Transfer Function Interpolation - Revision history

Will: /* Higher dimensional parameter spaces */

2013-06-17T08:20:15Z

Higher dimensional parameter spaces

← Older revision		Revision as of 08:20, 17 June 2013
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	Acta Numerica, vol. 13, pp. 147-269, 2004.</ref>, <ref>M. Griebel		Acta Numerica, vol. 13, pp. 147-269, 2004.</ref>, <ref>M. Griebel
	Sparse grids and related approximation schemes for higher dimensional problems		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		In Foundations of Computational Mathematics (FoCM05), Santander, pp. 106-161, 2006.</ref>, <ref>C. Zenger
	Sparse grids		Sparse grids
	In Parallel algorithms for partial differential equations		In Parallel algorithms for partial differential equations

Will: /* Higher dimensional parameter spaces */

2013-06-17T08:18:03Z

Higher dimensional parameter spaces

← Older revision		Revision as of 08:18, 17 June 2013
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	Sparse grids		Sparse grids
	In Parallel algorithms for partial differential equations		In Parallel algorithms for partial differential equations
	(Kiel, 1990), vol. 31 of Notes Numer. Fluid Mech., pp. 241-251, 1991.</ref>		(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		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.

	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"/>.		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==

Will: /* Description */

2013-06-17T08:14:56Z

Description

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

Baur: add contact

2013-05-31T10:19:29Z

add contact

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	<references/>		<references/>

			==Contact==

			[[User:Baur]]

Saak at 11:30, 30 May 2013

2013-05-30T11:30:55Z

← Older revision		Revision as of 11:30, 30 May 2013
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	The PMOR approach was originally proposed in <ref name="BauB09"> U. Baur and P. Benner, "<span class="plainlinks">[http://www.oldenbourg-link.com/doi/abs/10.1524/auto.2009.0787 Modellreduktion für parametrisierte Systeme durch balanciertes		The PMOR approach was originally proposed in <ref name="BauB09"> U. Baur and P. Benner, "<span class="plainlinks">[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)]</span>",
	Abschneiden und Interpolation (Model Reduction for Parametric Systems Using Balanced Truncation and Interpolation)]</span>",
	at-Automatisierungstechnik, vol. 57, no. 8, pp. 411-420, 2009</ref>		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.		using polynomial interpolation. It can simply be extended to a hybrid approach of [[Balanced Truncation\|balanced truncation]] applying different kinds of interpolation.
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	</math>		</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, "<span class="plainlinks">[http://link.springer.com/chapter/10.1007%2F3-7643-7356-3_3#		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, "<span class="plainlinks">[http://link.springer.com/chapter/10.1007%2F3-7643-7356-3_3# Recent developments in barycentric rational interpolation]</span>", in Trends and applications in constructive approximation,
	Recent developments in barycentric rational interpolation]</span>", in Trends and applications in constructive approximation,
	vol. 151 of Internat. Ser. Numer. Math., pp. 27-51. Birkhäuser, Basel, 2005.</ref>.		vol. 151 of Internat. Ser. Numer. Math., pp. 27-51. Birkhäuser, Basel, 2005.</ref>.

Baur: new method for PMOR

2013-05-28T22:20:59Z

new method for PMOR

New page

[[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/>