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 measurement, material property, damping of the systems or 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 models is much shorter than directly simulating the original large system. However, the time of constructing the reduced model increases with the dimension of the original system. If the original system is very large, the process of obtaining the reduced model could become extremely slow. The recycling algorithm considered in this paper tries to accelerate the above process and reduce the time of deriving the reduced model to a reasonable range.

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

$(E_0+s_1{E}_1+s_2{E}_2+\dots +s_p{E}_p)x=Bu(s_p),(1)y=L^{\mathrm{T}}x,$

where $s_1,s_2,\dots ,s_p$ 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 combination 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, respectively, the inputs and outputs of the system. To obtain the reduced model in (1), a projection matrix $V$ which is independent of all the parameters has to be computed.

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

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

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

$\tilde{E}=E_0+s_1^0{E}_1+s_2^0{E}_2+\cdots +s_p^0{E}_p,$

we can expand $x$ in (1) at $s_1,s_2,\dots ,s_p$ around a set of expansion points $p_0=[s_1^0,s_2^0,\cdots ,s_p^0]$ as below,

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

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. Failed to parse (syntax error): {\displaystyle 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 is suggested in~\cite{Daniel04}. The resulting $V$ is desired to expand the subspace:

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

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 (2)[1], a numerically robust method is proposed in [1] (the detailed algorithm is described in [3]), which combines the recursions in (4) 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,

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\{V\}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{R_0,R_1,\dots ,R_r\}.(3)$

It can be proved that the subspace in~(2) is included in the subspace in~(3). Due to the numerical stability properties of the repeated modified Gram-Schmidt process employed in [2][3], the reduced model derived from $V$ in~(3) is computed in a numerically stable and accurate way.

$R_0=B_M,R_1=[M_1{R}_0,\dots ,M_p{R}_0],R_2=[M_1{R}_1,\dots ,M_p{R}_1],⋮,R_r=[M_1{R}_{r-1},\dots ,M_p{R}_{r-1}]⋮.$

Furthermore, one can see that each moment matrix is actually several vectors multiplied by ${\tilde{E}}^{-1}$, and if the dimension of $\tilde{E}$ is very large, it is necessary to solve linear systems like

Failed to parse (unknown function "\label"): {\displaystyle \label{sequence} \tilde{E}x=w_i, \ i=1,2,\ldots, l }

to obtain ${\tilde{E}}^{-1}{w}_i$, where $\tilde{E}$ is generally non-symmetric and $w_i$ is a vector. Moreover, if quite a few of the moment matrices need to be computed (this is normal when system (1) contains more than 2 parameters), the number $l$ of the linear systems in (\ref{sequence}) will be very large.