A common interface for quadrature and numerical integration for the SciML scientific machine learning organization
Author SciML
52 Stars
Updated Last
5 Months Ago
Started In
August 2019


Build Status

Quadrature.jl is an instantiation of the DiffEqBase.jl common QuadratureProblem interface for the common quadrature packages of Julia. By using Quadrature.jl, you get a single predictable interface where many of the arguments are standardized throughout the various integrator libraries. This can be useful for benchmarking or for library implementations, since libraries which internally use a quadrature can easily accept a quadrature method as an argument.


For basic multidimensional quadrature we can construct and solve a QuadratureProblem:

using Quadrature
f(x,p) = sum(sin.(x))
prob = QuadratureProblem(f,ones(2),3ones(2))
sol = solve(prob,HCubatureJL(),reltol=1e-3,abstol=1e-3)

If we would like to parallelize the computation, we can use the batch interface to compute multiple points at once. For example, here we do allocation-free multithreading with Cubature.jl:

using Quadrature, Cubature, Base.Threads
function f(dx,x,p)
  Threads.@threads for i in 1:size(x,2)
    dx[i] = sum(sin.(@view(x[:,i])))
prob = QuadratureProblem(f,ones(2),3ones(2),batch=2)
sol = solve(prob,CubatureJLh(),reltol=1e-3,abstol=1e-3)

If we would like to compare the results against Cuba.jl's Cuhre method, then the change is a one-argument change:

using Cuba
sol = solve(prob,CubaCuhre(),reltol=1e-3,abstol=1e-3)


Quadrature.jl is a fully differentiable quadrature library. Thus, it adds the ability to perform automatic differentiation over any of the libraries that it calls. It integrates with ForwardDiff.jl for forward-mode automatic differentiation and Zygote.jl for reverse-mode automatic differentiation. For example:

using Quadrature, ForwardDiff, FiniteDiff, Zygote, Cuba
f(x,p) = sum(sin.(x .* p))
lb = ones(2)
ub = 3ones(2)
p = [1.5,2.0]

function testf(p)
    prob = QuadratureProblem(f,lb,ub,p)
dp1 = Zygote.gradient(testf,p)
dp2 = FiniteDiff.finite_difference_gradient(testf,p)
dp3 = ForwardDiff.gradient(testf,p)
dp1[1]  dp2  dp3


To use this package, you always construct a QuadratureProblem. This has a constructor:

                  nout=1, batch = 0, kwargs...)
  • f: Either a function f(x,p) for out-of-place or f(dx,x,p) for in-place.
  • lb: Either a number or vector of lower bounds.
  • ub: Either a number or vector of upper bounds.
  • p: The parameters associated with the problem.
  • nout: The output size of the function f. Defaults to 1, i.e., a scalar integral output.
  • batch: The preferred number of points to batch. This allows user-side parallelization of the integrand. If batch != 0, then each x[:,i] is a different point of the integral to calculate, and the output should be nout x batchsize. Note that batch is a suggestion for the number of points, and it is not necessarily true that batch is the same as batchsize in all algorithms.

Additionally, we can supply iip like QuadratureProblem{iip}(...) as true or false to declare at compile time whether the integrator function is in-place.


The following algorithms are available:

  • QuadGKJL: Uses QuadGK.jl. Requires nout=1 and batch=0.
  • HCubatureJL: Uses HCubature.jl. Requires batch=0.
  • VEGAS: Uses MonteCarloIntegration.jl. Requires nout=1.
  • CubatureJLh: h-Cubature from Cubature.jl. Requires using Cubature.
  • CubatureJLp: p-Cubature from Cubature.jl. Requires using Cubature.
  • CubaVegas: Vegas from Cuba.jl. Requires using Cuba.
  • CubaSUAVE: SUAVE from Cuba.jl. Requires using Cuba.
  • CubaDivonne: Divonne from Cuba.jl. Requires using Cuba.
  • CubaCuhre: Cuhre from Cuba.jl. Requires using Cuba.

Common Solve Keyword Arguments

  • reltol: Relative tolerance
  • abstol: Absolute tolerance
  • maxiters: The maximum number of iterations

Additionally, the extra keyword arguments are splatted to the library calls, so see the documentation of the integrator library for all of the extra details. These extra keyword arguments are not guaranteed to act uniformly.