additional small out-of-phase vibration
between two paddles on the same side of the bearing. This
out-of phase vibration is measured as the difference of the z-displacement
of the nodes with the red dots. Thus, measuring the displacement of two
adjacent paddles, the rotation velocity can be ascertained.

==Motivation of MOR==

When planning for and making decisions on future improvements of the Butterfly Gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==Description of the parametrized model==

Two parameters are of special interest for the model. The first one is the
quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by
finite element discretization of the parametrized model (in the form of partial differential equations) for the gyroscope. The details of constructing the parametrized system can be found in [2]. The system is of the following
form:

<math>
M(d)\ddot{x}+D(\theta)\dot{x}+T(d)x=B
y=Cx.
</math>

Here,
<math>M(d)=(M_1+dM_2)\in \mathbb R^{n\times n}</math> is the mass matrix,

<math>D(\theta)=\theta(D_1+dD_2)\in \mathbb R^{n\times n}</math> is the damping matrix,

<math>T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}</math> is the stiffness matrix,

<math>B \in \mathbb R^{n\times m_I}</math> is the load vector, <math>C \in \mathbb R^{m_O\times n}</math> is the output matrix, <math>x \in \mathbb R^{n}</math> is the state vector, and <math>y \in \mathbb R^{m_O}</math> is the output response.

The variables <math>d</math> and <math>\theta</math> are the parameters of the system, where
<math>d</math> is the width of the
bearing and <math>\theta</math> is the rotation velocity along the <math>x</math>-axis.

The interesting output <math>y</math> of the system is <math>\delta</math> which is the
difference of the displacement <math>z(t)</math> between the two red dots on
the same side of the bearing (see Fig.1). The number of degrees of freedom is <math>n=17913</math>.

The interesting range for the parameters are: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>. The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz.

==Data information==

The system matrices <math>M_1, \, M_2, \, D_1, \, D_2, \, T_1, \, T_2</math>, and <math> T_3</math> are in the MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[File: matrices.tgz]]. <math>M_1</math> corresponds to the file Bmass1.dat, <math>M_2</math> corresponds to the file Bmass2.dat., <math>D_1, \, D_2</math> correspond to the files Bdamp1.dat, Bdamp2.dat, respectively. <math>T_1, \, T_2, \, T_3</math> correspond to the files Bstiff1.dat, Bstiff2.dat, Bstiff3.dat, respectively.
The load vector <math>B</math> can be obtained from the file Bload.dat.

==References==

[1] J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "MEMS Compact Modeling Meets Model Order Reduction: Examples of the
Application of Arnoldi Methods to Microsystem Devices," Nanotech, 2004, pp. 303–306.

[2] C. Moosmann, "ParaMOR
Model Order Reduction for parameterized MEMS applications," PhD thesis, Department of Microsystems Engineering,
University of Freiburg, 2007.

[3] B. Salimbahrami, R. Eid, B. Lohmann, "Model Reduction by Second Order
Krylov Subspaces: Extensions, Stability and Proportional Damping,"
IEEE International Symposium on Intelligent Control, 2006, pp. 2997–3002.

Fig.1
[[File:Gyroscope.jpg]]

Microthruster Unit

2011-12-08T15:21:31Z

Will:

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:three parameters]]

This parametric model is collected in the Oberwolfach Model Reduction Benchmark Collection. The file describing the model is named "Boundary Condition Independent Thermal Model". The parametric model is a 2D-axisymmetric microthruster model with 3 independent parameters. By following this [http://simulation.uni-freiburg.de/downloads/benchmark/Thermal%20Model%20%2838865%29/ link], one can have the detailed description of the parameters and the model. The data of the model can also be downloaded there.

Microthruster Unit

2011-12-08T15:21:19Z

Will:

[[Category:benchmark, linear, time invariant, three physical parameters, first order system]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:three parameters]]

This parametric model is collected in the Oberwolfach Model Reduction Benchmark Collection. The file describing the model is named "Boundary Condition Independent Thermal Model". The parametric model is a 2D-axisymmetric microthruster model with 3 independent parameters. By following this [http://simulation.uni-freiburg.de/downloads/benchmark/Thermal%20Model%20%2838865%29/ link], one can have the detailed description of the parameters and the model. The data of the model can also be downloaded there.

Synthetic parametric model

2011-12-08T15:16:45Z

Will:

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:one parameter]]
[[Category:first order system]]
[[Category:synthetic model]]

== Introduction ==
On this page you will find a synthetic parametric model for which one can easily experiment with different system orders <math> n </math>, values of the parameter <math> \varepsilon </math>, as well as different poles and residues.

Also, the decay of the Hankel singular values can be changed indirectly through the parameter <math> \varepsilon </math>.

