Moment-matching method

The moment-matching methods are also called the Krylov subspace methods, as well as $Pade$ approximation methods. 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

$H(s_0+\sigma )=L^T[(s_0+\sigma )I-A{]}^{-1}B$

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

$=L^T[I-\sigma (s_{0}I-A{)}^{-1}{]}^{-1}[-(s_{0}I-A){]}^{-1}B$

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

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

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 $m_i(0)=L^{\mathrm{T}}(-A^{-1}{)}^{i+1}B$. For $s_0=\mathrm{\infty }$ the moments are also called Markov parameters which can be computed by $L^{\mathrm{T}}{A}^{i-1}B$.

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

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

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

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.,

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

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

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

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