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==Description== |
==Description== |
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| − | The moment-matching methods are also called the ''Krylov'' subspace methods |
+ | The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as |
| − | ''Padé'' approximation methods |
+ | ''Padé'' approximation methods<ref name="feldmann95"/>. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g. |
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</math> |
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| − | They are very efficient in many engineering applications, circuit simulation, Microelectromechanical systems(MEMS) simulation, etc.. |
+ | They are very efficient in many engineering applications, circuit simulation, Microelectromechanical systems (MEMS) simulation, etc.. |
The basic steps are as follows. First, the transfer function |
The basic steps are as follows. First, the transfer function |
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<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},</math> |
<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},</math> |
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| − | has order <math>r=k</math> and matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see |
+ | has order <math>r=k</math> and matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. |
It can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces |
It can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces |
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| − | which can easily be computed by Arnoldi or Lanczos methods |
+ | which can easily be computed by Arnoldi or Lanczos methods <ref name="freund03"/><ref name="feldmann95"/>. In |
these algorithms only matrix-vector multiplications are used which |
these algorithms only matrix-vector multiplications are used which |
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are simple to implement and the complexity of the resulting |
are simple to implement and the complexity of the resulting |
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==References== |
==References== |
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| + | </references> |
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Revision as of 11:39, 24 April 2013
Description
The moment-matching methods are also called the Krylov subspace methods[1], as well as Padé approximation methods[2]. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.
\( E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad y(t)=Cx(t), \quad \quad (1) \)
They are very efficient in many engineering applications, circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..
The basic steps are as follows. First, the transfer function
\(H(s)=Y(s)/U(s)=C(sE-A)^{-1}B\)
is expanded into a power series at an expansion point \(s_0\in\mathbb{C}\cup \infty\).
Let \(s=s_0+\sigma\), then, within the convergence radius of the series, we have
\(H(s_0 + \sigma)= C[(s_{0}+\sigma){E}-A]^{-1}B\)
\(=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B\)
\(=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}[(s_0{ E}-{ A})]^{-1}B\)
\(=C[{ I}-\sigma(s_0{ E}- A )^{-1}E+\sigma^2[(s_0{ E}-{ A})^{-1}E]^{2}+\ldots] s_0{E}-{ A})^{-1}B\)
\(=\sum \limits^\infty_{i=0}\underbrace{C[-(s_0{ E}-{A})^{-1}E]^i(s_0{ E}-{ A})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,\)
where \(m_i(s_0)\) are called the moments of the transfer function about \(s_0\) for \(i=0,1,2,\ldots\). If the expansion point is chosen as zero then the moments simplify to \(m_i(0)=C(A^{-1}E)^i(-A^{-1}B)\). For \(s_0=\infty\) the moments are also called Markov parameters which can be computed by \(C(E^{-1}A)^i(E^{-1}B)\) if E is invertable.
The goal in moment-matching model reduction is the construction of a reduced order system where some moments \(\hat m_i\) of the associated transfer function \(\hat H\) match some moments of the original transfer function \(H\).
The matrices \(V\) and \(W\) for model order reduction can be computed from the vectors which are associated with the moments, for example, using a single expansion point \(s_0=0\), by
\(\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\},\)
\(\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots ,(E^T{A}^{-T})^{r-1}C^T\},\)
where \(\tilde B=-A^{-1}B\). The reduced model is in the form of the system in (2) in Projection based MOR. The corresponding transfer function \(\hat H\) has good approximation properties around \(s_0\), which matches the first \(2r\) moments of \(H(s)\) at \(s_0\).
Using a set of \(k\) distinct expansion points \(\{s_1,\cdots,s_k\}\), the reduced model can be obtained by, e.g.,
\(\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \}\),
\(\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\)
has order \(r=k\) and matches the first two moments at each \(s_j\), \(j=1,\ldots,k\), see [3].
It can be seen that the columns of \(V\), \(W\) span Krylov subspaces which can easily be computed by Arnoldi or Lanczos methods [1][2]. In these algorithms only matrix-vector multiplications are used which are simple to implement and the complexity of the resulting methods is only \(O(n r^2)\).
References
- ↑ 1.0 1.1 R.W. Freund, "Model reduction methods based on Krylov subspaces". Acta Numerica, 12:267-319, 2003.
- ↑ 2.0 2.1 P. Feldmann and R.W. Freund, "Efficient linear circuit analysis by Pade approximation via the Lanczos process". IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 14:639-649, 1995.
- ↑ Cite error: Invalid
<ref>tag; no text was provided for refs namedgrimme97