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Fast jacobian computation through sparsity exploitation and matrix coloring
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This package is for exploiting sparsity in Jacobians and Hessians to accelerate computations. Matrix-free Jacobian-vector product and Hessian-vector product operators are provided that are compatible with AbstractMatrix-based libraries like IterativeSolvers.jl for easy and efficient Newton-Krylov implementation. It is possible to perform matrix coloring, and utilize coloring in Jacobian and Hessian construction.

Optionally, automatic and numerical differentiation are utilized.


Suppose we had the function

fcalls = 0
function f(dx,x)
  global fcalls += 1
  for i in 2:length(x)-1
    dx[i] = x[i-1] - 2x[i] + x[i+1]
  dx[1] = -2x[1] + x[2]
  dx[end] = x[end-1] - 2x[end]

For this function, we know that the sparsity pattern of the Jacobian is a Tridiagonal matrix. However, if we didn't know the sparsity pattern for the Jacobian, we could use the sparsity! function to automatically detect the sparsity pattern. This function is only available if you load SparsityDetection.jl as well. We declare that the function f outputs a vector of length 30 and takes in a vector of length 30, and sparsity! spits out a Sparsity object which we can turn into a SparseMatrixCSC:

using SparsityDetection
sparsity_pattern = sparsity!(f,output,input)
jac = Float64.(sparse(sparsity_pattern))

Now we call matrix_colors to get the colorvec vector for that matrix:

colors = matrix_colors(jac)

Since maximum(colors) is 3, this means that finite differencing can now compute the Jacobian in just 4 f-evaluations:

DiffEqDiffTools.finite_difference_jacobian!(jac, f, rand(30), colorvec=colors)
@show fcalls # 4

In addition, a faster forward-mode autodiff call can be utilized as well:

forwarddiff_color_jacobian!(jac, f, x, colorvec = colors)

If one only need to compute products, one can use the operators. For example,

u = rand(30)
J = JacVec(f,u)

makes J into a matrix-free operator which calculates J*v products. For example:

v = rand(30)
res = similar(v)
mul!(res,J,v) # Does 1 f evaluation

makes res = J*v. Additional operators for HesVec exists, including HesVecGrad which allows one to utilize a gradient function. These operators are compatible with iterative solver libraries like IterativeSolvers.jl, meaning the following performs the Newton-Krylov update iteration:

using IterativeSolvers


Matrix Coloring

This library extends the common ArrayInterface.matrix_colors function to allow for coloring sparse matrices using graphical techniques.

Matrix coloring allows you to reduce the number of times finite differencing requires an f call to maximum(colors)+1, or reduces automatic differentiation to using maximum(colors) partials. Since normally these values are length(x), this can be significant savings.

The API for computing the colorvec vector is:

matrix_colors(A::AbstractMatrix,alg::ColoringAlgorithm = GreedyD1Color();
              partition_by_rows::Bool = false)

The first argument is the abstract matrix which represents the sparsity pattern of the Jacobian. The second argument is the optional choice of coloring algorithm. It will default to a greedy distance 1 coloring, though if your special matrix type has more information, like is a Tridiagonal or BlockBandedMatrix, the colorvec vector will be analytically calculated instead. The keyword argument partition_by_rows allows you to partition the Jacobian on the basis of rows instead of columns and generate a corresponding coloring vector which can be used for reverse-mode AD. Default value is false.

The result is a vector which assigns a colorvec to each column (or row) of the matrix.

Colorvec-Assisted Differentiation

Colorvec-assisted differentiation for numerical differentiation is provided by DiffEqDiffTools.jl and for automatic differentiation is provided by ForwardDiff.jl.

For DiffEqDiffTools.jl, one simply has to use the provided colorvec keyword argument. See the DiffEqDiffTools Jacobian documentation for more details.

For forward-mode automatic differentiation, use of a colorvec vector is provided by the following function:

                            dx = nothing,
                            colorvec = eachindex(x),
                            sparsity = nothing)

This call wiil allocate the cache variables each time. To avoid allocating the cache, construct the cache in advance:

ForwardColorJacCache(f,x,_chunksize = nothing;
                              dx = nothing,
                              sparsity = nothing)

and utilize the following signature:


Jacobian-Vector and Hessian-Vector Products

Matrix-free implementations of Jacobian-Vector and Hessian-Vector products is provided in both an operator and function form. For the functions, each choice has the choice of being in-place and out-of-place, and the in-place versions have the ability to pass in cache vectors to be non-allocating. When in-place the function signature for Jacobians is f!(du,u), while out-of-place has du=f(u). For Hessians, all functions must be f(u) which returns a scalar

The functions for Jacobians are:

auto_jacvec!(du, f, x, v,
                      cache1 = ForwardDiff.Dual{DeivVecTag}.(x, v),
                      cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v))

auto_jacvec(f, x, v)

# If compute_f0 is false, then `f(cache1,x)` will be computed
num_jacvec!(du,f,x,v,cache1 = similar(v),
                     cache2 = similar(v);
                     compute_f0 = true)

For Hessians, the following are provided:

             cache1 = similar(v),
             cache2 = similar(v),
             cache3 = similar(v))


                 cache = ForwardDiff.GradientConfig(f,v),
                 cache1 = similar(v),
                 cache2 = similar(v))


                 cache1 = similar(v),
                 cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v),
                 cache3 = ForwardDiff.Dual{DeivVecTag}.(x, v))


In addition, the following forms allow you to provide a gradient function g(dx,x) or dx=g(x) respectively:

                     cache2 = similar(v),
                     cache3 = similar(v))


                     cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v),
                     cache3 = ForwardDiff.Dual{DeivVecTag}.(x, v))


The numauto and autonum methods both mix numerical and automatic differentiation, with the former almost always being more efficient and thus being recommended.

Optionally, if you load Zygote.jl, the following numback and autoback methods are available and allow numerical/ForwardDiff over reverse mode automatic differentiation respectively, where the reverse-mode AD is provided by Zygote.jl. Currently these methods are not competitive against numauto, but as Zygote.jl gets optimized these will likely be the fastest.

using Zygote # Required

                     cache1 = similar(v),
                     cache2 = similar(v))


# Currently errors! See
                     cache2 = ForwardDiff.Dual{DeivVecTag}.(x, v),
                     cache3 = ForwardDiff.Dual{DeivVecTag}.(x, v))


Jv and Hv Operators

The following produce matrix-free operators which are used for calculating Jacobian-vector and Hessian-vector products where the differentiation takes place at the vector u:


These all have the same interface, where J*v utilizes the out-of-place Jacobian-vector or Hessian-vector function, whereas mul!(res,J,v) utilizes the appropriate in-place versions. To update the location of differentiation in the operator, simply mutate the vector u: J.u .= ....

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