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Difference between revisions of "Projection based MOR"

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[[Category:method]]
 
[[Category:method]]
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[[Category:time invariant]]
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Consider the linear time invariant system
   
 
:<math>
All the existing model order reduction (MOR) methods are based on projection. That is to
 
 
E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
find a subspace <math>S_1</math> which approximates the manifold where the state
 
  +
y(t)=Cx(t), \quad \quad (1)
vector <math>x(t)</math> resides. Afterwards, <math>x(t)</math> is approximated by a vector <math>\tilde x(t)</math> in <math>S_1</math>. The reduced model is produced by Petrov-Galerkin projection onto a subspace <math>S_2</math>, or by Galerkin projection onto the same subspace <math>S_1</math>.
 
 
We use the system
 
 
<math>
 
E \frac{dx(t)}{dt}=A x(t)+B u(t),</math>
 
<math>
 
y(t)=Cx(t),
 
 
</math>
 
</math>
   
as an example to explain the basic idea. Assuming that an orthonormal
+
as an example.
  +
All the existing model order reduction (MOR) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>.
basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been
 
found, then the approximation <math>\tilde x(t)</math> in <math>S_1</math> can be represented by
+
That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides.
the basis as <math>\tilde x(t)=V z(t)</math>. Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>. Here <math>z</math> is a vector
+
Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>.
  +
The reduced model is produced by [[wikipedia:Petrov–Galerkin_method|Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[wikipedia:Galerkin_method|Galerkin projection]] using <math>S_1</math> as the test subspace.
of length $q \ll n$.
 
  +
Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\hat x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\hat{x} (t)=V z(t)</math>.
 
Once <math>z(t)</math> is computed, we can get an
+
Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.
approximate solution <math>\tilde x(t)=V z(t)</math> for <math>x(t)</math>. The vector <math>z(t)</math>
+
Here <math>z</math> is a vector of length <math>q \ll n</math>.
  +
Once <math>z(t)</math> is computed, an approximate solution <math>\hat x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.
can be computed from the reduced model which is derived by the
+
The vector <math>z(t)</math> can be computed from the reduced model, derived by the following two steps.
following two steps.
 
   
 
Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get
 
Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get
   
   
<math>E \frac{d{Vz}}{dt} \approx A Vz+Bu(t),</math>
+
:<math>E \frac{d{Vz}}{dt}=A Vz+Bu(t)+e,\quad \hat y(t)=CV z.</math>
<math>y(t) \approx CV z.</math>
 
   
   
Step 2. The residual is denoted as <math>e=AVz+Bu(t)-E \frac{d{Vz}}{dt}</math>. Force <math>e=0</math> in a properly chosen subspace <math>S_2</math> of <math>\mathbb {R}^n</math> leads to the Petrov-Galerkin projection: <math>W^T e=0</math>, where the columns of <math>W</math> are the basis of <math>S_2</math>. The the reduced model is
+
Step 2. The residual is denoted as <math>e=AVz+Bu(t)-E \frac{d{Vz}}{dt}</math>. Forcing <math>e=0</math> in a properly chosen subspace <math>S_2</math> of <math>\mathbb {R}^n</math> leads to the Petrov-Galerkin projection: <math>W^T e=0</math>, where the columns of <math>W</math> are the basis of <math>S_2</math>.
  +
Then we have,
   
<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t),</math>
+
:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>
<math>\hat{y}(t)=CVz.</math>
 
   
 
By defining <math>\hat{E}=W^TEV</math>, <math>\hat {A}=W^TAV, \hat{B}=W^TB</math>, <math>\hat{C}=CV</math>, we get the final reduced model
 
By defining <math>\hat{E}=W^TEV</math>, <math>\hat {A}=W^TAV, \hat{B}=W^TB</math>, <math>\hat{C}=CV</math>, we get the final reduced model
   
<math>\hat{E} \frac{dz(t)}{dt}&=&\hat{A}z(t)+\hat{B}u(t),</math>
+
:<math>\hat{E} \frac{dz(t)}{dt}=\hat{A}z(t)+\hat{B}u(t), \quad
<math>\hat{y}(t)& = &\hat{C}z(t).
+
\hat{y}(t)=\hat{C}z(t). \quad \quad (2)
 
</math>
 
</math>
   
  +
Notice that the approximation <math>\hat x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2).
Notice that the approximation $\tilde {\bf x}(t)=V{\bf z}(t)$ of ${\bf x}(t)$ can be obtained from ${\bf z}(t)$ by solving the system in~(\ref{sys3}). The system in~(\ref{sys3}) is much smaller than the system in~(\ref{sys1}) in the sense that there are many less equations in~(\ref{sys3}) than in~(\ref{sys1}). Therefore, the system in~(\ref{sys3}) is much easier to be solved, which is the so-called reduced model. In order to save the simulation time of solving~(\ref{sys1}), the system in~(\ref {sys3}) can be used to replace the original large system in~(\ref{sys1}) 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 output responses or the transfer functions of the two systems.
 
  +
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 <math>x(t), \hat x(t)</math>, or between the output responses <math>y(t), \hat y(t)</math>, or between the transfer functions of the two systems.
  +
It can be seen that once the two matrices <math>W</math> and <math>V</math> have been computed, the reduced model is obtained.
  +
While the Gramian based MOR methods (e.g. [[Balanced Truncation]]) usually compute <math>W</math> different from <math>V</math>, some methods use <math>W=V</math>, e.g. some of the [[Moment-matching method]]s, the [[Reduced Basis PMOR method]]s, and some of the [[POD method]]s etc.. When <math>W=V</math>, Petrov-Galerkin projection becomes Galerkin projection.
  +
MOR methods differ in the computation of the two matrices <math>W</math> and <math>V</math>.
  +
The Gramian based MOR methods compute <math>W</math> and <math>V</math> by the controllability and observability Gramians.
  +
Reduced basis methods and POD methods compute <math>V</math> from the snapshots of the state vector <math>x(t)</math> at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute :<math>W</math> and <math>V</math> from the moments of the transfer function.
  +
In eigenvalue based MOR methods, e.g. [[Modal truncation]], the columns of :<math>W</math> and <math>V</math> are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair <math>(A,E)</math>.
  +
  +
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.
  +
  +
==References==
  +
  +
<references/>

Latest revision as of 14:24, 11 May 2023

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[1]. 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 its projection \hat x(t) in S_1. The reduced model is produced by Petrov-Galerkin projection using a test subspace S_2, or by Galerkin projection using S_1 as the test subspace. Assuming that an orthonormal basis V=(v_1,v_2, \ldots, v_q) of the subspace S_1 has been found, then the approximation \hat 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 \hat x(t)=V z(t) for x(t) can be obtained. The vector z(t) can be computed from the reduced model, derived by the following two steps.

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


E \frac{d{Vz}}{dt}=A Vz+Bu(t)+e,\quad \hat y(t)=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 PMOR 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. In eigenvalue based MOR methods, e.g. Modal truncation, the columns of :W and V are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair (A,E).

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.

References

  1. Antoulas, A. C. "Approximation of large-scale dynamical systems". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3