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Old   December 5, 2005, 01:56
Default "PROPER ORTHOGONAL DECOMPOSITION"
  #1
dpshaka
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DOSE ANY BODY KNOW THE DETAIL OF THE "PROPER ORTHOGONAL DECOMPOSITION", I HAVE REVIEWED A LOT OF PAPERS ,BUT I STILL CAN NOT UNDERSTAND THE DETAIL OF THE NUMERICAL process of "POD".

IF SOME ONE IS FAMILIAR ABOUT "POD" , PLEASE HELP ME / REGARDS
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Old   December 5, 2005, 21:49
Default Re: "PROPER ORTHOGONAL DECOMPOSITION"
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Peter Attar
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Essentially you collect snapshots of the solution vector at various points in time for represented "forcings". You collect these into a matrix Q and then perform a singular value decomposition on it. You then say that the POD modes are a linear combination of the snapshot vectors(snapshot method of POD) so that the modes can be represented as [R]=[Q][V] where Q is the snapshot matrix and V are the right singular vectors. For a more detailed explanation on this look for papers by Thomas,Dowell et al and Beran et al.

Peter
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Old   December 8, 2005, 14:28
Default Re: "PROPER ORTHOGONAL DECOMPOSITION"
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richard
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You may want to check the following link:

http://dissertations.ub.rug.nl/FILES...azemier/c2.pdf

It doesn't seem to be too badly explained by this guy. For further literature, look for the method of Sirovich, which is an approximation of the continuous (aka. classic) POD.

Best regards, Richard
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Old   December 11, 2005, 05:01
Default Re: "PROPER ORTHOGONAL DECOMPOSITION"
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dpshaka
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thank you for your feedback, but would you please give me some detail information about the numerical procedure of the POD .

regards.

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Old   December 13, 2005, 19:35
Default Re: "PROPER ORTHOGONAL DECOMPOSITION"
  #5
richard
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Hopefully I got you right and you are looking for an implementation? Below you find a kind of step-by-step example (I am sorry for the looks. "_" and "^" are used for indecies and (f,g) is a scalar product):

N vector fileds u_k=u_k(t_k) (provided by e.g. PIV measurements) at different times t_k, k = 1...N.

u_k is representet by e.g. 2 matrices: u-component and v-component of velocity u_k, both of the size m x n.

Snapshot method by Sirovich: 1) create a vector (one dim. array) for each vector field e.g. U=[u_11,v_11,...u_mn,v_mn], where m and n are the dimensions of the vector field. The vector U has the size of 1 x 2nm.

2) Calculate the 2-point cross-correlation matrix Q: Q_ij = 1/N (U_i,U_j).

3) Solve the eigen value problem "Q q_i = lambda Q" to find the eigen vectors q_i. There are N eigen vectors q_i (i = 1...N) with the size of 1 x 2mn.

4) Calculate the orthonormal base, phi_i, (aka. eigen modes) of the POD: phi^i = SUM[n = 1...N](q_n^i U_n).

5) Reconstitute a vector field in the new orthonormal base: U_new = SUM[i = 1...N]((U_old,phi^i)phi^i).

I hope this will help you.

Regards
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Old   December 13, 2005, 22:31
Default Re: "PROPER ORTHOGONAL DECOMPOSITION"
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dpshaka
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Dear richard:

Thank you very much for your feedback.It gives me so much help.

I have seached a lot of paper, but the method that they used is different from what you have said. they all solve a eigenvalue problem in a integral equation form.

so would you mind give me your email address, i still have some problem about this POD technique.

ztdep@163.com best wishes

your DP

PRC XJTU

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Old   December 17, 2005, 02:38
Default Re: "PROPER ORTHOGONAL DECOMPOSITION"
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ztdep
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Dear richard:

i have send a email to you ,have you received it ?

regards?

yours
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