OREGO benchmark

Summary

The EK1 is able to solve mass-matrix DAEs. To achieve low error, use order 4 or higher.
The order-to-error-tolerance heuristic holds: lower tolerance level $\rightarrow$ higher order.

Code:

using LinearAlgebra, Statistics, Distributions
using DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots
using ModelingToolkit
using ProbNumDiffEq

Plots.theme(
    :dao;
    markerstrokewidth=0.5,
    legend=:outertopright,
    margin=5Plots.mm,
    xticks=10.0 .^ (-16:1:16),
)

function plot_chisq_interval!(df, q=0.01)
    dist = Chisq(df)
    low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)
    hline!([low, high], linestyle=:dash, color=:black, label="",
           fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)
    hline!([mid], linestyle=:solid, color=:black, label="")
end

Code:

@variables t y1(t)=1.0 y2(t)=2.0 y3(t)=3.0
@parameters p1=77.27 p2=8.375e-6 p3=0.161
D = Differential(t)
eqs = [
  D(y1) ~ p1*(y2+y1*(1-p2*y1-y2))
  D(y2) ~ (y3-(1+y1)*y2)/p1
  D(y3) ~ p3*(y1-y3)
]
@named sys = ODESystem(eqs)
simpsys = structural_simplify(sys)
mmprob = ODEProblem(sys,[],(0.0,30.0))
daeprob = DAEProblem(sys,[D(y1)=>77.26935286375,
                          D(y2)=>-0.012941633234114146,
                          D(y3)=>-0.322],[],(0.0,30.0))
odaeprob = ODAEProblem(simpsys,[],(0.0,30.0))

ref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14)

plot(ref_sol, title="OREGO Solution", legend=false, xticks=:auto)

`EK1` across orders

Code:

DENSE = false;
SAVE_EVERYSTEP = false;

_setups = [
  "EK1($order)" => Dict(:alg => EK1(order=order, smooth=DENSE))
  for order in 2:6
]

labels = first.(_setups)
setups = last.(_setups)

abstols = 1.0 ./ 10.0 .^ (6:10)
reltols = 1.0 ./ 10.0 .^ (3:7)

wp = WorkPrecisionSet(
    mmprob, abstols, reltols, setups;
    names = labels,
    appxsol = ref_sol,
    dense = DENSE,
    save_everystep = SAVE_EVERYSTEP,
    numruns = 10,
    maxiters = Int(1e7),
)

plot(wp, palette=Plots.palette([:blue, :red], length(_setups)))


_ref_setups = [
    "Rosenbrock23" => Dict(:alg => Rosenbrock23())
    "Rodas4P" => Dict(:alg => Rodas4P())
    "RadauIIA" => Dict(:alg => RadauIIA5())
]
ref_labels = first.(_ref_setups)
ref_setups = last.(_ref_setups)
ref_wp = WorkPrecisionSet(
    mmprob, abstols, reltols, ref_setups;
    names = ref_labels,
    appxsol = ref_sol,
    dense = DENSE,
    save_everystep = SAVE_EVERYSTEP,
    maxiters = Int(1e7),
)
plot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=:dash)

Calibration

Code:

plot(wp; x=:final, y=:chi2_final, yguide="Chi-squared (final)",
     palette=Plots.palette([:blue, :red], length(_setups)))
plot_chisq_interval!(3)

Appendix

Computer information:

using InteractiveUtils
InteractiveUtils.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: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)
  Threads: 17 on 12 virtual cores
Environment:
  JULIA_NUM_THREADS = auto

Package information:

using Pkg
Pkg.status()

Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.44.1
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⌃ [961ee093] ModelingToolkit v8.73.2
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Info Packages marked with ⌃ have new versions available and may be upgradable.

Full manifest:

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  [ffd25f8a] XZ_jll v5.4.5+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
  [8f1865be] ZeroMQ_jll v4.3.4+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
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.40+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+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
  [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
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

OREGO benchmark

EK1 across orders

Calibration

Appendix

`EK1` across orders