Automatic Jacobian sparsity detection using minimal scalar tracing and autodifferentiation
Author PALEOtoolkit
4 Stars
Updated Last
1 Year Ago
Started In
July 2021


Automatic Jacobian sparsity detection using minimal scalar tracing and autodifferentiation.

Limitations and alternatives

The detected sparsity pattern is that for the code path that is followed with the specified variable and parameter values. Any conditional logic that generates different code paths which result in different sparsity patterns is therefore not handled. As a workaround, either choose values that generate the least sparsity, or add 0.0*an_omitted_variable to conditional paths (relying on the fact that SparsityTracing includes structural zeros in the generated Jacobian), or call multiple times and merge the sparsity patterns (ie add the sparse Jacobians).

This package should be considered an interim solution until Symbolics.jl Jacobian tracing is fully implemented. Currently Symbolics v1.4.2 requires awkward workarounds for conditional logic (which will error), and is relatively slow (which may not be an issue for smaller models). See JuliaSymbolics/Symbolics.jl#326 for benchmark results showing that SparsityTracing is ~40x faster, and also demonstrating that the usage is very similar hence it is easy to just swap.

The earlier SparsityDetection.jl package is marked as deprecated.


julia> ] add SparsityTracing


julia> import SparsityTracing as AD

julia> function rober(du,u,p)
                   y₁,y₂,y₃ = u
                   k₁,k₂,k₃ = p
                   du[1] = -k₁*y₁+k₃*y₂*y₃
                   du[2] =  k₁*y₁-k₂*y₂^2-k₃*y₂*y₃
                   du[3] =  k₂*y₂^2

julia> u = [1.0, 2.0, 3.0];

julia> p = (0.04,3e7,1e4);

julia> u_ad = AD.create_advec(u);

julia> du_ad = similar(u_ad);

julia> rober(du_ad, u_ad, p)

julia> AD.deriv(du_ad[3])
2-element SparseArrays.SparseVector{Float64, Int64} with 1 stored entry:
  [2]  =  1.2e8

julia> Jad = AD.jacobian(du_ad, length(du_ad))
3×3 SparseArrays.SparseMatrixCSC{Float64, Int64} with 7 stored entries:
 -0.04  30000.0        20000.0
  0.04     -1.2003e8  -20000.0


The algorithm is essentially that of SparsLinC described in Bischof etal (1996), except with a simpler but less efficient data structure. Implements a scalar type SparsityTracing.ADval{T<:Real} that holds a value, and a binary tree of scalar derivatives including the element indices as leaf nodes. The tree is initialised to a Vector of leaf nodes by SparsityTracing.create_advec. It is populated with derivatives calculated by DiffRules.jl when a Julia function is called (eg a function y = f(x) calculating the RHS of an ODE). SparsityTracing.jacobian then walks the tree and calculates the Jacobian as a sparse matrix. This provides a robust way of detecting Jacobian sparsity (and for test purposes only, a very slow way of calculating the actual derivative). The sparsity pattern may then be used to generate matrix colouring for a fast AD package eg SparseDiffTools.jl.

Time taken increases approximately linearly with the number of scalar operations. Speed is approx 10 M scalar op/s (on a ~4Ghz laptop core) so 100 - 1000x slower than y = f(x) with primitive types (as each scalar operation generates a memory allocation to add a tree node).


Christian H. Bischof , Peyvand M. Khademi , Ali Buaricha & Carle Alan (1996) Efficient computation of gradients and Jacobians by dynamic exploitation of sparsity in automatic differentiation, Optimization Methods and Software, 7:1, 1-39, DOI: 10.1080/10556789608805642