DDE Fails to execute in parallel

[H-Sax]

Describe the bug 🐞

When performing a GSA on the Serial function runs quicker than the parallel counterpart. I have included the same test with an ODE version to highlight how the multithreaded version should execute more efficiently.

Expected behavior

The multithreaded version should run more efficiently than the serial version

Minimal Reproducible Example 👇

using DifferentialEquations

# Define DDE model 
begin
    function bc_model(du, u, h, p, t)
        p0, q0, v0, d0, p1, q1, v1, d1, d2, beta0, beta1, tau = p
        hist3 = h(p, t - tau)[3]
        du[1] = (v0 / (1 + beta0 * (hist3^2))) * (p0 - q0) * u[1] - d0 * u[1]
        du[2] = (v0 / (1 + beta0 * (hist3^2))) * (1 - p0 + q0) * u[1] +
                (v1 / (1 + beta1 * (hist3^2))) * (p1 - q1) * u[2] - d1 * u[2]
        du[3] = (v1 / (1 + beta1 * (hist3^2))) * (1 - p1 + q1) * u[2] - d2 * u[3]
    end

    h(p, t) = ones(3)

    tau = 1
    lags = [tau]

    p0 = 0.2;
    q0 = 0.3;
    v0 = 1;
    d0 = 5;
    p1 = 0.2;
    q1 = 0.3;
    v1 = 1;
    d1 = 1;
    d2 = 1;
    beta0 = 1;
    beta1 = 1;
    p = [p0, q0, v0, d0, p1, q1, v1, d1, d2, beta0, beta1, tau]
    tspan = (0.0, 10.0)
    u0 = [1.0, 1.0, 1.0]

    prob = DDEProblem(bc_model, u0, h, tspan, p; constant_lags = lags)
    alg = MethodOfSteps(Tsit5())
    @time sol = solve(prob, alg)
end 

using GlobalSensitivity, QuasiMonteCarlo, Statistics

samples = 30000
lb = 0.9*prob.p
ub = 1.1*prob.p
sampler = SobolSample()
A,B = QuasiMonteCarlo.generate_design_matrices(samples,lb,ub,sampler)


# Serial 
f1_s = function (p)
    prob1 = remake(prob;p=p)
    sol = solve(prob1,MethodOfSteps(Tsit5()))
    [mean(sol[1,:]), mean(sol[2,:])]
end
@time res = gsa(f1_s,Sobol(),A,B)
# Roughly 40s

## Para 

f1_p = function (p)
    prob_func(prob,i,repeat) = remake(prob;p=p[:,i], constant_lags = lags)
    ensemble_prob = EnsembleProblem(prob,prob_func=prob_func)
    sol = solve(ensemble_prob,MethodOfSteps(Tsit5()),EnsembleThreads();trajectories=size(p,2))
    # Now sol[i] is the solution for the ith set of parameters
    out = zeros(2,size(p,2))
    for i in 1:size(p,2)
      out[1,i] = mean(sol[i][1,:])
      out[2,i] = mean(sol[i][2,:])
    end
    out
end
@time res = gsa(f1_p,Sobol(),A,B,batch = true)
# Roughly 170s 

## ODE EXamples ##
function f(du,u,p,t)
    du[1] = p[1]*u[1] - p[2]*u[1]*u[2] #prey
    du[2] = -p[3]*u[2] + p[4]*u[1]*u[2] #predator
end
u0 = [1.0;1.0]
tspan = (0.0,10.0)
p = [1.5,1.0,3.0,1.0]
prob = ODEProblem(f,u0,tspan,p)
t = collect(range(0, stop=10, length=200))

using GlobalSensitivity, QuasiMonteCarlo, Statistics

samples = 50000
lb = 0.9*prob.p
ub = 1.1*prob.p
sampler = SobolSample()
A,B = QuasiMonteCarlo.generate_design_matrices(samples,lb,ub,sampler)

# Serial 
f1_s = function (p)
    prob1 = remake(prob;p=p)
    sol = solve(prob1,Tsit5(), saveat = t)
    [mean(sol[1,:]), mean(sol[2,:])]
end

@time res = gsa(f1_s,Sobol(),A,B)
# Roughly 35s

## Para 

f1_p = function (p)
    prob_func(prob,i,repeat) = remake(prob;p=p[:,i])
    ensemble_prob = EnsembleProblem(prob,prob_func=prob_func)
    sol = solve(ensemble_prob,Tsit5(), saveat = t,EnsembleThreads();trajectories=size(p,2))
    # Now sol[i] is the solution for the ith set of parameters
    out = zeros(2,size(p,2))
    for i in 1:size(p,2)
      out[1,i] = mean(sol[i][1,:])
      out[2,i] = mean(sol[i][2,:])
    end
    out
end
@time res = gsa(f1_p,Sobol(),A,B,batch = true)
# Roughly 18.5s - Quicker as expected with multiple threads

