Chemical Akzo Nobel Differential-Algebraic Equation (DAE) Work-Precision Diagrams
using OrdinaryDiffEq, DiffEqDevTools, Sundials, ModelingToolkit, ODEInterfaceDiffEq,
Plots, DASSL, DASKR
using LinearAlgebra
ModelingToolkit.@parameters begin
k₁=18.7
k₂=0.58
k₃=0.09
k₄=0.42
kbig=34.4
kla=3.3
ks=115.83
po2=0.9
hen=737
end
@variables begin
t
y₁(t) = 0.444
y₂(t) = 0.00123
y₃(t) = 0.0
y₄(t) = 0.007
y₅(t) = 1.0
y₆(t) = 115.83*0.444*0.007 # ks*y₁*y₄
end
D = Differential(t)
r₁ = k₁ * (y₁^4.)*sqrt(abs(y₂))
r₂ = k₂ * y₃ * y₄
r₃ = k₂/kbig * y₁ * y₅
r₄ = k₃*y₁*(y₄^2)
r₅ = k₄*(y₆^2)*sqrt(abs(y₂))
fin = kla*(po2/hen-y₂)
eqs = [
D(y₁) ~ -2. * r₁ + r₂ - r₃ - r₄
D(y₂) ~ -0.5 * r₁ - r₄ - 0.5*r₅ + fin
D(y₃) ~ r₁ - r₂ + r₃
D(y₄) ~ -r₂ + r₃ - 2. * r₄
D(y₅) ~ r₂ - r₃ + r₅
0. ~ ks * y₁ * y₄ - y₆
]
ModelingToolkit.@named sys = ModelingToolkit.ODESystem(eqs)
tspan = (0.0, 180.0)
mmprob = ODEProblem(sys, [], tspan)
sol = solve(mmprob, Rodas4(),abstol=1/10^14,reltol=1/10^14)
odaeprob = ODAEProblem(structural_simplify(sys),[],tspan)
ode_ref_sol = solve(odaeprob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14);
du = mmprob.f(mmprob.u0,mmprob.p,0.0)
du0 = D.(states(sys)) .=> du
daeprob = DAEProblem(sys,du0,[],tspan)
ref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14);
probs = [mmprob,daeprob,odaeprob]
refs = [ref_sol,ref_sol,ode_ref_sol];
plot(ode_ref_sol, vars = [y₁,y₂,y₃,y₄,y₅,y₆])
High Tolerances
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
#Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 1, :alg=>QNDF()),
Dict(:prob_choice => 1, :alg=>rodas()),
Dict(:prob_choice => 1, :alg=>radau()),
Dict(:prob_choice => 1, :alg=>RadauIIA5()),
Dict(:prob_choice => 2, :alg=>DFBDF()),
Dict(:prob_choice => 2, :alg=>IDA()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (2:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 3, :alg=>Rosenbrock23()),
Dict(:prob_choice => 3, :alg=>Rodas4()),
Dict(:prob_choice => 3, :alg=>CVODE_BDF()),
Dict(:prob_choice => 3, :alg=>TRBDF2()),
Dict(:prob_choice => 3, :alg=>KenCarp4()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (3:5);
setups = [Dict(:prob_choice => 3, :alg=>Rosenbrock23()),
Dict(:prob_choice => 3, :alg=>Rodas4()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 2, :alg=>DASSL.dassl()),
Dict(:prob_choice => 2, :alg=>DASKR.daskr()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
Timeseries Errors
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
#Dict(:prob_choice => 1, :alg=>FBDF()), # too slow
Dict(:prob_choice => 1, :alg=>QNDF()),
Dict(:prob_choice => 1, :alg=>rodas()),
Dict(:prob_choice => 1, :alg=>radau()),
Dict(:prob_choice => 1, :alg=>RadauIIA5()),
Dict(:prob_choice => 2, :alg=>DFBDF()),
Dict(:prob_choice => 2, :alg=>IDA()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;error_estimate = :l2,
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (2:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 3, :alg=>Rosenbrock23()),
Dict(:prob_choice => 3, :alg=>Rodas4()),
Dict(:prob_choice => 3, :alg=>CVODE_BDF()),
Dict(:prob_choice => 3, :alg=>TRBDF2()),
Dict(:prob_choice => 3, :alg=>KenCarp4()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;error_estimate = :l2,
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
Low Tolerances
This is the speed at lower tolerances, measuring what's good when accuracy is needed.
