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Difference between revisions of "Moment-matching PMOR method"

 
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[[Category:method]]
Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized
 
  +
[[Category:parametric]]
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 measurement, material property, damping of the
 
systems or 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 models is much shorter
 
than directly simulating the original large system. However, the
 
time of constructing the reduced model increases with the dimension
 
of the original system. If the original system is very large, the
 
process of obtaining the reduced model could become extremely slow.
 
The recycling algorithm considered in this paper tries to accelerate
 
the above process and reduce the time of deriving the reduced model
 
to a reasonable range.
 
   
The method introduced here is from [1][2], and applies to a linear parametrized system,
 
which has the following form in the frequency domain:
 
   
  +
==Description==
<math>
 
(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)x=Bu(s_p),
 
y=L^{\mathrm{T}}x, (1)
 
</math>
 
   
  +
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
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 combination of several physical parameters like <math>s_1=\rho v</math>, where <math>\rho</math> and <math>v</math> are two physical parameters.
 
  +
<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>x(t)\in \mathbb{R}^n</math> is the state vector, <math>u \in \mathbb{R}^{d_I}</math> and <math>y \in
 
 
(E_0+s_1 E+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad
\mathbb{R}^{d_O}</math> are, respectively, the inputs and outputs of the
 
  +
y=Cx, \quad \quad \quad \quad (1)
system. To obtain the reduced model in (1), a
 
  +
</math>
projection matrix <math>V</math> which is independent of all the parameters has
 
to be computed.
 
   
 
where <math>s_1=j2 \pi f</math> is the frequency domain variable, <math>f</math> is the frequency. <math>s_2, s_3, \ldots, s_{p}</math> are the parameters of the system. They can be any scalar functions of some source parameters, like <math>s_2=e^t</math>, where <math>t</math> is time, or combinations of several physical (geometrical) parameters like <math>s_2=\rho v</math>, where <math>\rho</math> and <math>v</math> are two independent physical (geometrical) 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
<math>V^T(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)Vx=V^TBu(s_p),
 
 
\mathbb{R}^{d_O}</math> are the inputs and outputs of the
y=L^{\mathrm{T}}Vx.
 
  +
system, respectively.
</math>
 
   
 
To obtain the reduced model in (2), a [[Projection_based_MOR|projection]] matrix
The matrix $V$ is derived by orthogonalizing a number of moment
 
  +
<math>V \in \mathbb{R}^{n \times r}, r\ll n</math> has to be computed.
matrices of the system in (1)[1][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>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>
 
\tilde{E}=E_0+s_1^0E_1+s_2^0E_2+\cdots+s_p^0E_p,
 
 
</math>
 
</math>
   
we can expand <math>x</math> in (1) at <math>s_1, s_2, \ldots, s_p</math> around a set of
+
The matrix <math>V</math> is derived by orthogonalizing a number of moment
  +
matrices of the system in (1) as follows, see <ref name="daniel04"/> or <ref name="feng07"/>.
expansion points <math>p_0=[s_1^0,s_2^0,\cdots,s_p^0]</math> as below,
 
   
<math>
+
By defining <math>
 
\tilde{E}=E_0+s_1^0E_1+s_2^0E_2+\cdots+s_p^0E_p,
x=[I-(\sigma_1M_1+\ldots +\sigma_pM_p]^{-1}B_Mu(s_p)
 
  +
</math> and <math>B_M=\tilde{E}^{-1}B, M_i=-\tilde{E}^{-1}E_i,i=1,2,\ldots,p</math>,
=\sum\limits_{i=0}^{\infty}(\sigma_1M_1+\ldots+\sigma_pM_p)^iB_Mu(s_p).
 
  +
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>
 
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>
 
</math>
   
 
Here <math>\sigma_i=s_i-s_i^0, i=1,2,\ldots,p</math>. We call the coefficients
 
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
 
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
+
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 of the transfer function are those moment
matrices multiplied by <math>L^{\mathrm{T}}</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 is suggested in~\cite{Daniel04}.
+
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:
   
<math>
+
:<math>
\mathop{\mathrm{range}}\{V\}=\mathop{\mathrm{span}}\{B_M, \ M_1B_M,\ldots, M_pB_M,\ M_1^2B_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>
 
 
M_p^2B_M, M_1^3B_M,\ldots, M_1^rB_M, \ldots,M_p^rB_M \}. \quad \quad \quad \quad (3)
<math>(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 \}. (2)
 
 
</math>
 
</math>
   
Line 71: Line 56:
 
due to numerical instability. Therefore, with this matrix <math>V</math> one
 
due to numerical instability. Therefore, with this matrix <math>V</math> one
 
cannot get an accurate reduced model which matches all the moments
 
cannot get an accurate reduced model which matches all the moments
included in the subspace.
+
algebraically included in the subspace.
   
