Modifications to elliptic eigenproblem example

[siuwuncheung]

This is a follow up PR for #266 to enable the followings:

1. Use eigenspace projection for error calculation
2. Add boundary condition switch
3. Clean up to retain only 4 problems, i.e. poisson, heterogeneous diffusion, Schrodinger equation with harmonic potential, Schrodinger equation with Gaussian potential well
4. Extend to 3D examples
5. Change some variable names

With the command lines,

./elliptic_eigenproblem_global_rom -offline -p 2 -rs 2 -id 0 -a 0 -n 4
./elliptic_eigenproblem_global_rom -offline -p 2 -rs 2 -id 1 -a 1 -n 4
./elliptic_eigenproblem_global_rom -merge -p 2 -rs 2 -ns 2 -n 4
./elliptic_eigenproblem_global_rom -fom -p 2 -rs 2 -a 0.5 -n 4 -visit
./elliptic_eigenproblem_global_rom -online -p 2 -rs 2 -a 0.5 -ef 1.0 -n 4 -visit

the output is

FOM solution for eigenvalue 0 = 19.878564
ROM solution for eigenvalue 0 = 19.878631
Absolute error of ROM solution for eigenvalue 0 = 6.6618756e-05
Relative error of ROM solution for eigenvalue 0 = 3.3512861e-06
FOM solution for eigenvalue 1 = 53.177886
ROM solution for eigenvalue 1 = 53.179662
Absolute error of ROM solution for eigenvalue 1 = 0.0017768914
Relative error of ROM solution for eigenvalue 1 = 3.3414104e-05
FOM solution for eigenvalue 2 = 53.177886
ROM solution for eigenvalue 2 = 53.179662
Absolute error of ROM solution for eigenvalue 2 = 0.0017769039
Relative error of ROM solution for eigenvalue 2 = 3.3414339e-05
FOM solution for eigenvalue 3 = 81.245811
ROM solution for eigenvalue 3 = 81.246105
Absolute error of ROM solution for eigenvalue 3 = 0.00029387266
Relative error of ROM solution for eigenvalue 3 = 3.6170808e-06
Relative l2 error of ROM eigenvector 0 = 0.0023815683
Relative l2 error of ROM eigenvector 1 = 0.0097810599
Relative l2 error of ROM eigenvector 2 = 0.0097816076
Relative l2 error of ROM eigenvector 3 = 0.0043291658

N.B.: It is a FOM issue rather than a ROM issue. If the Gaussian center falls on a grid point, the eigenvalue calculation for Schrodinger equation can be very unstable.

[chldkdtn]

I am getting the following dramatically different outputs in Ruby for ROMs:

Number of unknowns: 289
Opening file: elliptic_eigenproblem_basis.000000
spatial basis dimension is 289 x 14
Eigenvalue 0: = 26.893636
Eigenvalue 1: = 67.093674
Eigenvalue 2: = 69.162828
Eigenvalue 3: = 109.44197
Eigenvalue 4: = 136.84225
norm_ref[i] = 273.80793
norm_ev[i] = 270.02649
ip[i] = 1.8178431
norm_ref[i] = 259.39492
norm_ev[i] = 264.66499
ip[i] = -12.477348
norm_ref[i] = 264.55127
norm_ev[i] = 264.93123
ip[i] = -22.66543
norm_ref[i] = 264.64771
norm_ev[i] = 259.31325
ip[i] = -97.06303
norm_ref[i] = 269.74102
norm_ev[i] = 273.78523
ip[i] = -18.535971
FOM solution for eigenvalue 0 = 26.893632
ROM solution for eigenvalue 0 = 26.893636
Absolute error of ROM solution for eigenvalue 0 = 3.97966e-06
Relative error of ROM solution for eigenvalue 0 = 1.4797778e-07
FOM solution for eigenvalue 1 = 67.093653
ROM solution for eigenvalue 1 = 67.093674
Absolute error of ROM solution for eigenvalue 1 = 2.1396013e-05
Relative error of ROM solution for eigenvalue 1 = 3.1889772e-07
FOM solution for eigenvalue 2 = 69.162823
ROM solution for eigenvalue 2 = 69.162828
Absolute error of ROM solution for eigenvalue 2 = 4.853771e-06
Relative error of ROM solution for eigenvalue 2 = 7.0178902e-08
FOM solution for eigenvalue 3 = 109.44196
ROM solution for eigenvalue 3 = 109.44197
Absolute error of ROM solution for eigenvalue 3 = 4.1389821e-06
Relative error of ROM solution for eigenvalue 3 = 3.7818969e-08
FOM solution for eigenvalue 4 = 136.84157
ROM solution for eigenvalue 4 = 136.84225
Absolute error of ROM solution for eigenvalue 4 = 0.0006786573
Relative error of ROM solution for eigenvalue 4 = 4.9594382e-06
Relative l2 error of ROM eigenvector 0 = 0.51582637
Relative l2 error of ROM eigenvector 1 = 1.9924533
Relative l2 error of ROM eigenvector 2 = 1.8788293
Relative l2 error of ROM eigenvector 3 = 1.9193804
Relative l2 error of ROM eigenvector 4 = 0.18407987
Unable to connect to GLVis server at localhost:19916
GLVis visualization disabled.
Elapsed time for assembling ROM: 1.196660e-02 second
Elapsed time for solving ROM: 5.293224e-02 second

