Projection based MOR: Difference between revisions

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Revision as of 15:08, 12 March 2013

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 $𝐱 (t)$ 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 $S_{1}$ .

We use the system Failed to parse (syntax error): {\displaystyle E \frac{d{\bf x}}{dt}=A {\bf x}+B {\bf u}(t), \\ {\bf y}(t)=C{\bf x}+D{\bf 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 $\tilde{𝐱} (t)$ in $S_{1}$ can be represented by the basis as $\tilde{𝐱} (t) = V 𝐳 (t)$ . Therefore $𝐱 (t)$ can be approximated by $𝐱 (t) \approx V 𝐳 (t)$ . Here ${\bf z}$ is a vector of length $q \ll n$.

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

@@ Line 1: / Line 1: @@
 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
+find a subspace <math>S_1</math> which approximates the manifold where the state
-vector ${\bf x}(t)$ resides. Afterwards, ${\bf x}(t)$ is approximated by a vector $\tilde{\bf x}(t)$ 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 $S_1$.
+vector <math>{\bf x}(t)</math> resides. Afterwards, <math>{\bf x}(t)</math> is approximated by a vector <math>\tilde{\bf x}(t)</math> 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 <math>S_1</math>.
 We use the system
-<math>\begin{equation}
+<math>
-\label{sys1}
+E \frac{d{\bf x}}{dt}=A {\bf x}+B {\bf u}(t), \\
-\begin{array}{rcl}
+{\bf y}(t)=C{\bf x}+D{\bf u}(t).
-E \frac{d{\bf x}}{dt}&=&A {\bf x}+B {\bf u}(t), \\
+</math>
-{\bf y}(t)&=&C{\bf x}+D{\bf u}(t).
-\end{array}
-\end{equation}</math>
 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
+basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been
-found, then the approximation $\tilde{\bf x}(t)$ in $S_1$ can be represented by
+found, then the approximation <math>\tilde{\bf x}(t)</math> in <math>S_1</math> can be represented by
-the basis as $\tilde{\bf x}(t)=V{\bf z}(t)$. Therefore ${\bf x}(t)$ can be approximated by ${\bf x}(t) \approx V{\bf z}(t)$. Here ${\bf z}$ is a vector
+the basis as <math>\tilde{\bf x}(t)=V{\bf z}(t)</math>. Therefore <math>{\bf x}(t)</math> can be approximated by <math>{\bf x}(t) \approx V{\bf z}(t)</math>. Here ${\bf z}$ is a vector
 of length $q \ll n$.
-Once ${\bf z}(t)$ is computed, we can get an
+Once <math>{\bf z}(t)</math> is computed, we can get an
-approximate solution $\tilde{\bf x}(t)=V{\bf z}(t)$ for ${\bf x}(t)$. The vector ${\bf z}(t)$
+approximate solution <math>\tilde{\bf x}(t)=V{\bf z}(t)</math> for <math>{\bf x}(t)</math>. The vector <math>{\bf z}(t)</math>
 can be computed from the reduced model which is  derived by the
 following two steps.