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 Failed to parse (syntax error): {\bf x}(t) resides. Afterwards, Failed to parse (syntax error): {\bf x}(t) is approximated by a vector Failed to parse (syntax error): \tilde{\bf 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 $(E_0+s_1 E+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad y=Cx, \quad \quad \quad \quad (1)$

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

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

Once Failed to parse (syntax error): {\bf z}(t) is computed, we can get an approximate solution Failed to parse (syntax error): \tilde{\bf x}(t)=V{\bf z}(t) for Failed to parse (syntax error): {\bf x}(t) . The vector Failed to parse (syntax error): {\bf z}(t) can be computed from the reduced model which is derived by the following two steps.