Moment-matching method

Description

The moment-matching methods are also called the Krylov subspace methods, as well as Padé approximation methods. They belongs to the Projection based MOR 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, such as 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 $m_i(0)=C(A^{-1}E{)}^i(-A^{-1}B)$.

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

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

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(W)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{C^T,E^T{A}^{-T}{C}^T,(E^T{A}^{-T}{)}^2{C}^T,\dots ,(E^T{A}^{-T}{)}^{r-1}{C}^T\},(3)$

where $\tilde{B}=-A^{-1}B$.

The transfer function $\hat{H}$ of the reduced model 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 obtained by, e.g.,

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(V)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{(A-s_{1}E{)}^{-1}E\tilde{B},\dots ,(A-s_{k}E{)}^{-1}E\tilde{B}\}(4)$,

$\text{<mrow data-mjx-texclass="ORD">range</mrow>}(W)=\text{<mrow data-mjx-texclass="ORD">span</mrow>}\{E^T(A-s_{1}E{)}^{-T}{C}^T,\dots ,E^T(A-s_{k}E{)}^{-T}{C}^T\},(5)$

matches the first two moments at each $s_j$, $j=1,\dots ,k$, see . The reduced model is in the form as below

$W^{T}E\frac{dVz}{t}=W^{T}AVz+W^{T}Bu(t),\hat{y}(t)=CVz.$

For the case of one expansion point in (2),(3), it can be seen that the columns of $V$, $W$ span Krylov subspaces which can easily be computed by Arnoldi or Lanczos methods. The matrices $V$ and $W$ in (4),(5) can be computed with the rational Krylov algorithm in or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting methods is roughly $O(n{r}^2)$ for a sparse matrix $A$.