ROBER Differential-Algebraic Equation (DAE) Work-Precision Diagrams

using OrdinaryDiffEq, DiffEqDevTools, Sundials, ModelingToolkit, ODEInterfaceDiffEq,
      Plots, DASSL, DASKR
using LinearAlgebra

@variables t y₁(t)=1.0 y₂(t)=0.0 y₃(t)=0.0
@parameters k₁=0.04 k₂=3e7 k₃=1e4
D = Differential(t)
eqs = [
  D(y₁) ~ -k₁*y₁ + k₃*y₂*y₃
  D(y₂) ~  k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2
  0 ~  y₁ + y₂ + y₃ - 1
]
@named sys = ODESystem(eqs)
simpsys = structural_simplify(sys)
mmprob = ODEProblem(sys,[],(0.0,1e5))
daeprob = DAEProblem(sys,[D(y₁)=>-0.04,
                              D(y₂)=>0.04,
                              D(y₃)=>0.0],[],(0.0,1e5))
odaeprob = ODAEProblem(simpsys,[],(0.0,1e5))

ref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14);
ode_ref_sol = solve(odaeprob,CVODE_BDF(),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])

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","ROBERDAE.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`
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  [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