Moment-matching PMOR method

Parametric model order reduction (PMOR) methods are designed for model order reduction of parametrized 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 measurements, material properties, the damping of the system or the 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 model is much shorter than directly simulating the original large system.

The method introduced here is described in [1] and [2], and applies to a linear parametrized system, which has the following form in the frequency domain:

$(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad y=C{\mathrm{T}}x, \quad \quad \quad \quad (1)$

where $s_1, s_2, \ldots, s_{p}$ are the parameters of the system. They can be any scalar functions of some source parameters, like $s_1=e^t$, where $t$ is time, or combinations of several physical parameters like $s_1=\rho v$, where $\rho$ and $v$ are two physical 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 $V$ which is independent of all the parameters 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), see [1] or [2].

By defining $B_M=\tilde{E}^{-1}B, M_i=-\tilde{E}^{-1}E_i,i=1,2,\ldots,p$ and

$\tilde{E}=E_0+s_1^0E_1+s_2^0E_2+\cdots+s_p^0E_p,$

we can expand $x$ in (1) at $s_1, s_2, \ldots, s_p$ around a set of expansion points $p_0=[s_1^0,s_2^0,\cdots,s_p^0]$ 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 are those moment matrices multiplied by $L^{\mathrm{T}}$ 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 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 [3] ), 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 [3], the reduced model derived from $V$ in (4) is computed in a numerically stable and accurate way.

References

[1] 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] 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 and P.~Benner, "A robust algorithm for parametric model order reduction based on implicit moment matching," submitted.