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	<ref name="penzl06">T. Penzl, "<span class="plainlinks">[https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for model reduction of large dynamical systems]</span>", Linear Algebra Appl., 415(2--3): 322--343, 2006. (Reprint of Technical Report SFB393/99-40, TU Chemnitz, 1999.).</ref>		<ref name="penzl06">T. Penzl, "<span class="plainlinks">[https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for model reduction of large dynamical systems]</span>", Linear Algebra Appl., 415(2--3): 322--343, 2006. (Reprint of Technical Report SFB393/99-40, TU Chemnitz, 1999.).</ref>

	<ref name="benner18">P. Benner, C. Himpe, "<span class="plainlinks">[https://~~arxiv~~.org/~~abs~~/~~1809.08066~~ Cross-Gramian-Based Dominant Subspaces]</span>", ~~arXiv, math.OC~~: ~~1809.08066~~, ~~2018~~.</ref>		<ref name="benner18">P. Benner, C. Himpe, "<span class="plainlinks">[https://doi.org/10.1007/s10444-019-09724-7 Cross-Gramian-Based Dominant Subspaces]</span>", Advances in Computational Mathematics 45: 2533--2553, 2019.</ref>

	</references>		</references>

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New page

{{preliminary}} 

[[Category:method]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:linear algebra]]

==Dominant Subspace Projection Model Reduction Method==
The Dominant Subspace Projection Model Reduction Method (DSPMR), introduced in <ref name="penzl06"/>, or short '''Dominant Subspaces''' method is closely related to [[Balanced_Truncation|Balanced Truncation]], and can be seen as a simplification.
This is [[Projection_based_MOR|projection-based]] method is designed for linear time-invariant systems:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t), \\
y(t) &=& Cx(t).
\end{array}
</math>

Similar to [[Balanced_Truncation|Balanced Truncation]], the [[wikipedia:Controllability_Gramian|controllability Gramian matrix]] <math>W_C</math> and the [[wikipedia:Observability_Gramian|observability Gramian matrix]] <math>W_O</math> are computed:

:<math> AW_C+W_CA^T=-BB^T,</math>

:<math> A^TW_O+W_OA=-C^TC.</math>

As a second step, instead of balancing controllability and observability subspaces, the respective subspaces are conjoined.
Typically, (low-rank) [[wikipedia:Cholesky_decomposition|Cholesky factors]] of the system Gramians are computed directly,

:<math> W_C = L_C L_C^T,</math>

:<math> W_O = L_O L_O^T,</math>

which are then combined using a [[wikipedia:Singular_value_decomposition|singular value decomposition]] of the concatenated factors:

:<math> [L_C, L_O] = U D V</math>.

The principal left (or right) singular vectors, based on the magnitude of the singular values,
then serve as a [[wikipedia:Galerkin_method|Galerkin projection]].

Alternatively, different approaches for the combination of subspaces can be used.
For example in <ref name="li99"/> and <ref name="li01"/>, the singular vectors of the singular value decompositions of the system Gramians,

:<math> W_C = U_C D_C V_C,</math>

:<math> W_O = U_O D_O V_O,</math>

are concatenated and orthogonalized by a [[wikipedia:RRQR_factorization|rank-revealing QR decomposition]]:

:<math>[U_C, U_O] = Q R</math>,

with <math>Q</math> being the resulting Galerkin projection.

Furthermore, the singular vectors of the system Gramians can also be combined by a second singular value decomposition as in <ref name="benner18"/>.

==Variants==

===Refined DSPMR===

In <ref name="penzl06"/>, a variant called Refined Dominant Subspace Projection Model Reduction Method is presented.
For this variant the subspaces are weighted by their [[wikipedia:Matrix_norm#Frobenius_norm|Frobenius norm]] before combination, so for example:

:<math> \Big[\frac{1}{\|L_C\|_{\text{Fro}}} L_C, \frac{1}{\|L_O\|_{\text{Fro}}} L_O\Big] = U D V</math>.

:<math> \Big[\frac{1}{\|U_C\|_{\text{Fro}}} U_C, \frac{1}{\|U_O\|_{\text{Fro}}} U_O\Big] = Q R</math>.

===Cross-Gramian-Based DSPMR===

Instead of using controllability and observability Gramians, also the [[wikipedia:Cross_Gramian|cross Gramian]] <math>W_X</math> can be used for the DSPMR procedure <ref name="benner18"/>,

:<math> AW_X+W_XA=-BC,</math>.

The cross Gramian encodes controllability and observability, hence a singular value decompostion of the (the generally not-symmetric) cross Gramian matrix reveals dominant controllability and observability subspaces,

:<math> W_X = U_C D_X V_O,</math>

which can then be combined by another singular value decomposition:

:<math> [U_C, V_O] = U D V</math>.

The principal left (or right) singular values then represent the Galerkin projection.

This variant can also be refined, as proposed in <ref name="benner18"/>, by scaling the left and right singular vectors by their associated singular values:

:<math> [U_C D_X, V_O D_X] = U D V</math>.

==References==

<references>

<ref name="li99">J.-R. Li and J. White. "[https://doi.org/10.1109/ICCAD.1999.810679 Efficient model reduction of interconnect via approximate system Gramians]", In 1999 IEEE/ACM International Conference on Computer-Aided Design: 380--383, 1999.</ref>

<ref name="li01"> J.-R. Li and J. White. "[http://eudml.org/doc/207549 Reduction of large circuit models via low rank approximate Gramians]", Int. J. Appl. Math. Comput. Sci, 11(5):1151–1171, 2001</ref>

<ref name="penzl06">T. Penzl, "[https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for model reduction of large dynamical systems]", Linear Algebra Appl., 415(2--3): 322--343, 2006. (Reprint of Technical Report SFB393/99-40, TU Chemnitz, 1999.).</ref>

<ref name="benner18">P. Benner, C. Himpe, "[https://arxiv.org/abs/1809.08066 Cross-Gramian-Based Dominant Subspaces]", arXiv, math.OC: 1809.08066, 2018.</ref>

</references>

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