non-singular BIE solver, improved FFT interpolation for singular integral, and new magnetic field API

[unalmis]

(see #1360)

• All the solvers are thoroughly tested. See the shared paper to explain this PR. Significantly improved adjoint solve from earlier. It seems like this is still the only differentiable 3d laplace solver on the internet that works on multiply connected geometry.
• Tested and confirmed agreement with BIE3D package. plotter.py
• This PR includes my FFT improvements significantly improve free surface optimization (both old and new) and anything that involves singular integrals. Even the flaky GitHub ci benchmarks detect this.
• Non-invasive, backward compatible, new magnetic field api implemented. Without the new api, the code would be unreadable.
• There are other applications besides free surface.
• The linear solve to compute the objective function only takes 2 gmres iterations. Gradient based optimization is very fast. Using a fast multipole technique would greatly improve performance. There are jax libraries that do this, but that can be a later pull request to merge after this.

iterative plots

stop tolerance denotes typical stop tol in most applications. gmres is expected to always invert the linear system in 2 iterations. after stateful opt, my guess is it will be reduced to 1 iteration.

[github-actions]

Memory benchmark result

|               Test Name                |      %Δ      |    Master (MB)     |      PR (MB)       |    Δ (MB)    |    Time PR (s)     |  Time Master (s)   |
| -------------------------------------- | ------------ | ------------------ | ------------------ | ------------ | ------------------ | ------------------ |
  test_objective_jac_w7x                 |    1.86 %    |     3.976e+03      |     4.050e+03      |    73.79     |       31.75        |       29.03        |
  test_proximal_jac_w7x_with_eq_update   |   -0.28 %    |     6.578e+03      |     6.560e+03      |    -18.41    |       153.30       |       150.73       |
+ test_proximal_freeb_jac                |   -16.32 %   |     1.330e+04      |     1.113e+04      |   -2170.14   |       69.68        |       77.50        |
+ test_proximal_freeb_jac_blocked        |   -15.74 %   |     7.625e+03      |     6.425e+03      |   -1199.93   |       65.16        |       67.66        |
+ test_proximal_freeb_jac_batched        |   -16.27 %   |     7.670e+03      |     6.422e+03      |   -1247.70   |       66.47        |       67.47        |
  test_proximal_jac_ripple               |   -0.91 %    |     3.507e+03      |     3.475e+03      |    -31.98    |       53.55        |       54.87        |
  test_proximal_jac_ripple_bounce1d      |   -2.40 %    |     3.775e+03      |     3.685e+03      |    -90.53    |       68.92        |       69.61        |
  test_eq_solve                          |   -3.33 %    |     2.074e+03      |     2.005e+03      |    -69.16    |       86.31        |       86.42        |

For the memory plots, go to the summary of Memory Benchmarks workflow and download the artifact.

[github-actions]

