Projection based MOR

The basic idea of almost all the model order reduction (MOR) methods is to find a subspace $S_1$ which approximates the manifold where the state vector $\mathbf{𝐱}(t)$ resides. Afterwards, $\mathbf{𝐱}(t)$ is approximated by a vector $\widetilde{\mathbf{𝐱}}(t)$ 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 $S_1$.

We use the system $E\frac{d\mathbf{𝐱}}{dt}=A\mathbf{𝐱}+B\mathbf{𝐮}(t),\mathbf{𝐲}(t)=C\mathbf{𝐱}+D\mathbf{𝐮}(t).$

as an example to explain the basic idea. Assuming that an orthonormal basis $V=(v_1,v_2,\dots ,v_q)$ of the subspace $S_1$ has been found, then the approximation $\widetilde{\mathbf{𝐱}}(t)$ in $S_1$ can be represented by the basis as $\widetilde{\mathbf{𝐱}}(t)=V\mathbf{𝐳}(t)$. Therefore $\mathbf{𝐱}(t)$ can be approximated by $\mathbf{𝐱}(t)\approx V\mathbf{𝐳}(t)$. Here ${\bf z}$ is a vector of length $q \ll n$.

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