Instead of directly computing the moment matrices in (2)[1], a
+
Instead of directly computing the moment matrices in (3), a
numerically robust method is proposed in [1] (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 recursions in (4) 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>
 
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 \}. (3)
+
\mathop{\mathrm{range}}\{V\}=\mathop{\mathrm{span}}\{R_0, R_1,\ldots, R_r \}. \quad \quad \quad \quad (4)
 
</math>
 
</math>
   
  +
It can be proved that the subspace in(2) is included in the
 
 
:<math> R_0 =[B_M],</math>
subspace in(3). Due to the numerical stability properties of
 
  +
 
:<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
 
the repeated modified Gram-Schmidt process employed in
[2][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(3) is computed in a numerically stable and accurate way.
+
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>
  +
  +
<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>
   
  +
<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>
<math>
 
R_0=B_M, </math>
 
   
  +
<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>
<math>R_1=[M_1R_0,\ldots, M_pR_0], </math>
 
   
  +
<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>
<math>R_2=[M_1R_1,\ldots, M_pR_1],</math>
 
   
  +
<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>
<math>\vdots,</math>
 
   
  +
<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>
<math>R_r=[M_1R_{r-1},\ldots, M_pR_{r-1}]</math>
 
   
  +
</references>
<math>\vdots.</math>
 

Latest revision as of 10:34, 22 May 2013


Description

The method introduced here is described in [1] and [2], which is an extension of the moment-matching method for nonparametric systems (see [3], [4] 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:


(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)

where s_1=j2 \pi f is the frequency domain variable, f is the frequency. s_2, s_3, \ldots, s_{p} are the parameters of the system. They can be any scalar functions of some source parameters, like s_2=e^t, where t is time, or combinations of several physical (geometrical) parameters like s_2=\rho v, where \rho and v are two independent physical (geometrical) parameters. x(t)\in \mathbb{R}^n is the state vector, u \in \mathbb{R}^{d_I} and y \in
\mathbb{R}^{d_O} are the inputs and outputs of the system, respectively.

To obtain the reduced model in (2), a projection matrix V \in \mathbb{R}^{n \times r}, r\ll n has to be computed.

V^T(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)Vx=V^TBu(s_1,\ldots,s_p),
y=CVx.  \quad \quad \quad \quad  \quad \quad \quad \quad \quad \quad \quad \quad(2)

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

By defining 
\tilde{E}=E_0+s_1^0E_1+s_2^0E_2+\cdots+s_p^0E_p,
and B_M=\tilde{E}^{-1}B, M_i=-\tilde{E}^{-1}E_i,i=1,2,\ldots,p, we can expand x in (1) at s_1, s_2, \ldots, s_p around p_0=[s_1^0,s_2^0,\cdots,s_p^0] as below,


 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).

Here \sigma_i=s_i-s_i^0, i=1,2,\ldots,p. We call the coefficients in the above series expansion moment matrices of the parametrized system, i.e. 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. The corresponding moments of the transfer function are those moment matrices multiplied by C from the left. The matrix V can be generated by first explicitly computing some of the moment matrices and then orthogonalizing them as suggested in [1]. The resulting V is desired to expand the subspace:


\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)

However, V 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 V one cannot get an accurate reduced model which matches all the moments algebraically 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 [5] ), which combines the recursion in (5) with the modified Gram-Schmidt process to implicitly compute the moment matrices. The computed V is actually an orthonormal basis of the subspace as below,


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


 R_0 =[B_M],
R_1=[M_1R_0,\ldots, M_pR_0],
R_2=[M_1R_1,\ldots, M_pR_1], \quad \quad \quad \quad (5)
 \vdots,
R_r=[M_1R_{r-1},\ldots, M_pR_{r-1}]
 \vdots.

Due to the numerical stability properties of the repeated modified Gram-Schmidt process employed in [2] and [5], the reduced model derived from V in (4) is computed in a numerically stable and accurate way. Applications of the method in [2], [5] to the parametric models Gyroscope, Silicon nitride membrane, and Microthruster Unit, can be found in [6].

References

  1. 1.0 1.1 1.2 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. 2.0 2.1 2.2 2.3 2.4 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, 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.
  4. A. Odabasioglu, M. Celik, and L. T. Pileggi, "PRIMA: passive reduced-order interconnect macromodeling algorithm", IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.,17(8):645--654,1998.
  5. 5.0 5.1 5.2 L. Feng and P. Benner, "A robust algorithm for parametric model order reduction based on implicit moment matching", submitted.
  6. 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.