POD

Each method returns a tuple containing the pod basis of type PODBasis{T} and the corresponding singular values. The singular values are related to each modes importance to the dataset.

Method of snapshots

The eigen-decomposition based method of snapshots is the most commonly used method for fluid flow analysis where the number of datapoints is larger than the number of snapshots.

ProperOrthogonalDecomposition.PODeigen — Method

PODeigen(X)

Uses the eigenvalue method of snapshots to calculate the POD basis of X. Method of snapshots is efficient when number of data points n > number of snapshots m.

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ProperOrthogonalDecomposition.PODeigen! — Method

PODeigen!(X)

Same as PODeigen(X) but overwrites memory.

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Singular Value Decomposition based method

The SVD based approach is also available and is more robust against roundoff errors.

ProperOrthogonalDecomposition.PODsvd — Method

PODsvd(X)

Uses the SVD based decomposition technique to calculate the POD basis of X.

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ProperOrthogonalDecomposition.PODsvd! — Method

PODsvd!(X)

Same as PODsvd(X) but overwrites memory.

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Example

Here we will artifically create data which is PODed and then extract the first mode.

t, x = range(0, stop=30, length=50), range(-10, stop=30, length=120)

Xgrid = [i for i in x, j in t]
tgrid = [j for i in x, j in t]

f1 = sech.(Xgrid.-3.5) .* 10.0 .* cos.(0.5 .*tgrid)
f2 = cos.(Xgrid) .* 1.0 .* cos.(2.5 .*tgrid)
f3 = sech.(Xgrid.+5.0) .* 4.0 .* cos.(1.0 .*tgrid)

Y = f1+f2+f3

Our data Y looks like this

Now we POD the data and reconstruct the dataset using only the first mode.

res, singularvals  = POD(Y)
reconstructFirstMode = res.modes[:,1:1]*res.coefficients[1:1,:]

Note that the above used POD(Y) which defaults to the SVD based apparoch. The first mode over the time series looks like this