Ku/fourier bounce part1

[unalmis]

• Moves some stuff added in ripple and ku/fourier_bounce branches here to prevent those branches from diverging.
• Resolves test_compute_everything fails #1260

[github-actions]

|             benchmark_name             |         dt(%)          |         dt(s)          |        t_new(s)        |        t_old(s)        | 
| -------------------------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
 test_build_transform_fft_lowres         |     -6.77 +/- 4.26     | -3.83e-02 +/- 2.41e-02 |  5.28e-01 +/- 1.7e-02  |  5.67e-01 +/- 1.7e-02  |
 test_equilibrium_init_medres            |     -3.38 +/- 7.62     | -1.48e-01 +/- 3.35e-01 |  4.25e+00 +/- 2.2e-01  |  4.39e+00 +/- 2.5e-01  |
 test_equilibrium_init_highres           |     +1.51 +/- 10.59    | +8.58e-02 +/- 6.03e-01 |  5.78e+00 +/- 4.7e-01  |  5.69e+00 +/- 3.7e-01  |
 test_objective_compile_dshape_current   |     +0.07 +/- 4.24     | +2.87e-03 +/- 1.73e-01 |  4.09e+00 +/- 1.4e-01  |  4.08e+00 +/- 9.7e-02  |
-test_objective_compute_dshape_current   |    +17.77 +/- 4.30     | +6.50e-04 +/- 1.57e-04 |  4.31e-03 +/- 1.4e-04  |  3.66e-03 +/- 7.9e-05  |
 test_objective_jac_dshape_current       |     +0.97 +/- 4.22     | +4.17e-04 +/- 1.81e-03 |  4.34e-02 +/- 1.3e-03  |  4.30e-02 +/- 1.3e-03  |
 test_perturb_2                          |     +1.42 +/- 3.51     | +2.65e-01 +/- 6.55e-01 |  1.89e+01 +/- 4.5e-01  |  1.87e+01 +/- 4.8e-01  |
 test_proximal_freeb_jac                 |     +2.13 +/- 3.03     | +1.63e-01 +/- 2.31e-01 |  7.79e+00 +/- 2.1e-01  |  7.63e+00 +/- 1.0e-01  |
 test_solve_fixed_iter                   |     +0.36 +/- 58.39    | +1.83e-02 +/- 3.00e+00 |  5.15e+00 +/- 2.1e+00  |  5.13e+00 +/- 2.1e+00  |
 test_build_transform_fft_midres         |     +5.60 +/- 6.29     | +3.55e-02 +/- 3.99e-02 |  6.69e-01 +/- 1.3e-02  |  6.33e-01 +/- 3.8e-02  |
 test_build_transform_fft_highres        |     +4.15 +/- 2.62     | +4.26e-02 +/- 2.69e-02 |  1.07e+00 +/- 1.0e-02  |  1.03e+00 +/- 2.5e-02  |
 test_equilibrium_init_lowres            |     +8.96 +/- 7.37     | +3.58e-01 +/- 2.94e-01 |  4.35e+00 +/- 7.2e-02  |  3.99e+00 +/- 2.9e-01  |
 test_objective_compile_atf              |     +5.33 +/- 3.55     | +4.26e-01 +/- 2.83e-01 |  8.41e+00 +/- 1.8e-01  |  7.98e+00 +/- 2.2e-01  |
 test_objective_compute_atf              |     +1.27 +/- 3.77     | +1.40e-04 +/- 4.17e-04 |  1.12e-02 +/- 2.5e-04  |  1.11e-02 +/- 3.3e-04  |
 test_objective_jac_atf                  |     +2.84 +/- 1.74     | +5.64e-02 +/- 3.45e-02 |  2.04e+00 +/- 2.7e-02  |  1.99e+00 +/- 2.1e-02  |
 test_perturb_1                          |     +5.83 +/- 2.78     | +7.51e-01 +/- 3.58e-01 |  1.36e+01 +/- 3.0e-01  |  1.29e+01 +/- 1.9e-01  |
 test_proximal_jac_atf                   |     +0.31 +/- 1.11     | +2.54e-02 +/- 9.14e-02 |  8.25e+00 +/- 7.5e-02  |  8.22e+00 +/- 5.3e-02  |
 test_proximal_freeb_compute             |     -0.24 +/- 1.57     | -4.48e-04 +/- 2.92e-03 |  1.85e-01 +/- 1.9e-03  |  1.85e-01 +/- 2.2e-03  |

[codecov]

Codecov Report

Attention: Patch coverage is 95.16129% with 3 lines in your changes missing coverage. Please review.

Project coverage is 95.43%. Comparing base (89e7fe3) to head (41b6ec8).
Report is 15 commits behind head on master.

Files with missing lines	Patch %	Lines
desc/integrals/bounce_utils.py	91.42%	3 Missing ⚠️

Additional details and impacted files

@@             Coverage Diff             @@
##           master    #1259       +/-   ##
===========================================
+ Coverage   59.59%   95.43%   +35.84%     
===========================================
  Files          95       95               
  Lines       23425    23440       +15     
===========================================
+ Hits        13961    22371     +8410     
+ Misses       9464     1069     -8395     

Files with missing lines	Coverage Δ
desc/backend.py	90.18% <100.00%> (+1.29%)	⬆️
desc/equilibrium/coords.py	88.38% <100.00%> (+25.98%)	⬆️
desc/integrals/bounce_integral.py	100.00% <100.00%> (+32.69%)	⬆️
desc/integrals/interp_utils.py	100.00% <100.00%> (ø)
desc/integrals/quad_utils.py	100.00% <100.00%> (+8.10%)	⬆️
desc/integrals/bounce_utils.py	93.19% <91.42%> (+12.18%)	⬆️

... and 72 files with indirect coverage changes

[dpanici]

Clever of you to put the fix to #1260 here so I would have to review the rest of the PR as well

[f0uriest]

Not sure I totally understand this. The points are linearly spaced, and the weights are uniform, so isn't this just 1st order reimann sum?

[unalmis]

yes assuming the riemann sum starts and ends at the correct endpoints, it is the same as gaussian quadrature in a transformed variable y

[unalmis]

There's a longer comment on the ripple branch, but put another way simply, all quadrature tricks can be stated in the most general sense to make your function as smooth as possible before integrating. This ensures that a sample of the integrand at one point is the best approximation to stuff around it. Hence equivalently you can increase observed smoothness by putting more nodes in non-smooth areas of the function.

Secondary to the smoothness objective is the objective to minimize interpolation errors that result from weaknesses in the choice of basis functions. A Gaussian quadrature in x = 1/pitch sacrifices the first goal, for the secondary goal: minimizing runge. If the quantity to integrate is sufficiently smooth to begin with, then this choice is fine. Neither Gamma_c nor epsilon effective were sufficiently smooth for this. For the former it's easy to see from the expression, for the latter I can only claim that's what i see empirically

[unalmis]

feel free to merge whenever; codecov might block me