Add sparse AD comparison

[gdalle]

Add comparison between DifferentiationInterface and SparseDiffTools on the Brusselator Jacobian

Spoiler alert: for this very specific use case and AD backend the new pipeline is

• 100x faster for sparsity detection
• 10x faster for coloring
• ten percent slower for the compressed differentiation itself (but I haven't even profiled the code so there is clearly room for improvement)

[gdalle]

Results on my computer:

Sparse AD benchmarks.pdf

[ChrisRackauckas]

Can you also add the caching PDE one?

using PreallocationTools

const cs = Val{ForwardDiff.pickchunksize(length(u))}
const Ayu = FixedSizeDiffCache(zeros(N, N), cs)
const uAx = FixedSizeDiffCache(zeros(N, N), cs)
const Du = FixedSizeDiffCache(zeros(N, N), cs)
const Ayv = FixedSizeDiffCache(zeros(N, N), cs)
const vAx = FixedSizeDiffCache(zeros(N, N), cs)
const Dv = FixedSizeDiffCache(zeros(N, N), cs)
function gm3!(dr, r, p, t)
    a, α, ubar, β, D1, D2 = p
    Ayu = get_tmp(Ayu, dr)
    uAx = get_tmp(uAx, dr)
    Du = get_tmp(Du, dr)
    Ayv = get_tmp(Ayv, dr)
    vAx = get_tmp(vAx, dr)
    Dv = get_tmp(Dv, dr)
    u = @view r[:, :, 1]
    v = @view r[:, :, 2]
    du = @view dr[:, :, 1]
    dv = @view dr[:, :, 2]
    mul!(Ayu, Ay, u)
    mul!(uAx, u, Ax)
    mul!(Ayv, Ay, v)
    mul!(vAx, v, Ax)
    @. Du = D1 * (Ayu + uAx)
    @. Dv = D2 * (Ayv + vAx)
    @. du = Du + a * u * u ./ v + ubar - α * u
    @. dv = Dv + a * u * u - β * v
end
f(du,u) = f(du,u,p,0.0)

version from https://docs.sciml.ai/DiffEqDocs/stable/tutorials/faster_ode_example/#Example-Accelerating-Linear-Algebra-PDE-Semi-Discretization