abstols = 1.0 ./ 10.0 .^ (7:12)
reltols = 1.0 ./ 10.0 .^ (4:9)
setups = [Dict(:prob_choice => 1, :alg=>Rodas5()),
Dict(:prob_choice => 3, :alg=>Rodas5()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 3, :alg=>Rodas4()),
#Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 1, :alg=>QNDF()),
Dict(:prob_choice => 1, :alg=>rodas()),
Dict(:prob_choice => 1, :alg=>radau()),
Dict(:prob_choice => 1, :alg=>RadauIIA5()),
Dict(:prob_choice => 2, :alg=>DFBDF()),
Dict(:prob_choice => 2, :alg=>IDA()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
wp = WorkPrecisionSet(probs,abstols,reltols,setups;error_estimate = :l2,
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
DASKR fails at too low of tolerances, so pull back for a comparison.
abstols = 1.0 ./ 10.0 .^ (7:9)
reltols = 1.0 ./ 10.0 .^ (4:6)
setups = [Dict(:prob_choice => 1, :alg=>Rodas5()),
Dict(:prob_choice => 3, :alg=>Rodas5()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 3, :alg=>Rodas4()),
Dict(:prob_choice => 2, :alg=>DFBDF()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 2, :alg=>DASKR.daskr()),
]
gr()
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)
Conclusion
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/DAE","ChemicalAkzoNobel.jmd")
Computer Information:
Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DAE/Project.toml`
[165a45c3] DASKR v2.8.0
[e993076c] DASSL v2.6.0
[f3b72e0c] DiffEqDevTools v2.31.0
[961ee093] ModelingToolkit v8.18.0
[09606e27] ODEInterfaceDiffEq v3.10.1
[1dea7af3] OrdinaryDiffEq v6.19.2
[91a5bcdd] Plots v1.31.4
[31c91b34] SciMLBenchmarks v0.1.0
[c3572dad] Sundials v4.9.4
And the full manifest:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DAE/Manifest.toml`
[c3fe647b] AbstractAlgebra v0.27.0
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[aed1982a] XSLT_jll v1.1.34+0
[4f6342f7] Xorg_libX11_jll v1.6.9+4
[0c0b7dd1] Xorg_libXau_jll v1.0.9+4
[935fb764] Xorg_libXcursor_jll v1.2.0+4
[a3789734] Xorg_libXdmcp_jll v1.1.3+4
[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 v0.1.0+3
[c7cfdc94] Xorg_libxcb_jll v1.13.0+3
[cc61e674] Xorg_libxkbfile_jll v1.1.0+4
[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 v1.4.2+4
[33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
[c5fb5394] Xorg_xtrans_jll v1.4.0+3
[8f1865be] ZeroMQ_jll v4.3.4+0
[3161d3a3] Zstd_jll v1.5.2+0
[a4ae2306] libaom_jll v3.4.0+0
[0ac62f75] libass_jll v0.15.1+0
[f638f0a6] libfdk_aac_jll v2.0.2+0
[b53b4c65] libpng_jll v1.6.38+0
[a9144af2] libsodium_jll v1.0.20+0
[f27f6e37] libvorbis_jll v1.3.7+1
[1270edf5] x264_jll v2021.5.5+0
[dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v0.9.1+5
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8bb1440f] DelimitedFiles
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[b27032c2] LibCURL v0.6.3
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.8.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.0
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v0.5.2+0
[deac9b47] LibCURL_jll v7.81.0+0
[29816b5a] LibSSH2_jll v1.10.2+0
[c8ffd9c3] MbedTLS_jll v2.28.0+0
[14a3606d] MozillaCACerts_jll v2022.2.1
[4536629a] OpenBLAS_jll v0.3.20+0
[05823500] OpenLibm_jll v0.8.1+0
[efcefdf7] PCRE2_jll v10.40.0+0
[bea87d4a] SuiteSparse_jll v5.10.1+0
[83775a58] Zlib_jll v1.2.12+3
[8e850b90] libblastrampoline_jll v5.1.0+0
[8e850ede] nghttp2_jll v1.41.0+1
[3f19e933] p7zip_jll v17.4.0+0