Ag/pr vectorized second derivative volume

[andrewgiuliani]

This PR introduces a vectorized implementation of the second derivative of volume wrt to surface dofs calculation. This is useful for the BoozerLS calculation when volume is used as a surface label. Running the timing test:

import time
stellsym = True
nfp = 4
for mm in range(1, 15):
    mpol = mm
    ntor = mm
    phis = np.linspace(0, 1/nfp, 2*ntor+1, endpoint=False)
    thetas = np.linspace(0, 1, 2*mpol+1, endpoint=False)
    t0 = time.time()
    s = SurfaceXYZTensorFourier(
        nfp=nfp, stellsym=stellsym, mpol=mpol, ntor=ntor,
        quadpoints_phi=phis, quadpoints_theta=thetas)
    d2vol = s.d2volume_by_dcoeffdcoeff()
    t1 = time.time()
    print(mm, t1-t0)

I obtain the following timings for the original implementation:

mpol&ntor, time (s)
1 0.001640
2 0.003978
3 0.034043
4 0.108703
5 0.324571
6 0.867548
7 2.053304
8 4.371564
9 8.561552
10 15.6180
11 27.0530
12 out of memory
13 out of memory
14 out of memory

I run out of RAM for mpol, ntor >= 12. This is due (in part) to the line that computes d2nor_dcdc:

simsopt/src/simsoptpp/surface.cpp

Line 786 in dd1989d

auto d2nor_dcdc = this->d2normal_by_dcoeffdcoeff(); // uses a lot of memory for moderate surface complexity

This array stores the second derivative of the normal vector at all the surface quadrature points, totalling 3*nquadpoints_phi*nquadpoints_theta*ndofs*ndofs doubles. In the new vectorized implementation, I compute the entries of d2nor_dcdc on the fly, which alleviates memory requirements. Timings for the new vectorized implementation with a simd batchsize of 8 is:

mpol&ntor, time (s)
1 0.026728
2 0.005802
3 0.002354
4 0.004601
5 0.012529
6 0.040148
7 0.065161
8 0.123145
9 0.231217
10 0.41314
11 0.69693
12 1.10758
13 1.75156
14 2.73922

The original implementation is still used when xsimd is not available. I have not vectorized d2area_by_dcoeffdcoeff, because I typically do not use area as a surface label in the BoozerLS algorithm. However, it suffers from issues as mpol, ntor increase too.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 91.58%. Comparing base (3d1413b) to head (d6fab6a).
Report is 11 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #402      +/-   ##
==========================================
- Coverage   91.64%   91.58%   -0.06%     
==========================================
  Files          74       74              
  Lines       12987    12911      -76     
==========================================
- Hits        11902    11825      -77     
- Misses       1085     1086       +1     

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