ROM : POD Implementation

[emvalbuena]

Although in theory one can use the solution snapshots as a basis to project the operators, in practice these might define a basis with a large condition number.
Having it go through the POD, even if it is with the same number of basis vectors as snapshots, will produce an orthonormal basis.

Warning: watch-out for standardisations, this could introduce problems.

• Use SVD from scipy.
  • See https://github.com/KikeM/msc-thesis/issues/51#issuecomment-866179711
• Tests

Deliverable
Build POD space of solution snapshots.

References

• https://www.youtube.com/watch?v=axfUYYNd-4Y
• https://rtd.pymor.org/en/0.5.2/_modules/pymor/algorithms/pod.html#pod
• https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html
• https://github.com/users/KikeM/projects/2#card-54680317
• https://www.janheiland.de/post/fenics-burger-pod-dmd/
• https://deustotech.github.io/DyCon-Blog/tutorial/wp04/P0009

[emvalbuena]

For the moment I am using the scipy.linalg.orth.

According to docs, "Construct an orthonormal basis for the range of A using SVD".
I need to check again the POD procedure, I am not sure if it is equivalent to the use of this function.

That being said, the POD is quite butchery-like in terms of numerics, no novelty there to add on my side.

Therefore, I am going to leave it like that for the time being, until I polish out other tools of the ROM, such as the (M)DEIM procedure.

[emvalbuena]

Therefore, I am going to leave it like that for the time being, until I polish out other tools of the ROM, such as the (M)DEIM procedure.

Now that the DEIM and MDEIM implementations are robust, I should get back to this, because I am having some trouble with the reduction of the stiffness matrix in the ALE formulation, and I would like to know if the selection and the construction of the POD modes has anything to do with it.