Piecewise H2 Tangential Interpolation

2015-10-30T10:55:16Z

Baur: Created page with "Category:method Category:linear Category:parametric Category:time invariant '''Piecewise H2 tangential interpolation''' (PWH2TanInt) is an approach for parame..."

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

'''Piecewise H2 tangential interpolation''' (PWH2TanInt) is an approach for parametric model order reduction which is based on IRKA and a concatenation of the resulting local bases to a global basis for projection on a low-dimensional subspace.
<ref name="morBauBBetal11"> U. Baur, C. A. Beattie, P. Benner, and S. Gugercin,
"[http://epubs.siam.org/doi/abs/10.1137/090776925 Interpolatory projection methods for parameterized model reduction]",
SIAM J. Sci. Comput., 33(5):2489-2518, 2011</ref>

IRKA computes optimal (frequency) shifts <math>s_i</math> and corresponding tangential directions <math>b_{ij}</math> and <math>c_{ij}</math> such that

the reduced-order transfer function matches the p-gradient and p-Hessian of the original system response with respect to the parameters:

:<math>
\nabla_{\!p}c_{ij}^T G( s_i, p_j)b_{ij} =\nabla_{\!p}c_{ij}^T \hat G ( s_i, p_j)b_{ij}, \quad \nabla^2_{\!p}c_{ij}^T G( s_i, p_j)b_{ij} =\nabla^2_{\!p}c_{ij}^T \hat G ( s_i,p_j)b_{ij},

</math>
for <math>i=1,\ldots,r',\ j=1,\ldots, K</math>.

Additionally, the usual tangential interpolation properties hold:

:<math>
G( s_i, p_j)b_{ij} = \hat G( s_i, p_j)b_{ij},\quad c_{ij}^TG( s_i, p_j) = c_{ij}^T\hat G( s_i, p_j).
</math>

==References==

<references/>

==Contact==

[[User:Baur]]

File:Poster Baur MoRePaS.pdf

2015-10-30T10:36:14Z

Baur: PMOR Comparison from MoRePaS II

PMOR Comparison from MoRePaS II

File:Poster Baur.pdf

2015-10-30T10:34:19Z

Baur: PMOR Comparison from ModRedCIRM 'Model Reduction and Approximation for Complex Systems', Luminy, 2013

PMOR Comparison from ModRedCIRM 'Model Reduction and Approximation for Complex Systems', Luminy, 2013

Category:Method

2015-10-30T10:22:06Z

Baur:

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

A comparison of model order reduction methods can be found for example in<ref name="antoulas00">A.C. Antoulas; S. Gugercin, "[http://www.math.vt.edu/people/gugercin/ttpapers/GugercinC3.pdf A comparative study of 7 algorithms for model reduction]", Proceedings of the 39th IEEE Conference on Decision and Control, vol.3, pp. 2367--2372, 2000.</ref>, <ref name="antoulas01">
A.C. Antoulas; D.C. Sorensen; S. Gugercin, "[http://www.math.vt.edu/people/gugercin/papers/survey.pdf A survey of model reduction methods for large-scale systems]", Contemporary mathematics, vol.280, pp. 193--220, 2001.</ref> and cover:

* [[Balanced Truncation]]
* [[Balanced Truncation|Approximate Balancing]]
* Hankel Norm Approximation
* Singular Perturbation
* [[IRKA|Rational Krylov Method]]
* Lanczos Method
* Arnoldi Method

A comparison of parametric model order reduction methods is conducted in <ref name="baur15">U. Baur; P. Benner; B. Haasdonk; C. Himpe; I. Martini; M. Ohlberger, "[http://www2.mpi-magdeburg.mpg.de/preprints/2015/MPIMD15-01.pdf Comparison of methods for parametric model order reduction of instationary problems]", Preprint, 2015</ref> and covers:

* POD
* POD-Greedy
* Matrix Interpolation
* [[Transfer Function Interpolation]]
* Piecewise H2 Tangential Interpolation
* [[Moment-matching PMOR method|Multi Parameter Moment Matching]]
* [[Emgr|Empirical Linear Cross Gramian]]

== References ==

<references/>

----

Publications

2015-10-30T10:13:06Z

Baur: Frontpage for download of BibTeX file mor.bib

The BibTeX file mor.bib contains a list of references which are related to model order reduction.
The purpose of this BibTeX file is to facilitate the citation of MOR literature and to cite sources in a consistent manner.

The Bibtex keys for the entries are formed by 'mor' followed by the authors name(s) and
the year of publication as, e.g., in

3 leading characters of first authors last name (morXxx)
+1 leading character of second and third atuhros last names (YZ)
+year of publication as 2 digits (15)
+1 additional character for counting if the previous is non-unique
examples: morXxx08, morXxxYZ10, morXxxYZ15a, morXxxYZ15b, morXxxYZetal15

Everybody is encouraged to send missing references to xxx.

For a download of the full mor.bib including a list of BibTex Strings press ...

File:Anemometer.tar.gz

2015-02-13T09:05:44Z

Baur: Baur uploaded a new version of "File:Anemometer.tar.gz": changed: matrix E2 new: m file for generating the 3p anemometer

data for 1 parameter and 3 parameter anemometer example

MOR Wiki:Current events

2013-04-19T21:08:06Z

Baur: changes of categories

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear system]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:parametric 1 parameter]]
[[Category:parametric 2-5 parameters]]
[[Category:affine parameter representation]]
[[Category:first differential order]]

==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 - A1- p(A2 - A1))^{-1}B)
</math>

with the fluid velocity <math>p(=v)</math> as single parameter.
Here <math>E</math> is the heat capacitance matrix, <math>B</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the FEM discretization w.r.t. the spatial variables. <math>Ai</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection. Thus, for obtaining pure convection you have to compute <math>A2 - A1</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{(E_s + p_0 E_f)}_{E(p_0)} - \underbrace{( A_{d,s} + p_1 A_{d,f} + p_2 A_c )}_{A(p_1,p_2)} )^{-1}B)
</math>

with parameters <math>p_0, \, p_1, \, p_2</math> which are combinations of the original fluid parameters <math>\rho, \, c, \, \kappa, \, v: \quad p_0 = \rho c, \, p_1=\kappa,</math> and <math>p_2 =\rho c v,</math> see [5]. So far, we have considered the mass density as fixed, i.e. <math>\rho=1</math>.

==Origin==

IMTEK Freiburg, group of Jan Korvink.

==Data==

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

To test the quality of the reduced order systems, harmonic simulations as well as transient step responses could be computed, see [5].

Example with 1 parameter:

*.B: load vector
*.E: damping matrix
*.P: permutation matrix
*.A: stiffness matrices (2)

Example with 3 parameters:

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

==References==

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

[[File:Model_Color.pdf|600px|thumb|left|Schemantic: 2D-model-anemometer]]

[[file:ContourPlot30.pdf|600px|thumb|left|Calculated temperature profile for anemometer function]]

==Contact==

[[User:Baur]]