[ckendrick]

I am getting the same results as you. I think the sample output comments in the header are old and haven't been updated with new values from the changes made in this PR.

[chldkdtn]

If this is correct, then the relative l2 error for eigenvectors are pretty off.

[siuwuncheung]

I am updating some examples and parameters with outputing more information. Eventually, I want to use this example:

./elliptic_eigenproblem_global_rom -offline -p 2 -id 0 -a 0.0
./elliptic_eigenproblem_global_rom -offline -p 2 -id 1 -a 1.0
./elliptic_eigenproblem_global_rom -merge -p 2 -ns 2
./elliptic_eigenproblem_global_rom -fom -p 2 -a 0.5 -visit
./elliptic_eigenproblem_global_rom -online -p 2 -a 0.5 -visit

With the most recent commit 11d63b0, the results are

Warning: eigenvector 1 in FOM and ROM are not directly comparable.
Inner product = 0.78603098
TODO: Projection error of eigenvector.
Warning: eigenvector 2 in FOM and ROM are not directly comparable.
Inner product = 0.78603092
TODO: Projection error of eigenvector.
FOM solution for eigenvalue 0 = 19.921322
ROM solution for eigenvalue 0 = 19.92139
Absolute error of ROM solution for eigenvalue 0 = 6.7559202e-05
Relative error of ROM solution for eigenvalue 0 = 3.391301e-06
FOM solution for eigenvalue 1 = 52.415509
ROM solution for eigenvalue 1 = 52.416907
Absolute error of ROM solution for eigenvalue 1 = 0.0013989485
Relative error of ROM solution for eigenvalue 1 = 2.6689592e-05
FOM solution for eigenvalue 2 = 52.415509
ROM solution for eigenvalue 2 = 52.416908
Absolute error of ROM solution for eigenvalue 2 = 0.0013990685
Relative error of ROM solution for eigenvalue 2 = 2.6691881e-05
FOM solution for eigenvalue 3 = 81.244631
ROM solution for eigenvalue 3 = 81.245048
Absolute error of ROM solution for eigenvalue 3 = 0.00041707441
Relative error of ROM solution for eigenvalue 3 = 5.1335627e-06
Relative l2 error of ROM eigenvector 0 = 0.0010013981
Relative l2 error of ROM eigenvector 1 = 0.65416973
Relative l2 error of ROM eigenvector 2 = 0.65416982
Relative l2 error of ROM eigenvector 3 = 0.0025055664

Right now, the L2 error of the eigenvectors with non-simple eigenvalues (eigenvector 1 and 2 above) are large because of the way it is calculated, which is direct comparison of the FOM/ROM eigenvectors of the same index.

I will update the results in the header of the script and on the libROM webpage soon after the next PR.

[chldkdtn]

Looks good!