Moment-matching PMOR method - Revision history

Feng: /* Description */

2013-05-22T08:34:09Z

Description

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	==Description==		==Description==

	The method introduced here is described in <ref name="daniel04"/> and <ref name="feng07"/>, which is an extension of the [[~~Moment~~-matching method]] for nonparametric systems (see		The method introduced here is described in <ref name="daniel04"/> and <ref name="feng07"/>, which is an extension of the [[moment-matching method]] for nonparametric systems (see
	<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:		<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

Feng: /* Description */

2013-05-22T08:31:14Z

Description

← Older revision		Revision as of 08:31, 22 May 2013
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	system, respectively.		system, respectively.

	To obtain the reduced model in (2), a		To obtain the reduced model in (2), a [[Projection_based_MOR\|projection]] matrix
	projection matrix <math>V \in \mathbb{R}^{n \times r}, r\ll n</math> has to be computed.		<math>V \in \mathbb{R}^{n \times r}, r\ll n</math> 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>V^T(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)Vx=V^TBu(s_1,\ldots,s_p), </math>

Feng: /* Description */

2013-05-22T08:26:50Z

Description

← Older revision		Revision as of 08:26, 22 May 2013
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	==Description==		==Description==

	The method introduced here is described in <ref name="daniel04"/> and <ref name="feng07"/>, which is an extension of the ~~moment~~-matching ~~MOR~~ method for nonparametric systems (see		The method introduced here is described in <ref name="daniel04"/> and <ref name="feng07"/>, which is an extension of the [[Moment-matching method]] for nonparametric systems (see
	<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:		<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

Himpe: equation indent

2013-04-30T23:11:08Z

equation indent

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	<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:		<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

	<math>		:<math>
	(E_0+s_1 E+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad		(E_0+s_1 E+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad
	y=Cx, \quad \quad \quad \quad (1)		y=Cx, \quad \quad \quad \quad (1)
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	projection matrix <math>V \in \mathbb{R}^{n \times r}, r\ll n</math> has to be computed.		projection matrix <math>V \in \mathbb{R}^{n \times r}, r\ll n</math> 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>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=CVx. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad(2)		:<math>y=CVx. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad(2)
	</math>		</math>

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

	<math>		:<math>
	x=[I-(\sigma_1M_1+\ldots +\sigma_pM_p)]^{-1}B_Mu(s_1,\ldots,s_p)		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).		=\sum\limits_{i=0}^{\infty}(\sigma_1M_1+\ldots+\sigma_pM_p)^iB_Mu(s_1,\ldots,s_p).
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	The resulting <math>V</math> is desired to expand the subspace:		The resulting <math>V</math> is desired to expand the subspace:

	<math>		:<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, </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, </math>
	<math>		:<math>
	M_p^2B_M, M_1^3B_M,\ldots, M_1^rB_M, \ldots,M_p^rB_M \}. \quad \quad \quad \quad (3)		M_p^2B_M, M_1^3B_M,\ldots, M_1^rB_M, \ldots,M_p^rB_M \}. \quad \quad \quad \quad (3)
	</math>		</math>
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	is actually an orthonormal basis of the subspace as below,		is actually an orthonormal basis of the subspace as below,

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


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

	<math>R_1=[M_1R_0,\ldots, M_pR_0], </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>R_2=[M_1R_1,\ldots, M_pR_1], \quad \quad \quad \quad (5) </math>

	<math> \vdots,</math>		:<math> \vdots,</math>

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

	<math> \vdots.</math>		:<math> \vdots.</math>

	Due to the numerical stability properties of		Due to the numerical stability properties of

Himpe: references with cite, reference link

2013-04-24T12:03:43Z

references with cite, reference link

← Older revision		Revision as of 12:03, 24 April 2013
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	==Description==		==Description==

	The method introduced here is described in ~~[1]~~ and ~~[2]~~, which is an extension of the moment-matching MOR method for nonparametric systems (see ~~[5][6]~~ for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:		The method introduced here is described in <ref name="daniel04"/> and <ref name="feng07"/>, which is an extension of the moment-matching MOR method for nonparametric systems (see
			<ref name="feng13a"/>, <ref name="oda98"/> for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

	<math>		<math>
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	The matrix <math>V</math> is derived by orthogonalizing a number of moment		The matrix <math>V</math> is derived by orthogonalizing a number of moment
	matrices of the system in (1) as follows, see ~~[1]~~ or ~~[2]~~.		matrices of the system in (1) as follows, see <ref name="daniel04"/> or <ref name="feng07"/>.

