Some performance improvements in bivariate interpolation

[DanielVandH]

I changed the small vectors to just be Tuples, removing all allocations. The way I did it also has the benefit that it'll be type stable for non-Float64. I also made some better use of the DelaunayTriangulation.jl API.

I don't think there are any 2D tests or I missed them, but I tested some random values and they are the same between master and this branch.

Master:

julia> Random.seed!(123)
TaskLocalRNG()

julia> x, y, z, dzdx, dzdy = rand(100), rand(100), rand(100), rand(100), rand(100);

julia> spl2d = BivariateCHSInterpolation(x, y, z, dzdx, dzdy);

julia>  spl2d(x[5], y[5])
0.5320809801118653

julia> interp(spl2d, x[6], y[6])
0.042869724439189925

julia>  grad(spl2d, x[10], y[10])
(0.9563604567029627, 0.6881752317560613)

julia> grad(spl2d, 0.49211, 0.9492)
(-10.075789358336532, 17.877296736109372)

julia> spl2d(0.123, 0.99992)
NaN

julia> spl2d(0.123, 0.30491)
0.7794865485348513

This PR:

julia> Random.seed!(123)
TaskLocalRNG()

julia> x, y, z, dzdx, dzdy = rand(100), rand(100), rand(100), rand(100), rand(100);

julia> spl2d = BivariateCHSInterpolation(x, y, z, dzdx, dzdy);

julia> spl2d(x[5], y[5])
0.5320809801118653

julia> interp(spl2d, x[6], y[6])
0.042869724439189925

julia> grad(spl2d, x[10], y[10])
(0.9563604567029627, 0.6881752317560613)

julia> grad(spl2d, 0.49211, 0.9492)
(-10.075789358336532, 17.877296736109372)

julia> spl2d(0.123, 0.99992)
NaN

julia> spl2d(0.123, 0.30491)
0.7794865485348513

Some benchmarks to show that this is an actual improvement:

Master:

julia> using BenchmarkTools

julia> @benchmark $spl2d(x, y) setup=(x=rand(); y = rand())
BenchmarkTools.Trial: 10000 samples with 10 evaluations.
 Range (min … max):  1.320 μs … 29.960 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.500 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.790 μs ±  1.108 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

   ▂▁       ▄▇█▇▆▅▄▅▅▄▄▃▃▃▃▂▂▁▁▁ ▁▁                          ▂
  ▆██▇▅▇▅▅▅▅█████████████████████████▇▇▇▇▇▇▆▆▅▆▄▅▃▄▄▄▄▅▄▄▂▄▂ █
  1.32 μs      Histogram: log(frequency) by time     6.41 μs <

 Memory estimate: 80 bytes, allocs estimate: 1.

julia> @benchmark $spl2d.($x, $y)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  123.100 μs …  1.487 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     263.650 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   274.048 μs ± 86.035 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

                   ██▁
  ▂▃▃▅▃▂▂▂▂▂▁▁▁▃▃▂▃███▆▅▄▃▃▃▂▂▂▂▂▂▂▂▂▂▂▁▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  123 μs          Histogram: frequency by time          574 μs <

 Memory estimate: 32.12 KiB, allocs estimate: 401.

julia> @benchmark $grad.($(spl2d,), $x, $y)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  130.100 μs … 200.146 ms  ┊ GC (min … max): 0.00% … 99.78%
 Time  (median):     293.800 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   343.414 μs ±   2.015 ms  ┊ GC (mean ± σ):  7.34% ±  2.55%

              █    ▂
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  130 μs           Histogram: frequency by time          688 μs <

 Memory estimate: 56.45 KiB, allocs estimate: 701.

This PR:

julia> using BenchmarkTools

julia> @benchmark $spl2d(x, y) setup=(x=rand(); y = rand())
BenchmarkTools.Trial: 10000 samples with 10 evaluations.
 Range (min … max):  1.100 μs … 69.380 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     1.370 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   1.485 μs ±  1.015 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

      ▆█▆▁
  ▁▃▅▆████▅▄▃▃▃▃▄▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  1.1 μs         Histogram: frequency by time        3.54 μs <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark $spl2d.($x, $y)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  110.200 μs …  1.155 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     128.100 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   137.327 μs ± 38.222 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

  ▃▆▄▄██▆▄▄▄▃▃▃▂▂▂▁▁                                           ▂
  █████████████████████▇█▇█▅▇▅▇▆▆▆▆▆▆▄▆▆▆▆▄▆▆▄▆▄▄▅▄▅▅▅▅▄▄▆▄▅▂▅ █
  110 μs        Histogram: log(frequency) by time       319 μs <

 Memory estimate: 896 bytes, allocs estimate: 1.

julia> @benchmark $grad.($(spl2d,), $x, $y)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  112.900 μs …  1.621 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     130.500 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   144.823 μs ± 51.121 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

      █
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  113 μs          Histogram: frequency by time          341 μs <

 Memory estimate: 1.77 KiB, allocs estimate: 1.

[liuyxpp]

Wow, huge thanks! I like your revisions, especially those DT methods which I am not aware of. I will also try to add some tests for 2D. Thanks again!