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

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⌃ [0c5d862f] Symbolics v5.22.1
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.11.1
⌅ [6aa5eb33] TaylorSeries v0.15.4
  [62fd8b95] TensorCore v0.1.1
  [8290d209] ThreadingUtilities v0.5.2
  [ac1d9e8a] ThreadsX v0.1.12
⌅ [731e570b] TiffImages v0.6.8
  [a759f4b9] TimerOutputs v0.5.23
  [0796e94c] Tokenize v0.5.28
⌃ [3bb67fe8] TranscodingStreams v0.10.3
  [28d57a85] Transducers v0.4.81
  [592b5752] Trapz v2.0.3
  [d5829a12] TriangularSolve v0.1.20
  [410a4b4d] Tricks v0.1.8
  [981d1d27] TriplotBase v0.1.0
  [781d530d] TruncatedStacktraces v1.4.0
  [9d95972d] TupleTools v1.5.0
  [9d95f2ec] TypedTables v1.4.6
  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.19.0
  [45397f5d] UnitfulLatexify v1.6.3
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.65
  [19fa3120] VertexSafeGraphs v0.2.0
  [ea10d353] WeakRefStrings v1.4.2
  [efce3f68] WoodburyMatrices v1.0.0
  [76eceee3] WorkerUtilities v1.6.1
  [0b7ba130] Blosc_jll v1.21.5+0
  [6e34b625] Bzip2_jll v1.0.8+1
  [4e9b3aee] CRlibm_jll v1.0.1+0
⌃ [83423d85] Cairo_jll v1.16.1+1
  [7bc98958] Cubature_jll v1.0.5+0
  [5ae413db] EarCut_jll v2.2.4+0
  [2702e6a9] EpollShim_jll v0.0.20230411+0
  [2e619515] Expat_jll v2.5.0+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [f5851436] FFTW_jll v3.3.10+0
  [a3f928ae] Fontconfig_jll v2.13.93+0
  [d7e528f0] FreeType2_jll v2.13.1+0
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.9+0
⌅ [d2c73de3] GR_jll v0.73.2+0
  [1b77fbbe] GSL_jll v2.7.2+0
  [78b55507] Gettext_jll v0.21.0+0
⌃ [7746bdde] Glib_jll v2.76.5+0
  [3b182d85] Graphite2_jll v1.3.14+0
⌃ [0234f1f7] HDF5_jll v1.14.2+1
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [e33a78d0] Hwloc_jll v2.10.0+0
  [905a6f67] Imath_jll v3.1.7+0
  [1d5cc7b8] IntelOpenMP_jll v2024.0.2+0
  [aacddb02] JpegTurbo_jll v3.0.2+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [1d63c593] LLVMOpenMP_jll v15.0.7+0
  [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.6.0+0
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.17.0+0
⌃ [4b2f31a3] Libmount_jll v2.35.0+0
⌅ [89763e89] Libtiff_jll v4.5.1+1
⌃ [38a345b3] Libuuid_jll v2.36.0+0
  [5ced341a] Lz4_jll v1.9.4+0
  [856f044c] MKL_jll v2024.0.0+0
  [7cb0a576] MPICH_jll v4.2.0+0
  [f1f71cc9] MPItrampoline_jll v5.3.2+0
  [9237b28f] MicrosoftMPI_jll v10.1.4+2
  [e7412a2a] Ogg_jll v1.3.5+1
  [18a262bb] OpenEXR_jll v3.1.4+0
  [fe0851c0] OpenMPI_jll v5.0.2+0
  [458c3c95] OpenSSL_jll v3.0.13+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [32165bc3] PMIx_jll v4.2.7+0
⌃ [36c8627f] Pango_jll v1.50.14+0
  [30392449] Pixman_jll v0.42.2+0
  [c0090381] Qt6Base_jll v6.5.3+1
  [f50d1b31] Rmath_jll v0.4.0+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.21.0+1
  [2381bf8a] Wayland_protocols_jll v1.31.0+0
  [02c8fc9c] XML2_jll v2.12.5+0
  [aed1982a] XSLT_jll v1.1.34+0
⌃ [ffd25f8a] XZ_jll v5.6.0+0
  [f67eecfb] Xorg_libICE_jll v1.0.10+1
  [c834827a] Xorg_libSM_jll v1.2.3+0
  [4f6342f7] Xorg_libX11_jll v1.8.6+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.11+0
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.4+0
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
  [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.6+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.0+0
  [3161d3a3] Zstd_jll v1.5.5+0
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
  [9a68df92] isoband_jll v0.2.3+0
⌃ [477f73a3] libaec_jll v1.0.6+1
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [1080aeaf] libevent_jll v2.1.13+1
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
⌃ [b53b4c65] libpng_jll v1.6.42+0
  [075b6546] libsixel_jll v1.10.3+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+0
  [eb928a42] prrte_jll v3.0.2+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [8bf52ea8] CRC32c
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.5+1
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+2
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2

• Output of versioninfo()

Julia Version 1.10.0
Commit 3120989f39b (2023-12-25 18:01 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 8 × 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, tigerlake)
  Threads: 5 on 8 virtual cores
Environment:
  JULIA_EDITOR = code
  JULIA_NUM_THREADS = 4

Additional context

This is on the example from the DDE website

[ChrisRackauckas]

DDEs have more memory usage required because of how it is required to handle the history. That would effect threading performance.

[H-Sax]

Would larger memory lead to better threading performance?

[H-Sax]

Does anyone know if there is any literature on this? I presume people have worked round this problem previously give DDE prevalence in systems biology.

[ChrisRackauckas]

If there was a maximal delay that was known, then it could optimize to that.