MOR Wiki - User contributions [en]

MOR Toolbox

2014-01-31T09:46:42Z

Grundel: Created page with "Test"

Test

Main Page

2014-01-30T14:05:51Z

Grundel:

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

The purpose of the Model Order Reduction (MOR) Wiki is to bring together experts in the area of model reduction, 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 ([[:Wikipedia:MEMS|MEMS]]) design, and control design. The processes or devices can be modeled by partial differential equations ([[:Wikipedia:Partial_differential_equation|PDEs]]). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations ([[:Wikipedia:Ordinary_differential_equation|ODEs]]), or differential algebraic equations ([[:Wikipedia:Differential_algebraic_equation|DAEs]]).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate such large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR (see [[Projection based MOR]] for the basic idea) 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.

[[: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, 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 [[:Category:Method|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.

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Main Page

2014-01-30T14:04:46Z

Grundel:

Main Page

2014-01-30T14:04:33Z

Grundel:

Main Page

2014-01-30T14:04:08Z

Grundel: TEst

{{Infobox State Representative
| name =Abbott Barnes Rice
| image =
| caption =
| state_house1 =Massachusetts
| district1 = Newton, Middlesex
| term_start1 = 1919
| term_end1 = 1922
| preceded1 =
| succeeded1 =
| party1 =
| state_senate2 =Massachusetts
| district2 = Newton, Middlesex
| term_start2 = 1923
| term_end2 = 1926
| preceded2 =
| succeeded2 =
| party2 =
| birth_date ={{birth date|1862|4|17}}
| birth_place = [[Hopkinton, Massachusetts]]
| death_date = {{death date and age |1926|10|10|1862|4|17}}
| death_place = [[Newton, Massachusetts]]
| alma_mater = [[Brown University]] [[Bachelor of arts|A.B.]] 1884 and [[Master of Arts (postgraduate)|A.M.]] 1889
| profession = merchant, state legislator
| spouse = Amy Thurber Bridges (m. 29 August 1890)
| children = Adams Thurber Rice (1892-1976) [[Willard Rice|Willard Wadsworth Rice]] (1895-1967) Lawrence Bridges Rice (1898-1992)
| residence =[[Hopkinton, Massachusetts]], [[Newton, Massachusetts]]
| religion =[[Congregational church|Congregational]]
}}

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

The purpose of the Model Order Reduction (MOR) Wiki is to bring together experts in the area of model reduction, 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 ([[:Wikipedia:MEMS|MEMS]]) design, and control design. The processes or devices can be modeled by partial differential equations ([[:Wikipedia:Partial_differential_equation|PDEs]]). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations ([[:Wikipedia:Ordinary_differential_equation|ODEs]]), or differential algebraic equations ([[:Wikipedia:Differential_algebraic_equation|DAEs]]).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate such large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR (see [[Projection based MOR]] for the basic idea) 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.

[[: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, 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 [[:Category:Method|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.

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Main Page

2014-01-30T13:59:50Z

Grundel:

Main Page

2014-01-30T13:57:48Z

Grundel: twidget input

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

The purpose of the Model Order Reduction (MOR) Wiki is to bring together experts in the area of model reduction, 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 ([[:Wikipedia:MEMS|MEMS]]) design, and control design. The processes or devices can be modeled by partial differential equations ([[:Wikipedia:Partial_differential_equation|PDEs]]). To simulate such models, spatial discretization via e.g. finite element discretization is necessary, which results in a system of ordinary differential equations ([[:Wikipedia:Ordinary_differential_equation|ODEs]]), or differential algebraic equations ([[:Wikipedia:Differential_algebraic_equation|DAEs]]).

After spatial discretization, the number of degrees of freedom is usually very high. It is therefore very time consuming to simulate such large-scale systems of ODEs or DAEs. Developed from well-established mathematical theories and robust numerical algorithms, MOR (see [[Projection based MOR]] for the basic idea) 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.

[[: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, 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 [[:Category:Method|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.

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Projection based MOR

2013-12-10T16:16:03Z

Grundel:

[[Category:method]]
[[Category:time invariant]]
Consider the linear time invariant system

:<math>
E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
y(t)=Cx(t), \quad \quad (1)
</math>

as an example.
All the existing model order reduction ([[List_of_abbreviations#MOR|MOR]]) methods are based on projection<ref>Antoulas, A. C. "[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>.
That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides.
Afterwards, <math>x(t)</math> is approximated by a vector <math>\tilde x(t)</math> in <math>S_1</math>.
The reduced model is produced by [[Petrov-Galerkin projection]] onto a subspace <math>S_2</math>, or by [[Galerkin projection]] onto the same subspace <math>S_1</math>.
Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\tilde x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\hat{z} (t)=V z(t)</math>.
Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.
Here <math>z</math> is a vector of length <math>q \ll n</math>.
Once <math>z(t)</math> is computed, an approximate solution <math>\tilde x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.
The vector <math>z(t)</math> can be computed from the reduced model which is derived by the following two steps.

Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get

:<math>E \frac{d{Vz}}{dt} \approx A Vz+Bu(t),\quad y(t) \approx CV z.</math>

Step 2. The residual is denoted as <math>e=AVz+Bu(t)-E \frac{d{Vz}}{dt}</math>. Forcing <math>e=0</math> in a properly chosen subspace <math>S_2</math> of <math>\mathbb {R}^n</math> leads to the Petrov-Galerkin projection: <math>W^T e=0</math>, where the columns of <math>W</math> are the basis of <math>S_2</math>.
Then we have,

:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

By defining <math>\hat{E}=W^TEV</math>, <math>\hat {A}=W^TAV, \hat{B}=W^TB</math>, <math>\hat{C}=CV</math>, we get the final reduced model

:<math>\hat{E} \frac{dz(t)}{dt}=\hat{A}z(t)+\hat{B}u(t), \quad
\hat{y}(t)=\hat{C}z(t). \quad \quad (2)
</math>

Notice that the approximation <math>\hat x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2).
The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1).
Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model.
In order to save the simulation time of solving (1), the reduced model can be used to replace (1) to attain a fast simulation. Furthermore, the error between the two systems should be within acceptable tolerance.
The error can be measured through the error between the the state vectors <math>x(t), \hat x(t)</math>, or between the output responses <math>y(t), \hat y(t)</math>, or between the transfer functions of the two systems.
It can be seen that once the two matrices <math>W</math> and <math>V</math> have been computed, the reduced model is obtained.
While the Gramian based MOR methods (e.g. [[Balanced Truncation]]) usually compute <math>W</math> different from <math>V</math>, some methods use <math>W=V</math>, e.g. some of the [[Moment-matching method]]s, the [[Reduced Basis PMOR method]]s, and some of the [[POD method]]s etc.. When <math>W=V</math>, Petrov-Galerkin projection becomes Galerkin projection.
MOR methods differ in the computation of the two matrices <math>W</math> and <math>V</math>.
The Gramian based MOR methods compute <math>W</math> and <math>V</math> by the controllability and observability Gramians.
Reduced basis methods and [[List_of_abbreviations#POD|POD]] methods compute <math>V</math> from the snapshots of the state vector <math>x(t)</math> at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute :<math>W</math> and <math>V</math> from the moments of the transfer function.
In eigenvalue based MOR methods, e.g. [[Modal truncation]], the columns of :<math>W</math> and <math>V</math> are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair <math>(A,E)</math>.

One common goal of all MOR methods is that the behavior of the reduced model should be sufficiently "close" to that of the original model guaranteed through the above mentioned error measurements.

==References==

<references/>

User:Grundel

2013-06-13T06:49:20Z

Grundel:

Sara Grundel 
Computational Methods in Systems and Control Theory, 
Max Planck Institute for Dynamics of Complex Technical Systems, 
Sandtorstr. 1, 
39106 Magdeburg 
Tel.: +49-391-6110-805 
Fax: +49-391-6110-453 
E-mail: grundel@mpi-magdeburg.mpg.de 
http://www.mpi-magdeburg.mpg.de/mpcsc/grundel/

2013-03-25T12:38:01Z

Grundel:

[[Category:method]]

An important projection model reduction method which delivers high quality reduced models by making an extra effort in choosing the projection subspaces.

A stable system <math>\Sigma</math> , realized by (A,B,C,D) is called balanced, if the Gramians, i.e. the solutions P,Q of the Lyapunov equations

<math> AP+PA^T+BB^T=0,\quad A^TQ+QA+C^TC=0</math>

satisfy <math> P=Q=diag(\sigma_1,\dots,\sigma_n)</math> with <math> \sigma_1\geq\sigma_2\geq \dots\geq\sigma_n\geq0</math>

The spectrum of <math> (PQ)^{\frac{1}{2}}</math> which is <math>\{\sigma_1,\dots,\sigma_n\}</math> are the Hankel singular values.

