Dominant Subspace Projection Model Reduction Method
The Dominant Subspace Projection Model Reduction Method (DSPMR), introduced in [1], or short Dominant Subspaces method is closely related to Balanced Truncation, and can be seen as a simplification. This is projection-based method is designed for linear time-invariant systems:
\[ \begin{array}{rcl} \dot{x}(t) &=& Ax(t) + Bu(t), \\ y(t) &=& Cx(t). \end{array} \]
Similar to Balanced Truncation, the controllability Gramian matrix \(W_C\) and the observability Gramian matrix \(W_O\) are computed:
\[ AW_C+W_CA^T=-BB^T,\]
\[ A^TW_O+W_OA=-C^TC.\]
As a second step, instead of balancing controllability and observability subspaces, the respective subspaces are conjoined. Typically, (low-rank) Cholesky factors of the system Gramians are computed directly,
\[ W_C = L_C L_C^T,\]
\[ W_O = L_O L_O^T,\]
which are then combined using a singular value decomposition of the concatenated factors:
\[ [L_C, L_O] = U D V\].
The principal left (or right) singular vectors, based on the magnitude of the singular values, then serve as a Galerkin projection.
Alternatively, different approaches for the combination of subspaces can be used. For example in [2] and [3], the singular vectors of the singular value decompositions of the system Gramians,
\[ W_C = U_C D_C V_C,\]
\[ W_O = U_O D_O V_O,\]
are concatenated and orthogonalized by a rank-revealing QR decomposition:
\[[U_C, U_O] = Q R\],
with \(Q\) 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 [4].
Variants
Refined DSPMR
In [1], a variant called Refined Dominant Subspace Projection Model Reduction Method is presented. For this variant the subspaces are weighted by their Frobenius norm before combination, so for example:
\[ \Big[\frac{1}{\|L_C\|_{\text{Fro}}} L_C, \frac{1}{\|L_O\|_{\text{Fro}}} L_O\Big] = U D V\].
\[ \Big[\frac{1}{\|U_C\|_{\text{Fro}}} U_C, \frac{1}{\|U_O\|_{\text{Fro}}} U_O\Big] = Q R\].
Cross-Gramian-Based DSPMR
Instead of using controllability and observability Gramians, also the cross Gramian \(W_X\) can be used for the DSPMR procedure [4],
\[ AW_X+W_XA=-BC,\].
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,
\[ W_X = U_C D_X V_O,\]
which can then be combined by another singular value decomposition:
\[ [U_C, V_O] = U D V\].
The principal left (or right) singular values then represent the Galerkin projection.
This variant can also be refined, as proposed in [4], by scaling the left and right singular vectors by their associated singular values:
\[ [U_C D_X, V_O D_X] = U D V\].
References
- ↑ 1.0 1.1 T. Penzl, "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.).
- ↑ J.-R. Li and J. White. "Efficient model reduction of interconnect via approximate system Gramians", In 1999 IEEE/ACM International Conference on Computer-Aided Design: 380--383, 1999.
- ↑ J.-R. Li and J. White. "Reduction of large circuit models via low rank approximate Gramians", Int. J. Appl. Math. Comput. Sci, 11(5):1151–1171, 2001
- ↑ 4.0 4.1 4.2 P. Benner, C. Himpe, "Cross-Gramian-Based Dominant Subspaces", Advances in Computational Mathematics 45: 2533--2553, 2019.