Single-stage optimization example notebook

[IssraAli]

Created a single-stage optimization example notebook.

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|             benchmark_name             |         dt(%)          |         dt(s)          |        t_new(s)        |        t_old(s)        | 
| -------------------------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
 test_build_transform_fft_midres         |     +0.80 +/- 4.85     | +4.89e-03 +/- 2.97e-02 |  6.17e-01 +/- 2.6e-02  |  6.12e-01 +/- 1.4e-02  |
 test_build_transform_fft_highres        |     -0.09 +/- 1.83     | -8.93e-04 +/- 1.85e-02 |  1.01e+00 +/- 1.4e-02  |  1.01e+00 +/- 1.2e-02  |
 test_equilibrium_init_lowres            |     +0.83 +/- 1.57     | +3.20e-02 +/- 6.04e-02 |  3.88e+00 +/- 3.7e-02  |  3.85e+00 +/- 4.8e-02  |
 test_objective_compile_atf              |     +0.10 +/- 3.06     | +8.03e-03 +/- 2.41e-01 |  7.89e+00 +/- 1.9e-01  |  7.88e+00 +/- 1.4e-01  |
 test_objective_compute_atf              |     +1.05 +/- 3.12     | +1.11e-04 +/- 3.30e-04 |  1.07e-02 +/- 2.6e-04  |  1.06e-02 +/- 2.0e-04  |
 test_objective_jac_atf                  |     -2.75 +/- 2.93     | -5.37e-02 +/- 5.72e-02 |  1.90e+00 +/- 3.4e-02  |  1.95e+00 +/- 4.6e-02  |
 test_perturb_1                          |     +0.28 +/- 1.94     | +3.56e-02 +/- 2.48e-01 |  1.28e+01 +/- 2.3e-01  |  1.28e+01 +/- 9.1e-02  |
 test_proximal_jac_atf                   |     -0.88 +/- 1.15     | -7.04e-02 +/- 9.22e-02 |  7.92e+00 +/- 5.7e-02  |  7.99e+00 +/- 7.2e-02  |
 test_proximal_freeb_compute             |     +0.77 +/- 0.74     | +1.41e-03 +/- 1.35e-03 |  1.83e-01 +/- 8.7e-04  |  1.81e-01 +/- 1.0e-03  |
 test_build_transform_fft_lowres         |     +0.91 +/- 4.63     | +4.80e-03 +/- 2.43e-02 |  5.30e-01 +/- 2.1e-02  |  5.25e-01 +/- 1.2e-02  |
 test_equilibrium_init_medres            |     +1.53 +/- 1.47     | +6.33e-02 +/- 6.09e-02 |  4.20e+00 +/- 3.7e-02  |  4.14e+00 +/- 4.9e-02  |
 test_equilibrium_init_highres           |     +0.71 +/- 1.16     | +3.91e-02 +/- 6.33e-02 |  5.51e+00 +/- 4.3e-02  |  5.47e+00 +/- 4.6e-02  |
 test_objective_compile_dshape_current   |     +0.86 +/- 5.50     | +3.29e-02 +/- 2.12e-01 |  3.88e+00 +/- 4.4e-02  |  3.84e+00 +/- 2.1e-01  |
 test_objective_compute_dshape_current   |     +0.68 +/- 3.03     | +2.46e-05 +/- 1.09e-04 |  3.63e-03 +/- 9.9e-05  |  3.60e-03 +/- 4.6e-05  |
 test_objective_jac_dshape_current       |     +0.10 +/- 8.15     | +4.04e-05 +/- 3.38e-03 |  4.15e-02 +/- 2.2e-03  |  4.14e-02 +/- 2.6e-03  |
 test_perturb_2                          |     +0.41 +/- 1.27     | +7.30e-02 +/- 2.25e-01 |  1.77e+01 +/- 2.0e-01  |  1.76e+01 +/- 9.7e-02  |
 test_proximal_freeb_jac                 |     -0.27 +/- 1.06     | -2.02e-02 +/- 7.99e-02 |  7.50e+00 +/- 6.0e-02  |  7.52e+00 +/- 5.3e-02  |
 test_solve_fixed_iter                   |     +0.24 +/- 58.01    | +1.19e-02 +/- 2.84e+00 |  4.90e+00 +/- 2.0e+00  |  4.89e+00 +/- 2.0e+00  |

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Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 95.53%. Comparing base (831e7bd) to head (98f0fe0).
Report is 61 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1236   +/-   ##
=======================================
  Coverage   95.52%   95.53%           
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  Files          96       96           
  Lines       24010    24010           
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+ Hits        22936    22937    +1     
+ Misses       1074     1073    -1     

see 2 files with indirect coverage changes

[ddudt]

Noting that this will have to wait for #1222 and possibly also #1205

[dpanici]

Noting that this will have to wait for #1222 and possibly also #1205

IDT it needs to wait for #1205 if it is vacuum, though yea we should wait for both of these if we want a finite beta case also in this

[IssraAli]

I wanted to mention, certain combinations of hyperparameters gave very bad outputs, see below for an example. I didn't really systematically document this, but I can do that if you guys are interested (sorry for the bad pic):

The optimizer also converged to designs where the coils were all together in the centre of the torus all tangled up. Perhaps it would be worth it to create a constraint function that ensures that the coils are within a certain "winding volume" padding the torus to prevent these kinds of outputs.

I generally got much more normal looking coils with the coil-related losses weighted on the order of 100x higher than the QS loss. However, there seems to be quite a lot of field error.