In order to do balanced truncation one has to first compute a balanced realization via state-space transformation

<math> (A,B,C,D)\Rightarrow (TAT^{-1},TB,CT^{-1},D)=\left (\begin{bmatrix}A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix},\begin{bmatrix}B_1\\B_2\end{bmatrix}\begin{bmatrix} C_1 &C_2 \end{bmatrix},D\right)</math>

The truncated reduced system is then given by

<math> (\hat{A},\hat{B},\hat{C},\hat{D})=(A_{11},B_1,C_1,D) </math>

== Implementation: SR Method==
One computes it for example by the SR Method.
First one computes the (Cholesky) factors of the gramians <math>P=S^TS,\; Q=R^TR</math>. Then we compute the singular value decomposition of <math> SR^T\;</math>

<math> SR^T=\begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}</math>

Then the reduced order model is given by <math>(W^TAV,W^TB,CV,D)\;</math> where

<math> W=R^T V_1\Sigma_1^{-\frac{1}{2}},\quad V= S^T U_1 \Sigma_1^{-\frac{1}{2}}.</math>

We get then that <math>V^TW=I_r</math> which makes <math> VW^T</math> an oblique projector and hence balanced trunctation a Petrov-Galerkin projection method. The reduced model is stable with Hankel Singular Values given by <math>\sigma_1,\dots,\sigma_r</math>, where r is the order of the reduced system. It is possible to choose r via the computable error bound

<math> \|y-\hat{y}\|_2\leq (2\sum_{k=r+1}^n\sigma_k)\|u\|_2. </math>

==References==

Balanced Truncation

2013-03-25T12:37:05Z

2013-03-08T12:37:28Z

Grundel: Start the BT page

[[Category:method]]

An important projection model reduction method which delivers high quality reduced models by making an extra effort in choosing the projection subspaces.

==References==

Branchline Coupler

2012-11-27T07:57:53Z

Grundel: /* Model Description */

[[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 equation.
A 2-section branchline coupler consists of four strip line ports, coupled to each other by two transversal bridges.
The energy excited at one port is coupled almost in equal shares to the two opposite ports, when considered as a MIMO-system.
Here, only the SISO case is considered.
The branchline coupler with 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.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]]''

Branchline Coupler

2012-11-27T07:57:16Z

Grundel: /* Model Description */

[[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 equation.
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.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]]''

MESS

2012-11-27T07:41:46Z

Grundel:

[[Category:Software]]
[[Category:Software_Linear_Algebra]]
[[Category:Software_Sparse_Methods]]

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

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

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

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

FitzHugh-Nagumo System

2012-11-20T16:32:48Z

Grundel: /* Reformulation as a quadratic-bilinear system */

[[Category:benchmark]]
[[Category:non-linear system]]
[[Category:time invariant]]
[[Category:first order system]]

==Description==

The FitzHugh-Nagumo system describes a prototype of an excitable system (e.g., a neuron).
If the external stimulus <math>i_0(t)</math> exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables <math>v</math> and <math>w</math> relax back to their rest values. This behaviour is typical for spike generations (=short elevation of membrane voltage <math>v</math>) in a neuron after stimulation by an external input current.

Here, we present the setting from [1], where the equations for the dynamical system read

<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05,\;i_0(t)=5\cdot
10^4t^3 \exp(-15t).</math> In [1], the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>n=1024. </math> The following picture shows the typical limit cycle behaviour described above.

[[File:FHN.png]]

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u(t) + b u(t),
</math>

with <math>E,A,N \in \mathbb R^{n\times n},\ H \in \mathbb R^{n\times n^2} </math> and <math> b \in \mathbb R.</math> The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to [2], where it is successfully applied to several smooth nonlinear control-affine systems. As it is discussed in [3], the previously mentioned introduction of <math>z</math> yields a quadratic-bilinear control of dimension <math> N = 3\cdot k</math> with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from [2], see also [3] for more details on the implementation.

==Data==

[[File:FHN.tar.gz]]

All matrices of the quadratic-bilinear formulation discretized with <math>k=512</math> are in the Matrix Market format (http://math.nist.gov/MatrixMarket/). The matrix name is used as an extension of the matrix file. For more information on the discretization details, see [1].

==References==

[1] S. Chaturantabut and D.C. Sorensen, Nonlinear Model Reduction via Discrete Empirical Interpolation, SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.

[2] C. Gu, QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems, IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.

[3] P. Benner and T. Breiten, Two-sided moment matching methods for nonlinear model reduction, 2012, Preprint MPIMD/12-12.

Contact information:

'' [[User:Breiten]]''

User:Grundel

2012-11-20T16:10:40Z

Grundel:

Sara Grundel 
Computational Methods in Systems and Control Theory, 
Max Planck Institute for Dynamics of Complex Technical Systems, 
Sandtorstr. 1, 
39106 Madgeburg 
Tel.: +49-391-6110-805 
Fax: +49-391-6110-453 
E-mail: grundel@mpi-magdeburg.mpg.de 
http://www.mpi-magdeburg.mpg.de/mpcsc/grundel/

User:Grundel

2012-11-20T16:09:47Z

Grundel:

Sara Grundel 
Computational Methods in Systems and Control Theory, 
Max Planck Institute for Dynamics of Complex Technical Systems, 
Sandtorstr. 1, 
39106 Madgeburg 
Tel.: +49-391-6110-805 
Fax: +49-391-6110-453 
E-mail: grundel@mpi-magdeburg.mpg.de