Make Jacobian calculation with coloring compatible with GPUs

[ChrisRackauckas]

            for i in 1:length(cols_index)
                if color[cols_index[i]]==color_i
                    J[rows_index[i],cols_index[i]] = dx[rows_index[i]]
                end
            end

is the only part that doesn't play nicely with the GPU. If that can be made into a broadcast expression, then GPU-based Jacobians would be a reality. This would be very nice since finite differencing really doesn't do well with GPUs, and we know that if the code is GPU compatible then it is likely to be pure-Julia code and thus AD-compatible.

[ChrisRackauckas]

@YingboMa if you have any sparse time this might be something to look at. It might already GPU better than I think.

[pkj-m]

for color_i in 1:maximum(color)
        for i in eachindex(x)
            if color[i]==color_i
                partials_array[i,color_i] = true
            else
                partials_array[i,color_i] = false
            end
        end
end 

Won't we have to vectorize this block of code as well before we can call the whole package truly GPU-based?

[ChrisRackauckas]

We will likely want color to live on the CPU, so this is fine on the CPU, as long as we make p automatically get sent to the GPU. Over time we may want this all on the GPU as well.

[ChrisRackauckas]

The hard thing with the GPU support is that we want to support GPUs without explicitly requiring GPU libraries. The way this is done is by using Requires.jl. This looks like:

• Add Requires.jl
• Use an __init__ function to add dispatches at precompile. https://github.com/JuliaDiffEq/DiffEqBase.jl/blob/master/src/init.jl#L1 is an example. What you do is create a stub function. Here, make it is_gpu(x) = false. Then in the init, you add a dispatch is_gpu(x::CuArray) = true. Now you have a way of telling if something is a GPU array without requiring GPU arrays.
• if is_gpu(J), then make this path occur: Vectorized decompression #38
• Where the cu stuff exists in https://github.com/JuliaDiffEq/SparseDiffTools.jl/pull/39/files#diff-de9340ebf40cd94cabb99e3bcdf7f466R42 , use Adapt.jl to make it automatically send the cache variables to the GPU.