Moment-matching method

Description

The moment-matching methods are also called the Krylov subspace methods [1], as well as Padé approximation methods [2]. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.

$E\frac{dx(t)}{dt}=Ax(t)+Bu(t),y(t)=Cx(t),(1)$

They are very efficient in many engineering applications, circuit simulation, Microelectromechanical systems(MEMS) simulation, etc..

The basic steps are as follows. First, the transfer function

$H(s)=Y(s)/U(s)=C(sE-A{)}^{-1}B$

is expanded into a power series at an expansion point $s_0\in \mathbb{ℂ}\cup \mathrm{\infty }$.

Let $s=s_0+\sigma$, then, within the convergence radius of the series, we have

$H(s_0+\sigma )=C[(s_0+\sigma )E-A{]}^{-1}B$

$=C[\sigma E+(s_{0}E-A){]}^{-1}B$

$=C[I+\sigma (s_{0}E-A{)}^{-1}E{]}^{-1}[(s_{0}E-A){]}^{-1}B$

$=C[I-\sigma (s_{0}E-A{)}^{-1}E+{\sigma }^2[(s_{0}E-A{)}^{-1}E{]}^2+\dots ]s_{0}E-A{)}^{-1}B$

$=\sum _{i=0}^{\mathrm{\infty }}\underset{:=m_i(s_0)}{\underbrace{C[-(s_{0}E-A{)}^{-1}E{]}^i(s_{0}E-A{)}^{-1}B}}{\sigma }^i,$

where $m_i(s_0)$ are called the moments of the transfer function about $s_0$ for $i=0,1,2,\dots$. If the expansion point is chosen as zero then the moments simplify to $m_i(0)=C(A^{-1}E{)}^i(-A^{-1}B)$. For $s_0=\mathrm{\infty }$ the moments are also called Markov parameters which can be computed by $C(E^{-1}A{)}^i(E^{-1}B)$ if E is invertable.

The goal in moment-matching model reduction is the construction of a reduced order system where some moments ${\hat{m}}_i$ of the associated transfer function $\hat{H}$ match some moments of the original transfer function $H$.

The matrices $V$ and $W$ for model order reduction can be computed from the vectors which are associated with the moments, for example, using a single expansion point $s_0=0$, by

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(V)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{\tilde{B},(A^{-1}E{)}^2\tilde{B},\dots ,(A^{-1}E{)}^r\tilde{B}\},$

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(W)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{C^T,E^T{A}^{-T}{C}^T,(E^T{A}^{-T}{)}^2{C}^T,\dots ,(E^T{A}^{-T}{)}^{r-1}{C}^T\},$

where $\tilde{B}=-A^{-1}B$. The reduced model is in the form of the system in (2) in Projection based MOR. The corresponding transfer function $\hat{H}$ has good approximation properties around $s_0$, which matches the first $2r$ moments of $H(s)$ at $s_0$.

Using a set of $k$ distinct expansion points $\{s_1,\cdots ,s_k\}$, the reduced model can be obtained by, e.g.,

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(V)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{(A-s_{1}E{)}^{-1}E\tilde{B},\dots ,(A-s_{k}E{)}^{-1}E\tilde{B}\}$,

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(W)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{E^T(A-s_{1}E{)}^{-T}{C}^T,\dots ,E^T(A-s_{k}E{)}^{-T}{C}^T\},$

has order $r=k$ and matches the first two moments at each $s_j$, $j=1,\dots ,k$, see [3].

It can be seen that the columns of $V$, $W$ span Krylov subspaces which can easily be computed by Arnoldi or Lanczos methods [1][2]. In these algorithms only matrix-vector multiplications are used which are simple to implement and the complexity of the resulting methods is only $O(n{r}^2)$.

References

[1] R.W. Freund, Model reduction methods based on Krylov subspaces. Acta Numerica, 12:267-319, 2003.

[2] P. Feldmann and R.W. Freund, Efficient linear circuit analysis by Pade approximation via the Lanczos process. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 14:639-649, 1995.

[3] E.J. Grimme, Krylov projection methods for model reduction. PhD thesis, Univ. Illinois, Urbana-Champaign, 1997.