|             benchmark_name             |         dt(%)          |         dt(s)          |        t_new(s)        |        t_old(s)        | 
| -------------------------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
 test_build_transform_fft_lowres         |     +4.56 +/- 9.57     | +3.91e-02 +/- 8.20e-02 |  8.96e-01 +/- 8.0e-02  |  8.57e-01 +/- 1.7e-02  |
 test_equilibrium_init_medres            |     -2.76 +/- 7.52     | -2.06e-01 +/- 5.63e-01 |  7.28e+00 +/- 5.3e-01  |  7.48e+00 +/- 2.0e-01  |
 test_equilibrium_init_highres           |     -2.28 +/- 6.58     | -1.85e-01 +/- 5.34e-01 |  7.93e+00 +/- 4.8e-01  |  8.11e+00 +/- 2.2e-01  |
 test_objective_compile_dshape_current   |     -1.59 +/- 2.23     | -6.64e-02 +/- 9.28e-02 |  4.10e+00 +/- 7.5e-02  |  4.17e+00 +/- 5.5e-02  |
 test_objective_compute_dshape_current   |     -2.49 +/- 7.26     | -1.57e-05 +/- 4.56e-05 |  6.13e-04 +/- 3.4e-05  |  6.29e-04 +/- 3.0e-05  |
 test_objective_jac_dshape_current       |     -5.96 +/- 28.21    | -1.57e-03 +/- 7.44e-03 |  2.48e-02 +/- 5.7e-03  |  2.64e-02 +/- 4.8e-03  |
 test_perturb_2                          |     +4.41 +/- 3.09     | +8.52e-01 +/- 5.96e-01 |  2.02e+01 +/- 3.3e-01  |  1.93e+01 +/- 5.0e-01  |
 test_proximal_jac_atf_with_eq_update    |     +2.99 +/- 2.09     | +3.76e-01 +/- 2.63e-01 |  1.30e+01 +/- 2.2e-01  |  1.26e+01 +/- 1.4e-01  |
+test_proximal_freeb_jac                 |    -14.62 +/- 2.31     | -6.93e-01 +/- 1.09e-01 |  4.04e+00 +/- 5.7e-02  |  4.74e+00 +/- 9.3e-02  |
 test_solve_fixed_iter_compiled          |     +3.91 +/- 3.12     | +3.39e-01 +/- 2.71e-01 |  9.01e+00 +/- 1.8e-01  |  8.67e+00 +/- 2.0e-01  |
 test_LinearConstraintProjection_build   |    +10.67 +/- 7.47     | +6.98e-01 +/- 4.89e-01 |  7.24e+00 +/- 4.7e-01  |  6.55e+00 +/- 1.4e-01  |
 test_objective_compute_ripple_bounce1d  |     +1.71 +/- 4.94     | +4.78e-03 +/- 1.38e-02 |  2.84e-01 +/- 7.1e-03  |  2.79e-01 +/- 1.2e-02  |
 test_objective_grad_ripple_bounce1d     |     +0.00 +/- 1.58     | +2.08e-05 +/- 1.54e-02 |  9.79e-01 +/- 9.5e-03  |  9.79e-01 +/- 1.2e-02  |
 test_build_transform_fft_midres         |     -0.14 +/- 3.36     | -1.25e-03 +/- 2.97e-02 |  8.84e-01 +/- 2.2e-02  |  8.85e-01 +/- 1.9e-02  |
 test_build_transform_fft_highres        |     -0.39 +/- 1.69     | -4.64e-03 +/- 1.99e-02 |  1.17e+00 +/- 1.3e-02  |  1.18e+00 +/- 1.5e-02  |
 test_equilibrium_init_lowres            |     -0.70 +/- 2.58     | -4.45e-02 +/- 1.65e-01 |  6.35e+00 +/- 1.2e-01  |  6.40e+00 +/- 1.1e-01  |
 test_objective_compile_atf              |     -0.03 +/- 2.81     | -1.65e-03 +/- 1.74e-01 |  6.22e+00 +/- 6.0e-02  |  6.22e+00 +/- 1.6e-01  |
 test_objective_compute_atf              |     -1.02 +/- 18.79    | -2.09e-05 +/- 3.85e-04 |  2.03e-03 +/- 3.0e-04  |  2.05e-03 +/- 2.4e-04  |
 test_objective_jac_atf                  |     +0.81 +/- 3.01     | +1.28e-02 +/- 4.78e-02 |  1.60e+00 +/- 3.3e-02  |  1.59e+00 +/- 3.4e-02  |
 test_perturb_1                          |     -0.12 +/- 2.25     | -1.88e-02 +/- 3.57e-01 |  1.58e+01 +/- 1.0e-01  |  1.59e+01 +/- 3.4e-01  |
 test_proximal_jac_atf                   |     +0.39 +/- 1.69     | +2.05e-02 +/- 8.81e-02 |  5.24e+00 +/- 5.5e-02  |  5.22e+00 +/- 6.9e-02  |
 test_proximal_freeb_compute             |     -6.48 +/- 3.12     | -1.08e-02 +/- 5.18e-03 |  1.55e-01 +/- 2.8e-03  |  1.66e-01 +/- 4.4e-03  |
 test_solve_fixed_iter                   |     -0.60 +/- 1.86     | -1.55e-01 +/- 4.80e-01 |  2.57e+01 +/- 3.7e-01  |  2.59e+01 +/- 3.0e-01  |
 test_objective_compute_ripple           |     -0.77 +/- 3.90     | -1.81e-03 +/- 9.18e-03 |  2.33e-01 +/- 7.0e-03  |  2.35e-01 +/- 6.0e-03  |
 test_objective_grad_ripple              |     -0.09 +/- 2.68     | -7.71e-04 +/- 2.32e-02 |  8.66e-01 +/- 2.1e-02  |  8.66e-01 +/- 1.0e-02  |

Github CI performance can be noisy. When evaluating the benchmarks, developers should take this into account.

[codecov]

Codecov Report

❌ Patch coverage is 94.73684% with 43 lines in your changes missing coverage. Please review.
✅ Project coverage is 94.39%. Comparing base (c119da0) to head (69d556f).

Files with missing lines	Patch %	Lines
desc/compute/_laplace.py	93.77%	15 Missing ⚠️
desc/integrals/singularities.py	90.56%	15 Missing ⚠️
desc/magnetic_fields/_laplace.py	93.54%	6 Missing ⚠️
desc/backend.py	84.00%	4 Missing ⚠️
desc/compute/utils.py	96.49%	2 Missing ⚠️
desc/basis.py	92.30%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2213      +/-   ##
==========================================
+ Coverage   94.34%   94.39%   +0.05%     
==========================================
  Files         101      103       +2     
  Lines       28845    29375     +530     
==========================================
+ Hits        27213    27729     +516     
- Misses       1632     1646      +14     

Files with missing lines	Coverage Δ
desc/coils.py	97.96% <ø> (ø)
desc/compute/__init__.py	100.00% <ø> (ø)
desc/compute/_basis_vectors.py	100.00% <100.00%> (ø)
desc/compute/_core.py	100.00% <100.00%> (ø)
desc/compute/_metric.py	100.00% <100.00%> (ø)
desc/compute/_surface.py	100.00% <100.00%> (ø)
desc/compute/data_index.py	95.89% <ø> (ø)
desc/equilibrium/equilibrium.py	95.96% <ø> (ø)
desc/geometry/core.py	95.30% <100.00%> (ø)
desc/grid.py	94.76% <100.00%> (+0.04%)	⬆️
... and 15 more

... and 3 files with indirect coverage changes

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