	By defining <math>		By defining <math>
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	matrices multiplied by <math>C</math> from the left. The matrix <math>V</math> can be		matrices multiplied by <math>C</math> from the left. The matrix <math>V</math> can be
	generated by first explicitly computing some of the moment matrices		generated by first explicitly computing some of the moment matrices
	and then orthogonalizing them as suggested in ~~[1]~~.		and then orthogonalizing them as suggested in <ref name="daniel04"/>.
	The resulting <math>V</math> is desired to expand the subspace:		The resulting <math>V</math> is desired to expand the subspace:

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	Instead of directly computing the moment matrices in (3), a		Instead of directly computing the moment matrices in (3), a
	numerically robust method is proposed in ~~[2]~~ ( the		numerically robust method is proposed in <ref name="feng07"/> ( the
	detailed algorithm is described in ~~[3]~~ ), which combines		detailed algorithm is described in <ref name="fengXX"/> ), which combines
	the recursion in (5) with the modified Gram-Schmidt		the recursion in (5) with the modified Gram-Schmidt
	process to implicitly compute the moment matrices. The computed <math>V</math>		process to implicitly compute the moment matrices. The computed <math>V</math>
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	Due to the numerical stability properties of		Due to the numerical stability properties of
	the repeated modified Gram-Schmidt process employed in		the repeated modified Gram-Schmidt process employed in
	~~[2]~~ and ~~[3]~~, the reduced model derived from <math>V</math>		<ref name="feng07"/> and <ref name="fengXX"/>, the reduced model derived from <math>V</math>
	in (4) is computed in a numerically stable and accurate way. Applications of the method in ~~[2][3]~~ to the parametric models [[Gyroscope]], [[Silicon nitride membrane]], and [[Microthruster Unit]], can be found in ~~[4]~~.		in (4) is computed in a numerically stable and accurate way. Applications of the method in <ref name="feng07"/>, <ref name="fengXX"/> to the parametric models [[Gyroscope]], [[Silicon nitride membrane]], and [[Microthruster Unit]], can be found in <ref name="feng13"/>.

	==References==		==References==

	~~[1] L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J.~White. "A~~		<references>
	~~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~~		<ref name="daniel04">L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J.~White. "<span class="plainlinks">[http://dx.doi.org/10.1109/TCAD.2004.826583 A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models]</span>", IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst, 22(5): 678--693, 2004.</ref>
	~~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~~		<ref name="feng07">L. Feng and P. Benner, "<span class="plainlinks">[http://dx.doi.org/10.1002/pamm.200700749 A Robust Algorithm for Parametric Model Order Reduction]</span>", In Proc. Applied Mathematics and Mechanics (ICIAM 2007), 7(1): 10215.01--02, 2007.</ref>
	~~order reduction based on implicit moment matching~~,~~" submitted~~.

	~~[4]~~ L. Feng, P. Benner, J.~~G Korvink, "Subspace recycling accelerates the~~ parametric ~~macromodeling of MEMS~~" ~~International Journal for Numerical Methods in Engineering, 94(1): 84-110~~, ~~2013~~.		<ref name="fengXX">L. Feng and P. Benner, "<span class="plainlinks">[http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=64CF520F4D47C5E63F6BA178288BE18F?doi=10.1.1.154.4365&rep=rep1&type=pdf A robust algorithm for parametric model order reduction based on implicit moment matching]</span>", submitted.</ref>

	~~[5]~~ L. Feng, P. Benner, ~~and~~ J.G Korvink, "~~System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background~~. ~~In T~~. ~~Bechtold, G~~. ~~Schrag, and L~~. ~~Feng, editors, System-Level Modeling~~ of MEMS, ~~Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9~~, ~~Wiley~~-~~VCH~~, 2013.		<ref name="feng13">L. Feng, P. Benner, J.G Korvink, "<span class="plainlinks">[http://dx.doi.org/10.1002/nme.4449 Subspace recycling accelerates the parametric macromodeling of MEMS]</span>", International Journal for Numerical Methods in Engineering, 94(1): 84-110, 2013.</ref>

