Squared singular values to calculate POD variance

[axla-io]

Current libROM implementation of BasisGenerator::finalSummary sums the singular values to calculate the total POD energy, but this should actually be the sum of squared singular values, compare to Chatterjee (2000) chapter 6.4. This short PR adresses that in libROM.

Test mixed nonlinear diffusion

To test with mixed nonlinear diffusion, run:

./mixed_nonlinear_diffusion -p 1 -offline -id 0 -sh 0.25
./mixed_nonlinear_diffusion -p 1 -offline -id 1 -sh 0.15
// Merge
./mixed_nonlinear_diffusion -p 1 -merge -ns 2

On the current master, the file mergedSV_FR reads

For energy fraction: 0.9, take first 5 of 200 basis vectors
For energy fraction: 0.99, take first 8 of 200 basis vectors
For energy fraction: 0.999, take first 12 of 200 basis vectors
For energy fraction: 0.9999, take first 16 of 200 basis vectors
For energy fraction: 0.99990, take first 16 of 200 basis vectors

In this PR, the file reads

For energy fraction: 0.9, take first 3 of 200 basis vectors
For energy fraction: 0.99, take first 5 of 200 basis vectors
For energy fraction: 0.999, take first 7 of 200 basis vectors
For energy fraction: 0.9999, take first 8 of 200 basis vectors
For energy fraction: 0.99990, take first 8 of 200 basis vectors

Test nonlinear elasticity

To test with nonlinear elasticity, run:

./nonlinear_elasticity_global_rom -offline -dt 0.01 -tf 5.0 -s 14 -vs 100 -sc 0.9 -id 0
./nonlinear_elasticity_global_rom -offline -dt 0.01 -tf 5.0 -s 14 -vs 100 -sc 1.1 -id 1

// Merge:
./nonlinear_elasticity_global_rom -merge -ns 2 -dt 0.01 -tf 5.0

On the current master, the file mergedSV_H.txt reads

For energy fraction: 0.9, take first 12 of 1002 basis vectors
For energy fraction: 0.99, take first 29 of 1002 basis vectors
For energy fraction: 0.999, take first 59 of 1002 basis vectors
For energy fraction: 0.9999, take first 96 of 1002 basis vectors
For energy fraction: 0.99999, take first 134 of 1002 basis vectors
For energy fraction: 0.999999, take first 165 of 1002 basis vectors
For energy fraction: 0.9999999, take first 189 of 1002 basis vectors
For energy fraction: 0.99999990, take first 189 of 1002 basis vectors

In this PR, the file reads

For energy fraction: 0.9, take first 5 of 1002 basis vectors
For energy fraction: 0.99, take first 11 of 1002 basis vectors
For energy fraction: 0.999, take first 19 of 1002 basis vectors
For energy fraction: 0.9999, take first 26 of 1002 basis vectors
For energy fraction: 0.99999, take first 35 of 1002 basis vectors
For energy fraction: 0.999999, take first 49 of 1002 basis vectors
For energy fraction: 0.9999999, take first 67 of 1002 basis vectors
For energy fraction: 0.99999990, take first 67 of 1002 basis vectors

[dylan-copeland]

Using pow(x,2) can be slower than x*x, but that is only a concern within a bottleneck. Here it shouldn't matter.