Introduce OctaminimalGaussianArray; smaller, faster but less accurate

[milankl]

Motivated by #594 this reduces the number of grid points on the OctahedralGaussianGrid by starting at 4 points around the pole instead of 20. At nlat_half = 24, the default resolution, this yields 2400 grid points instead of 3168, i.e. -25%, hopefully making the parameterizations 25% faster, and the whole model by 12.5%-ish as parameterizations are currently more than 50% of total runtime.

Number of grid points per ring are (old, new)

julia> [(16+4i, 4i) for i in 1:24]
24-element Vector{Tuple{Int64, Int64}}:
 (20, 4)
 (24, 8)
 (28, 12)
 (32, 16)
 (36, 20)
 (40, 24)
 (44, 28)
 (48, 32)
 (52, 36)
 (56, 40)
 (60, 44)
 (64, 48)
 (68, 52)
 (72, 56)
 (76, 60)
 (80, 64)
 (84, 68)
 (88, 72)
 (92, 76)
 (96, 80)
 (100, 84)
 (104, 88)
 (108, 92)
 (112, 96)

[milankl]

The "exact" tests with OctaminimalGaussianGrid pass once it's >T63 and single precision, at the moment the transform error is therefore only really present at low resolutions, see here

Test Summary:                              |   Pass  Fail   Total  Time
Transform: Individual Legendre polynomials | 665163   117  665280  7.6s
  trunc = 31                               |  33579    21   33600  0.7s
    NF = Float32                           |  16795     5   16800  0.3s
      Grid = FullGaussianGrid              |   3360          3360  0.1s
      Grid = FullClenshawGrid              |   3360          3360  0.0s
      Grid = OctahedralGaussianGrid        |   3360          3360  0.0s
      Grid = OctahedralClenshawGrid        |   3360          3360  0.0s
      Grid = OctaminimalGaussianGrid       |   3355     5    3360  0.1s
    NF = Float64                           |  16784    16   16800  0.4s
      Grid = FullGaussianGrid              |   3360          3360  0.1s
      Grid = FullClenshawGrid              |   3360          3360  0.0s
      Grid = OctahedralGaussianGrid        |   3360          3360  0.0s
      Grid = OctahedralClenshawGrid        |   3360          3360  0.0s
      Grid = OctaminimalGaussianGrid       |   3344    16    3360  0.2s
  trunc = 63                               | 128608    32  128640  1.5s
    NF = Float32                           |  64320         64320  0.6s
    NF = Float64                           |  64288    32   64320  0.9s
      Grid = FullGaussianGrid              |  12864         12864  0.1s
      Grid = FullClenshawGrid              |  12864         12864  0.1s
      Grid = OctahedralGaussianGrid        |  12864         12864  0.1s
      Grid = OctahedralClenshawGrid        |  12864         12864  0.1s
      Grid = OctaminimalGaussianGrid       |  12832    32   12864  0.5s
  trunc = 127                              | 502976    64  503040  5.4s
    NF = Float32                           | 251520        251520  2.1s
    NF = Float64                           | 251456    64  251520  3.3s
      Grid = FullGaussianGrid              |  50304         50304  0.4s
      Grid = FullClenshawGrid              |  50304         50304  0.8s
      Grid = OctahedralGaussianGrid        |  50304         50304  0.4s
      Grid = OctahedralClenshawGrid        |  50304         50304  0.4s
      Grid = OctaminimalGaussianGrid       |  50240    64   50304  1.3s

[milankl]

Interpolation tests all pass once we switch to staggered longitudes, i.e. with an offset (not starting at 0˚E) as are also the HEALPix grids defined, first ring

  45.0
 135.0
 225.0
 315.0

second ring

  22.5
  67.5
 112.5
 157.5
 202.5
 247.5
 292.5
 337.5

etc

[milankl]

Created a little comparison

moving to 4 points around the poles introduces weird grid cells around the poles because ECMWF defines Gaussian grids with 0 offset from prime meridian

with offset like HEALPix grids also use them that goes away

[milankl]

So a candidate for a new default grid could be this one 2400 grid points, that's what I'd define as the OctaminimalGaussianGrid (octahedral Gaussian starting with 4 points, with offset)

vs a 24-ring octahedral Gaussian grid as ECMWF defines it, starting with 20 points, no offset, which has, however, 3168 grid points, i.e. +32%

[milankl]

And one last

[milankl]

No proper benchmarking here, but with the OctaminimalGaussianGrid I'm hitting some 700SYPD compared to 500-550SYPD on the octahedral Gaussian grid. Nice one 🚀

[milankl]

@hottad just merged, another grid in our grid zoo, the OctaminimalGaussianGrid, looks like the OctaHEALPixGrid, but uses Gaussian latitudes instead, haven't checked the error properly, but I suspect it's somewhere in between the exact grids and the healpix grids -- maybe a nice tradeoff. Definitely way more equal area than the OctahedralGaussianGrid!