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The basic idea of almost all the model order reduction (MOR) methods is to |
The basic idea of almost all the model order reduction (MOR) methods is to |
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− | find a subspace |
+ | find a subspace <math>S_1</math> which approximates the manifold where the state |
− | vector |
+ | 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 |
We use the system |
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− | <math> |
+ | <math> |
⚫ | |||
− | \label{sys1} |
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⚫ | |||
− | \begin{array}{rcl} |
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+ | </math> |
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⚫ | |||
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− | \end{array} |
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− | \end{equation}</math> |
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as an example to explain the basic idea. Assuming that an orthonormal |
as an example to explain the basic idea. Assuming that an orthonormal |
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− | basis |
+ | basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been |
− | found, then the approximation |
+ | found, then the approximation <math>\tilde{\bf x}(t)</math> in <math>S_1</math> can be represented by |
− | the basis as |
+ | 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 $q \ll n$. |
of length $q \ll n$. |
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− | Once |
+ | Once <math>{\bf z}(t)</math> is computed, we can get an |
− | approximate solution |
+ | 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 |
can be computed from the reduced model which is derived by the |
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following two steps. |
following two steps. |
Revision as of 17:08, 12 March 2013
The basic idea of almost all the model order reduction (MOR) methods is to
find a subspace 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
.
We use the system Failed to parse (syntax error): 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 of the subspace
has been
found, then the approximation Failed to parse (syntax error): \tilde{\bf x}(t)
in
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.