Performance with nested QuadGK integrals

General Usage Performance
1

I encountered a performance issue whilst working. I’ve managed to produce a minimal example of my code that reproduces a performance issue. I have a file bug.jl:

import QuadGK

function inner_integrand()
    function integrand(t :: Float64)
        return 0.1
    end
    term1 = QuadGK.quadgk(integrand, 1, 2)[1]
    return term1
end

function outer_integrand()
    p = inner_integrand()
    return 0.5
end

function evaluate_double_integral()
    function integrand(theta_0)
        return outer_integrand()
    end
    return QuadGK.quadgk(integrand, 0, 0.1)[1] 
end

function compute_solution()
    println("Inside computation function")
    integro_diff_mtrx = 
        map(
            in->evaluate_double_integral(),
            [0]
        )
    println("Finished computation function")
    return
end

I can use this with a script, testa.jl:

include("bug.jl")
println("Before computation function")
compute_solution()
println("After computation function")

I can run this from the command line: julia.exe testa.jl.
It takes a whole minute and 21 seconds to run.

Oddly, I can make it run much faster by making evaluating the inner integrand once before we attempt the double integral: testb.jl:

include("bug.jl")
println("Evaluating a function pointlessly.")
inner_integrand()
println("Before computation function")
compute_solution()
println("After computation function")

Now it takes 1.6 seconds. Much faster (but still surprisingly slow?).

Does this count as a bug, or is there something I’m missing?
I’m using Windows 10 x64. Issue in Julia 1.0.3.

2

I haven’t had a chance to look at your particular case, but I should say that in general I recommend against doing nested calls to any adaptive 1d quadrature routine. You can easily end up wasting lots of time refining one of the inner integrals even though it makes a negligible contribution to the overall 2d integral. You are usually better off with a 2d quadrature routine like in the HCubature package.

3

You’re probably right, but for now the code that this is based on is easier to read with nested quadratures. I’ll be sure to look at HCubature if I optimization becomes necessary. But I’m more interested / surprised about the reasoning for the massive difference in performance. The Julia runtime spends a long time working before it even prints “Inside computation function”.

As an aside, I am finding your exponential integral implementation useful. Thanks!