== Model description ==

The parameter <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> H(s,\varepsilon) = \sum_{i=1}^{n}\frac{r_i}{s-p_i} = \sum_{i=1}^{n}\frac{r_i}{s-(\varepsilon a_i+jb_i)} </math>

we can write down the state-space realization

:<math> H(s,\varepsilon) = \widehat{C}\Big(sI-\varepsilon \widehat{A}_\varepsilon - \widehat{A}_0\Big)^{-1}\widehat{B}+D</math>

with system matrices defined as

:<math>\varepsilon \widehat{A}_\varepsilon + \widehat{A}_0 = \varepsilon \left[\begin{array}{ccc} a_1 & & \\ & \ddots & \\ & & a_n\end{array}\right] +\left[\begin{array}{ccc} jb_1 & & \\ & \ddots & \\ & & jb_n\end{array}\right] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

:<math> p_1 = \varepsilon a_1+jb_1, p_2 = \varepsilon a_1-jb_1, \ldots, p_{n-1} = \varepsilon a_k+jb_k, p_n = \varepsilon a_k-jb_k, </math>

and the residues also form complex conjugate pairs

:<math> r_1 = c_1+jd_1, r_2 = c_1-jd_1, \ldots, r_{n-1} = c_k+jd_k, r_n = c_k-jd_k. </math>

Then a realization with matrices having real entries is given by

:<math> A_\varepsilon = \left[\begin{array}{ccc} A_{\varepsilon,1} & & \\ & \ddots & \\ & & A_{\varepsilon,k}\end{array}\right], \quad A_0 = \left[\begin{array}{ccc} A_{0,1} & & \\ & \ddots & \\ & & A_{0,k}\end{array}\right], \quad B = \left[\begin{array}{c} B_1 \\ \vdots \\ B_k\end{array}\right], \quad C = \left[\begin{array}{ccc} C_1 & \cdots & C_k\end{array}\right], \quad D = 0,</math>

with <math> A_{\varepsilon,i} = \left[\begin{array}{cc} a_i& 0 \\ 0 & a_i \end{array}\right] </math>,
<math> A_{0,i} = \left[\begin{array}{cc} 0& b_i \\ -b_i & 0 \end{array}\right] </math>,
<math> B_{i} = \left[\begin{array}{c} 2 \\ 0 \end{array}\right] </math>,
<math> C_{i} = \left[\begin{array}{cc} c_i& d_i\end{array}\right] </math>.

== Numerical values ==

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

* <math>a_i </math> equally spaced in <math> [-10^3, -10]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB, the system matrices are easily formed as follows:

<source lang="matlab">
n = 100;
a = -linspace(1e1,1e3,n/2).'; b = linspace(1e1,1e3,n/2).';
c = ones(n/2,1); d = zeros(n/2,1);
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;
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);
</source>

The above system matrices <math>A_\varepsilon, A_0, B, C</math> are also available in [http://math.nist.gov/MatrixMarket/formats.html MatrixMarket] format [[Media:Synth_matrices.tar.gz|Synth_matrices.tar.gz]].

== Plots ==

We plot the frequency response <math>H(s,\varepsilon) = C\big(sI-\varepsilon A_\varepsilon - A_0\big)^{-1}B</math> and poles for parameter values <math>\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] </math>.

:[[Image:synth_freq_resp.png|frame|border|left|Frequency response of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]
[[Image:synth_poles.png|frame|center|border|Poles of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]

In MATLAB, the plots are generated using the following commands:

<source lang="matlab">
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
for j = 1:length(ep)
p(:,j) = [ep(j)*a+1j*b; ep(j)*a-1j*b]; % poles
[jww,pp] = meshgrid(jw,p(:,j));
Hjw(j,:) = (r.')*(1./(jww-pp)); % freq. resp.
end
figure, loglog(imag(jw),abs(Hjw),'LineWidth',2)
axis tight, xlim([6 1200])
xlabel('frequency (rad/sec)')
ylabel('magnitude')
title('Frequency response for different \epsilon')
figure, plot(real(p),imag(p),'.')
title('Poles for different \epsilon')
</source>

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

Next, for <math>\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] </math>, we also plot the decay of the Hankel singular values. Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

[[Image:synth_hsv.png|frame|border|center|Hankel singular values of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]

'' [[User:Ionita|Ionita]] 14:38, 29 November 2011 (UTC) ''

Synthetic parametric model

2011-12-08T15:15:56Z

Will:

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:physical parameters]]
[[Category:one parameter]]
[[Category:first order system]]
[[Category:synthetic model]]

== Introduction ==
On this page you will find a synthetic parametric model for which one can easily experiment with different system orders <math> n </math>, values of the parameter <math> \varepsilon </math>, as well as different poles and residues.

Also, the decay of the Hankel singular values can be changed indirectly through the parameter <math> \varepsilon </math>.

== Model description ==

The parameter <math>\varepsilon</math> scales the real part of the system poles, that is, <math>p_i=\varepsilon a_i+jb_i</math>.
For a system in pole-residue form

:<math> H(s,\varepsilon) = \sum_{i=1}^{n}\frac{r_i}{s-p_i} = \sum_{i=1}^{n}\frac{r_i}{s-(\varepsilon a_i+jb_i)} </math>

we can write down the state-space realization

:<math> H(s,\varepsilon) = \widehat{C}\Big(sI-\varepsilon \widehat{A}_\varepsilon - \widehat{A}_0\Big)^{-1}\widehat{B}+D</math>

with system matrices defined as

:<math>\varepsilon \widehat{A}_\varepsilon + \widehat{A}_0 = \varepsilon \left[\begin{array}{ccc} a_1 & & \\ & \ddots & \\ & & a_n\end{array}\right] +\left[\begin{array}{ccc} jb_1 & & \\ & \ddots & \\ & & jb_n\end{array}\right] ,</math>

:<math>\widehat{B} = [1,\ldots,1]^T,\quad \widehat{C} = [r_1,\ldots,r_n],\quad D = 0.</math>

Notice that the system matrices have complex entries.

