Restructure POD interface

[FHoltorf]

Hey,

I revisited some of the POD interface and realized there were a few problems:

First of all, the type specs in the POD type are problematic: For example, specifiying integer-valued data will error as the rel. energy and the reduced basis is then constrained to be integer-valued as well (obviously not generally true):

mutable struct POD{FT} <: AbstractDRProblem
    snapshots::Union{Vector{Vector{FT}}, Matrix{FT}}
    nmodes::Int
    rbasis::Matrix{FT}
    renergy::FT

Further, the interface did not accommodate a typical workflow for POD. Specifically, one often would like to reduce the dimensionality to capture a specified minimum relative energy (say 95%). Right now we require the dimension to be specified directly. So a user would effectively need to take the SVD of their data, look at the spectrum and decide on the cutoff, just to then take the svd all over again within ModelOrderReduction. I adjusted the POD type to also carry fields min_renergy, min_nmodes, max_nmodes, referring to the min relative energy captured as well as the minimum and maximum number of modes retained, respectively. I updated the reduction step accordingly to choose the reduction to be inline with these specifications. Moreover, I added a field spectrum that carries the singular spectrum (which is often of interest for a posterior analysis).

Lastly, I changed the reduce statement to dispatch directly on the type of the data to be reduced. Consequently, POD can be applied via this interface to any data set which admits computation of a singular value decomposition via the specified routine. That way, everything composes better with the greater ecosystem.

Let me know!
Flemming

[ChrisRackauckas]

Redundant?

[FHoltorf]

absolutely

[ChrisRackauckas]

This makes it piracy because it's then adding a function on Base types. It would be safer to just shadow or define a _tsvd etc. inside of the package for the Vector{Vector} form.

[FHoltorf]

good point. Didn't think about that whatsoever. Should be fixed now.

[codecov]

Codecov Report

Merging #25 (16198a8) into main (4910f4a) will increase coverage by 9.60%.
The diff coverage is 87.30%.

@@            Coverage Diff             @@
##             main      #25      +/-   ##
==========================================
+ Coverage   72.34%   81.94%   +9.60%     
==========================================
  Files           3        3              
  Lines          47       72      +25     
==========================================
+ Hits           34       59      +25     
  Misses         13       13              

Impacted Files	Coverage Δ
src/DataReduction/POD.jl	75.92% <82.60%> (+8.42%)	⬆️
src/ErrorHandle.jl	100.00% <100.00%> (ø)
src/Types.jl	100.00% <100.00%> (ø)

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[ChrisRackauckas]

Other than the shadowing issue, I think this is good.