Moment-matching PMOR method

The method introduced here is described in [1] and [2], which is an extension of the moment-matching MOR method for nonparametric systems (see [5][6] for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

$(E_0+s_{1}E+s_2{E}_2+\dots +s_p{E}_p)x=Bu(s_1,\dots ,s_p),y=Cx,(1)$

where $s_1=2\pi jf$ is the frequency domain variable, $f$ is the frequency. $s_2,s_3,\dots ,s_p$ 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 (geometrical) parameters like $s_1=\rho v$, where $\rho$ and $v$ are two independent physical (geometrical) 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 $V\in {\mathbb{ℝ}}^{n\times r},r\ll n$ has to be computed.

$V^T(E_0+s_1{E}_1+s_2{E}_2+\dots +s_p{E}_p)Vx=V^{T}Bu(s_1,\dots ,s_p),$

$y=CVx.(2)$

The matrix $V$ is derived by orthogonalizing a number of moment matrices of the system in (1) as follows, see [1] or [2].

By defining $\tilde{E}=E_0+s_1^0{E}_1+s_2^0{E}_2+\cdots +s_p^0{E}_p,$ and $B_M={\tilde{E}}^{-1}B,M_i=-{\tilde{E}}^{-1}{E}_i,i=1,2,\dots ,p$, we can expand $x$ in (1) at $s_1,s_2,\dots ,s_p$ around $p_0=[s_1^0,s_2^0,\cdots ,s_p^0]$ as below,

$x=[I-({\sigma }_1{M}_1+\dots +{\sigma }_p{M}_p){]}^{-1}{B}_{M}u(s_1,\dots ,s_p)=\sum _{i=0}^{\mathrm{\infty }}({\sigma }_1{M}_1+\dots +{\sigma }_p{M}_p{)}^i{B}_{M}u(s_1,\dots ,s_p).$

Here ${\sigma }_i=s_i-s_i^0,i=1,2,\dots ,p$. We call the coefficients in the above series expansion moment matrices of the parametrized system, i.e. $B_M,M_1{B}_M,\dots ,M_p{B}_M,M_1^2{B}_M,(M_1{M}_2+M_2{M}_1)B_M,\dots ,(M_1{M}_p+M_p{M}_1)B_M,M_p^2{B}_M,M_1^3{B}_M,\dots$. The corresponding moments of the transfer function are those moment matrices multiplied by $C$ 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:

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\{V\}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{B_M,M_1{B}_M,\dots ,M_p{B}_M,M_1^2{B}_M,(M_1{M}_2+M_2{M}_1)B_M,\dots ,(M_1{M}_p+M_p{M}_1)B_M,$ $M_p^2{B}_M,M_1^3{B}_M,\dots ,M_1^r{B}_M,\dots ,M_p^r{B}_M\}.(3)$

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 algebraically 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,

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\{V\}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{R_0,R_1,\dots ,R_r\}.(4)$

$R_0=[B_M],$

$R_1=[M_1{R}_0,\dots ,M_p{R}_0],$

$R_2=[M_1{R}_1,\dots ,M_p{R}_1],(5)$

$⋮,$

$R_r=[M_1{R}_{r-1},\dots ,M_p{R}_{r-1}]$

$⋮.$

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. Application of the method in [2][3] to the parametric models Gyroscope, Silicon nitride membrane, and Microthruster Unit, can be found in [4].

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.

[4] L. Feng, P. Benner, J.G Korvink, "Subspace recycling accelerates the parametric macromodeling of MEMS" International Journal for Numerical Methods in Engineering, accepted.

[5] L. Feng, P. Benner, and J.G Korvink, "System-level modeling of MEMS by means of model order reduction (mathematical approximation)--mathematical background. In T. Bechtold, G. Schrag, and L. Feng, editors, System-Level Modeling of MEMS, Advanced Micro & Nanosystems. ISBN 978-3-527-31903-9, Wiley-VCH, 2013.

[6] A. Odabasioglu, M. Celik, and L. T. Pileggi, "PRIMA: passive reduced-order interconnect macromodeling algorithm," IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.,17(8):645--654,1998.