Projection based MOR

Consider the linear time invariant system

$E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad y(t)=Cx(t), \quad \quad (1)$

as an example. All the existing model order reduction (MOR) methods are based on projection. That is to find a subspace $S_1$ which approximates the manifold where the state vector $x(t)$ resides. Afterwards, $x(t)$ is approximated by a vector $\tilde x(t)$ in $S_1$. The reduced model is produced by Petrov-Galerkin projection onto a subspace $S_2$, or by Galerkin projection onto the same subspace $S_1$.

Assuming that an orthonormal basis $V=(v_1,v_2, \ldots, v_q)$ of the subspace $S_1$ has been found, then the approximation $\tilde x(t)$ in $S_1$ can be represented by the basis as $\hat x(t)=V z(t)$. Therefore $x(t)$ can be approximated by $x(t) \approx V z(t)$. Here $z$ is a vector of length $q \ll n$.

Once $z(t)$ is computed, an approximate solution $\tilde x(t)=V z(t)$ for $x(t)$ can be obtained. The vector $z(t)$ can be computed from the reduced model which is derived by the following two steps.

Step 1. By replacing $x$ in (1) with $Vz$, we get

$E \frac{d{Vz}}{dt} \approx A Vz+Bu(t),\quad y(t) \approx CV z.$

Step 2. The residual is denoted as $e=AVz+Bu(t)-E \frac{d{Vz}}{dt}$. Forcing $e=0$ in a properly chosen subspace $S_2$ of $\mathbb {R}^n$ leads to the Petrov-Galerkin projection: $W^T e=0$, where the columns of $W$ are the basis of $S_2$. Then we have,

$W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.$

By defining $\hat{E}=W^TEV$, $\hat {A}=W^TAV, \hat{B}=W^TB$, $\hat{C}=CV$, we get the final reduced model

$\hat{E} \frac{dz(t)}{dt}=\hat{A}z(t)+\hat{B}u(t), \quad \hat{y}(t)=\hat{C}z(t). \quad \quad (2)$

Notice that the approximation $\hat x(t)=Vz(t)$ of $x(t)$ can be obtained from $z(t)$ by solving the system in (2). The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1). Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model. In order to save the simulation time of solving (1), the reduced model can be used to replace (1) to attain a fast simulation. Furthermore, the error between the two systems should be within acceptable tolerance. The error can be measured through the error between the the state vectors $x(t), \hat x(t)$, or between the output responses $y(t), \hat y(t)$, or between the transfer functions of the two systems.

It can be seen that once the two matrices $W$ and $V$ have been computed, the reduced model is obtained. While the Gramian based MOR methods (e.g. Balanced Truncation) usually compute $W$ different from $V$, some methods use $W=V$, e.g. some of the Moment-matching methods, the Reduced Basis Methods, and some of the POD methods etc.. When $W=V$, Petrov-Galerkin projection becomes Galerkin projection.

MOR methods differ in the computation of the two matrices $W$ and $V$. The Gramian based MOR methods compute $W$ and $V$ by the controllability and observability Gramians. Reduced basis methods and POD methods compute $V$ from the snapshots of the state vector $x(t)$ at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute $W$ and $V$ from the moments of the transfer function.

One common goal of all MOR methods is that the behavior of the reduced model should be sufficiently "close" to that of the original model guaranteed through the above mentioned error measurements.