For simplicity, assume that <math> n </math> is even, <math> n=2k </math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.

:<math> p_1 = \varepsilon a_1+jb_1, p_2 = \varepsilon a_1-jb_1, \ldots, p_{n-1} = \varepsilon a_k+jb_k, p_n = \varepsilon a_k-jb_k, </math>

and the residues also form complex conjugate pairs

:<math> r_1 = c_1+jd_1, r_2 = c_1-jd_1, \ldots, r_{n-1} = c_k+jd_k, r_n = c_k-jd_k. </math>

Then a realization with matrices having real entries is given by

:<math> A_\varepsilon = \left[\begin{array}{ccc} A_{\varepsilon,1} & & \\ & \ddots & \\ & & A_{\varepsilon,k}\end{array}\right], \quad A_0 = \left[\begin{array}{ccc} A_{0,1} & & \\ & \ddots & \\ & & A_{0,k}\end{array}\right], \quad B = \left[\begin{array}{c} B_1 \\ \vdots \\ B_k\end{array}\right], \quad C = \left[\begin{array}{ccc} C_1 & \cdots & C_k\end{array}\right], \quad D = 0,</math>

with <math> A_{\varepsilon,i} = \left[\begin{array}{cc} a_i& 0 \\ 0 & a_i \end{array}\right] </math>,
<math> A_{0,i} = \left[\begin{array}{cc} 0& b_i \\ -b_i & 0 \end{array}\right] </math>,
<math> B_{i} = \left[\begin{array}{c} 2 \\ 0 \end{array}\right] </math>,
<math> C_{i} = \left[\begin{array}{cc} c_i& d_i\end{array}\right] </math>.

== Numerical values ==

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

* <math>a_i </math> equally spaced in <math> [-10^3, -10]</math>,

* <math>b_i </math> equally spaced in <math>[10, 10^3]</math>,

* <math> c_i = 1</math>,

* <math> d_i = 0</math>,

* <math>\varepsilon</math><math> \in [1/50,1]</math>.

In MATLAB, the system matrices are easily formed as follows:

<source lang="matlab">
n = 100;
a = -linspace(1e1,1e3,n/2).'; b = linspace(1e1,1e3,n/2).';
c = ones(n/2,1); d = zeros(n/2,1);
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;
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);
</source>

The above system matrices <math>A_\varepsilon, A_0, B, C</math> are also available in [http://math.nist.gov/MatrixMarket/formats.html MatrixMarket] format [[Media:Synth_matrices.tar.gz|Synth_matrices.tar.gz]].

== Plots ==

We plot the frequency response <math>H(s,\varepsilon) = C\big(sI-\varepsilon A_\varepsilon - A_0\big)^{-1}B</math> and poles for parameter values <math>\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] </math>.

:[[Image:synth_freq_resp.png|frame|border|left|Frequency response of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]
[[Image:synth_poles.png|frame|center|border|Poles of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]

In MATLAB, the plots are generated using the following commands:

<source lang="matlab">
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
for j = 1:length(ep)
p(:,j) = [ep(j)*a+1j*b; ep(j)*a-1j*b]; % poles
[jww,pp] = meshgrid(jw,p(:,j));
Hjw(j,:) = (r.')*(1./(jww-pp)); % freq. resp.
end
figure, loglog(imag(jw),abs(Hjw),'LineWidth',2)
axis tight, xlim([6 1200])
xlabel('frequency (rad/sec)')
ylabel('magnitude')
title('Frequency response for different \epsilon')
figure, plot(real(p),imag(p),'.')
title('Poles for different \epsilon')
</source>

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

Next, for <math>\varepsilon \in [1/50, 1/20, 1/10, 1/5, 1/2, 1] </math>, we also plot the decay of the Hankel singular values. Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

[[Image:synth_hsv.png|frame|border|center|Hankel singular values of synthetic parametric system, for parameter values 1/50 (blue), 1/20 (green), 1/10 (red), 1/5 (teal), 1/2 (purple), 1 (yellow).]]

'' [[User:Ionita|Ionita]] 14:38, 29 November 2011 (UTC) ''

Category:Time invariant

2011-12-08T15:12:14Z

Will: Created page with 'Category:parametric Category:linear system Category:first order system Category:second order system Category:non-linear system'

[[Category:parametric]]
[[Category:linear system]]
[[Category:first order system]]
[[Category:second order system]]
[[Category:non-linear system]]

Category:Parametric

2011-12-08T15:07:59Z

Will: Created page with 'Category:benchmark'

[[Category:benchmark]]