I prefer to add a wiki page for the cross Gramian approach and delete the direct truncation part (where does the name come from?) here. - U. Baur

OK with me. The source of the name I don't remember, I guess some talk, I have been using it for some time. It seems fitting, as without the balancing the truncation happens directly, thus my initial append of the BT article due to the close relation. - C. Himpe

I agree with U. Baur. This direct truncation is usually referred to as cross Gramian balanced truncation for which several extra research articles can be found.
Hence, it should be called that way, or it should get a wiki page of its own. Either way some references are required. - P. Kürschner

Please add a reference that introduces / uses the term "Cross Gramian Balancing". Thanks. - C. Himpe

The only reference with the term "Cross Gramian Balancing" I was able to find is "Model reduction using semidefinite programming", which is touches the cross gramian topic with only 5 lines of text and cites a Sorensen paper from 2002 which uses the term "Approximate Balancing". I did find no reference to "Cross Gramian Balanced Truncation". In both cases I just might not have looked in right places. I personally would prefer a naming that focusses on the Truncation and not the Balancing, since there is no balancing procedure (as with WC and WO) involved, thus my usage of "Direct Truncation". - C. Himpe

The origin is exactly this Sorensen, Anthoulas paper. The term, approximate reduction is, however, commonly associated with approaches where the Gramians are in some way approximated.
It does to my knowledge not refer to this particular approach. - P. Kürschner

To make it clearer, I think it is not bad to add some sentences like: The original system is directly truncated by the Cross Gramian approach without any balancing. The reduced model is thus not balanced. -L. Feng

Power system examples

2013-05-29T06:32:15Z

Padé approximation methods

2013-04-29T10:07:48Z

Feng:

2013-04-29T09:52:05Z

Feng: /* Description */

[[Category:method]]
[[Category:DAE order unspecified]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:linear algebra]]

==Description==
The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as
''Padé'' approximation methods<ref name="feldmann95"/>. They belongs to the [[Projection based MOR]] methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.

<math>
E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
y(t)=Cx(t), \quad \quad (1)
</math>

They are very efficient in many engineering applications, such as circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..

The basic steps are as follows. First, the transfer function

<math>H(s)=Y(s)/U(s)=C(sE-A)^{-1}B</math>

is expanded into a power series at an expansion point <math>s_0\in\mathbb{C}\cup \infty</math>.

Let <math>s=s_0+\sigma</math>, then, within the convergence radius of the series, we have

<math>H(s_0 + \sigma)= C[(s_{0}+\sigma){E}-A]^{-1}B</math>

<math>=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B</math>

<math>=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}[(s_0{ E}-{ A})]^{-1}B</math>

<math>=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</math>

<math>=\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,</math>

where <math>m_i(s_0)</math> are called the moments of the transfer function about <math>s_0</math> for <math>i=0,1,2,\ldots</math>.
If the expansion point is chosen as zero, then the moments simplify to <math>m_i(0)=C(A^{-1}E)^i(-A^{-1}B)</math>.

The goal in moment-matching model reduction is the construction of a reduced order
system where some moments <math>\hat m_i</math> of the associated transfer function <math>\hat H</math> match some moments
of the original transfer function <math>H</math>.

The matrices <math>V</math> and <math>W</math> for model order reduction can be computed
from the vectors which are associated with the moments, for
example, using a single expansion point <math>s_0=0</math>, by

<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (2) </math>

<math>\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\}, \quad \quad (3) </math>

where <math>\tilde B=-A^{-1}B</math>.

The transfer function <math>\hat H</math> of the reduced model has good approximation properties around <math>s_0</math>, which matches the first <math>2r</math> moments of <math>H(s)</math> at <math>s_0</math>.

Using a set of <math>k</math> distinct expansion points <math>\{s_1,\cdots,s_k\}</math>, the reduced model obtained by, e.g.,

<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \} \quad \quad \quad \quad \quad \quad \quad \quad (4)</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 \},\quad \quad \quad \quad \quad (5) </math>

matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. The reduced model is in the form as below

<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

For the case of one expansion point in (2)(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces
which can easily be computed by Arnoldi or Lanczos methods. The matrices <math>V</math> and <math>W</math> in (4)(5) can be computed with the rational Krylov algorithm in<ref name="grimme97"/> or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting
methods is roughly <math>O(n r^2)</math> for sparse matrices <math>A, E</math>.

==References==

<references>

<ref name="freund03">R.W. Freund, "[http://dx.doi.org/10.1017/S0962492902000120 Model reduction methods based on Krylov subspaces]". Acta Numerica, 12:267-319, 2003.</ref>

<ref name="feldmann95">P. Feldmann and R.W. Freund, "[http://dx.doi.org/10.1109/43.384428 Efficient linear circuit analysis by Pade approximation via the Lanczos process]". IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 14:639-649, 1995.</ref>

<ref name="grimme97">E.J. Grimme, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.9254&rep=rep1&type=pdf Krylov projection methods for model reduction]. PhD thesis, Univ. Illinois, Urbana-Champaign, 1997.

</references>