	[6] A. Odabasioglu, M. Celik, and L. T. Pileggi, "PRIMA: passive		<ref name="feng13a">L. Feng, P. Benner, and J.G Korvink, "<span class="plainlinks">[http://dx.doi.org/%2010.1002/9783527647132.ch3 System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background]</span>". In T. Bechtold, G. Schrag, and L. Feng, editors, System-Level Modeling of MEMS, Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9, Wiley-VCH, 2013.</ref>
	reduced-order interconnect macromodeling algorithm,"
	IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.,17(8):645--654,1998.		<ref name="oda98">A. Odabasioglu, M. Celik, and L. T. Pileggi, "<span class="plainlinks">[http://dx.doi.org/10.1109/ICCAD.1997.643366 PRIMA: passive reduced-order interconnect macromodeling algorithm]</span>", IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.,17(8):645--654,1998.</ref>

			</references>

Grundel at 09:19, 19 April 2013

2013-04-19T09:19:16Z

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


	==Description==		==Description==

Feng: /* References */

2013-03-28T08:36:58Z

References

← Older revision		Revision as of 08:36, 28 March 2013
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	order reduction based on implicit moment matching," submitted.		order reduction based on implicit moment matching," submitted.

	[4] L. Feng, P. Benner, J.G Korvink, "Subspace recycling accelerates the parametric macromodeling of MEMS" International Journal for Numerical Methods in Engineering, ~~to appear~~.		[4] L. Feng, P. Benner, J.G Korvink, "Subspace recycling accelerates the parametric macromodeling of MEMS" International Journal for Numerical Methods in Engineering, 94(1): 84-110, 2013.

	[5] L. Feng, P. Benner, and J.G Korvink, "System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background. In T. Bechtold, G. Schrag, and L. Feng, editors, System-Level Modeling of MEMS, Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9, Wiley-VCH, 2013.		[5] L. Feng, P. Benner, and J.G Korvink, "System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background. In T. Bechtold, G. Schrag, and L. Feng, editors, System-Level Modeling of MEMS, Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9, Wiley-VCH, 2013.

Himpe at 11:42, 25 March 2013

2013-03-25T11:42:50Z

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

			==Description==

	The method introduced here is described in [1] and [2], which is an extension of the moment-matching MOR method for nonparametric systems (see [5][6] for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:		The method introduced here is described in [1] and [2], which is an extension of the moment-matching MOR method for nonparametric systems (see [5][6] for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

Feng: /* References */

2012-12-20T14:09:36Z

References

← Older revision		Revision as of 14:09, 20 December 2012
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	[2] L. Feng and P. Benner, "A Robust Algorithm for Parametric Model		[2] L. Feng and P. Benner, "A Robust Algorithm for Parametric Model
	Order Reduction," In Proc. Applied Mathematics and		Order Reduction," In Proc. Applied Mathematics and
	Mechanics (ICIAM 2007)}, 7(1): 10215.01--02, 2007.		Mechanics (ICIAM 2007), 7(1): 10215.01--02, 2007.

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

	[4] L. Feng, P. Benner, J.G Korvink, "Subspace recycling accelerates the parametric macromodeling of MEMS" International Journal for Numerical Methods in Engineering, ~~accepted~~.		[4] L. Feng, P. Benner, J.G Korvink, "Subspace recycling accelerates the parametric macromodeling of MEMS" International Journal for Numerical Methods in Engineering, to appear.

	[5] L. Feng, P. Benner, and J.G Korvink, "System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background. In T. Bechtold, G. Schrag, and L. Feng, editors, System-Level Modeling of MEMS, Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9, Wiley-VCH, 2013.		[5] L. Feng, P. Benner, and J.G Korvink, "System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background. In T. Bechtold, G. Schrag, and L. Feng, editors, System-Level Modeling of MEMS, Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9, Wiley-VCH, 2013.

Feng at 10:49, 23 November 2012

2012-11-23T10:49:56Z

← Older revision		Revision as of 10:49, 23 November 2012
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	the repeated modified Gram-Schmidt process employed in		the repeated modified Gram-Schmidt process employed in
	[2] and [3], the reduced model derived from <math>V</math>		[2] and [3], the reduced model derived from <math>V</math>
	in (4) is computed in a numerically stable and accurate way. ~~Application~~ of the method in [2][3] to the parametric models [[Gyroscope]], [[Silicon nitride membrane]], and [[Microthruster Unit]], can be found in [4].		in (4) is computed in a numerically stable and accurate way. Applications of the method in [2][3] to the parametric models [[Gyroscope]], [[Silicon nitride membrane]], and [[Microthruster Unit]], can be found in [4].

	==References==		==References==