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

using OrdinaryDiffEq, DiffEqDevTools, Sundials, ModelingToolkit, ODEInterfaceDiffEq,
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 => 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()),
]
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 => 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 => 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()),
]
gr()
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

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
[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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[8ba89e20] Distributed
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[b27032c2] LibCURL v0.6.3
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[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