Fast jacobian computation through sparsity exploitation and matrix coloring
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March 2019


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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(y,x) # in-place
  global fcalls += 1
  for i in 2:length(x)-1
    y[i] = x[i-1] - 2x[i] + x[i+1]
  y[1] = -2x[1] + x[2]
  y[end] = x[end-1] - 2x[end]

function g(x) # out-of-place
  global fcalls += 1
  y = zero(x)
  for i in 2:length(x)-1
    y[i] = x[i-1] - 2x[i] + x[i+1]
  y[1] = -2x[1] + x[2]
  y[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 Symbolics.jacobian_sparsity function to automatically detect the sparsity pattern. We declare that the function f outputs a vector of length 30 and takes in a vector of length 30, and jacobian_sparsity returns a SparseMatrixCSC:

using Symbolics
input = rand(30)
output = similar(input)
sparsity_pattern = Symbolics.jacobian_sparsity(f,output,input)
jac = Float64.(sparsity_pattern)

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

using SparseDiffTools
colors = matrix_colors(jac)

Since maximum(colors) is 3, this means that finite differencing can now compute the Jacobian in just 4 f-evaluations. Generating the sparsity pattern used 1 (pseudo) f-evaluation, so the total number of times that f is called to compute the sparsity pattern plus the entire 30x30 Jacobian is 5 times:

using FiniteDiff
FiniteDiff.finite_difference_jacobian!(jac, f, rand(30), colorvec=colors)
@show fcalls # 5

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

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

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

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

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 ArrayInterfaceCore.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::SparseDiffToolsColoringAlgorithm = 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 FiniteDiff.jl and for automatic differentiation is provided by ForwardDiff.jl.

For FiniteDiff.jl, one simply has to use the provided colorvec keyword argument. See the FiniteDiff 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)

Notice that if a sparsity pattern is not supplied then the built Jacobian will be the compressed Jacobian: sparsity must be a sparse matrix or a structured matrix (Tridiagonal, Banded, etc. conforming to the ArrayInterfaceCore.jl specs) with the appropriate sparsity pattern to allow for decompression.

This call will 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:


dx is a pre-allocated output vector which is used to declare the output size, and thus allows for specifying a non-square Jacobian.

If one is using an out-of-place function f(x), then the alternative form ca be used:

jacout = forwarddiff_color_jacobian(g, x,
                                    dx = similar(x),
                                    colorvec = 1:length(x),
                                    sparsity = nothing,
                                    jac_prototype = nothing)

Note that the out-of-place form is efficient and compatible with StaticArrays. One can specify the type and shape of the output Jacobian by giving an additional jac_prototype to the out-of place forwarddiff_color_jacobian function, otherwise it will become a dense matrix. If jac_prototype and sparsity are not specified, then the oop Jacobian will assume that the function has a square Jacobian matrix. If it is not the case, please specify the shape of output by giving dx.

Similar functionality is available for Hessians, using finite differences of forward derivatives. Given a scalar function f(x), a vector value for x, and a color vector and sparsity pattern, this can be accomplished using numauto_color_hessian or its in-place form numauto_color_hessian!.

H = numauto_color_hessian(f, x, colorvec, sparsity)
numauto_color_hessian!(H, f, x, colorvec, sparsity)

To avoid unnecessary allocations every time the Hessian is computed, construct a ForwardColorHesCache beforehand:

hescache = ForwardColorHesCache(f, x, colorvec, sparsity)
numauto_color_hessian!(H, f, x, hescache)

By default, these methods use a mix of numerical and automatic differentiation, namely by taking finite differences of gradients calculated via ForwardDiff.jl. Alternatively, if you have your own custom gradient function g!, you can specify it as an argument to ForwardColorHesCache:

hescache = ForwardColorHesCache(f, x, colorvec, sparsity, g!)

Note that any user-defined gradient needs to have the signature g!(G, x), i.e. updating the gradient G in place.

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!(dy, 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!(dy,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(dy,x) or dy=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 .= ....