Squared singular values in calculation of variance

[axla-io]

Hi @dreamer2368 , I'm back!

Similar to llnl/libROM#306 this PR is a fix so that we use the squared singular values for the energy fraction calculation.

Example:

cd examples/poisson
../../bin/main -i poisson.full.yml -f main/mode=sample_generation

nb=1 # Number of bases in ROM

# Previous behavior
../../bin/main -i poisson.full.yml -f main/mode=train_rom:basis/svd/square_sv=false:basis/tags/0/number_of_basis=$nb:basis/tags/1/number_of_basis=$nb:basis/tags/2/number_of_basis=$nb:basis/tags/3/number_of_basis=$nb

# Current behavior
../../bin/main -i poisson.full.yml -f main/mode=train_rom:basis/tags/0/number_of_basis=$nb:basis/tags/1/number_of_basis=$nb:basis/tags/2/number_of_basis=$nb:basis/tags/3/number_of_basis=$nb

On my local device, the first training prints:

Singular values: 2.103E-01      1.928E-01       3.835E-02       8.536E-03       5.104E-04
Energy fraction with 1 basis: 46.6832645%
Creating file: poisson0_square-circle.000000
Singular values: 2.858E-01      1.391E-01       2.535E-02       2.736E-03       1.710E-04
Energy fraction with 1 basis: 63.0686903%
Creating file: poisson0_square-square.000000
Singular values: 3.545E-01      3.122E-01       6.106E-02       1.244E-02       6.307E-04
Energy fraction with 1 basis: 47.8501513%
Creating file: poisson0_square-star.000000
Singular values: 3.396E-01      3.038E-01       8.137E-02       2.957E-02       1.826E-03
Energy fraction with 1 basis: 44.9135469%

And the second one prints:

Singular values: 2.103E-01      1.928E-01       3.835E-02       8.536E-03       5.104E-04
Energy fraction with 1 basis: 53.3251752%
Creating file: poisson0_square-circle.000000
Singular values: 2.858E-01      1.391E-01       2.535E-02       2.736E-03       1.710E-04
Energy fraction with 1 basis: 80.3314430%
Creating file: poisson0_square-square.000000
Singular values: 3.545E-01      3.122E-01       6.106E-02       1.244E-02       6.307E-04
Energy fraction with 1 basis: 55.3531120%
Creating file: poisson0_square-star.000000
Singular values: 3.396E-01      3.038E-01       8.137E-02       2.957E-02       1.826E-03
Energy fraction with 1 basis: 53.6178019%

Let me know if you have any comments!

Best,
Axel