Van der Pol benchmark

Summary

Van der Pol is a low-dimensional, stiff, second-order ODE. We see that:

The EK1 is very well able to solve stiff problems.
Since Van der Pol is actually a second-order ODE, do solve it as a second-order ODE.
Use the TaylorInit or ForwardDiffInit initialization. While SimpleInit works well for lower orders, it fails for higher orders. And since Taylor-mode initialization is fast and works well, there is no reason not to use it.

Code:

using LinearAlgebra, Statistics, Distributions
using DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Plots
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:

function vanderpol!(du, u, p, t)
    du[1] = u[2]
    du[2] = p[1] * ((1 - u[1]^2) * u[2] - u[1])
end
p = [1e5]
tspan = (0.0, 2.0)
u0 = [2.0, 0.0]
prob = ODEProblem(vanderpol!, u0, tspan, p)

test_sol = solve(prob, RadauIIA5(), abstol=1/10^14, reltol=1/10^14)
plot(test_sol, title="Van der Pol Solution", legend=false, ylims=(-5, 5), xticks=:auto)

`EK1` across orders

Code:

DENSE = false;
SAVE_EVERYSTEP = false;

_setups = [
  "EK1($order)" => Dict(:alg => EK1(order=order, smooth=DENSE))
  for order in 3:7
]

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

abstols = 1.0 ./ 10.0 .^ (6:11)
reltols = 1.0 ./ 10.0 .^ (3:8)

wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = labels,
    appxsol = test_sol,
    dense = DENSE,
    save_everystep = SAVE_EVERYSTEP,
    maxiters = Int(1e7),
    numruns = 5,
)

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

_ref_setups = [
    "Rosenbrock23" => Dict(:alg => Rosenbrock23())
    "Rodas4P" => Dict(:alg => Rodas4P())
    "RadauIIA5" => Dict(:alg => RadauIIA5())
]
ref_labels = first.(_ref_setups)
ref_setups = last.(_ref_setups)
ref_wp_final = WorkPrecisionSet(
    prob, abstols, reltols, ref_setups;
    names = ref_labels,
    appxsol = test_sol,
    dense = false,
    save_everystep = false,
    maxiters = Int(1e7),
)
ref_wp_dense = WorkPrecisionSet(
    prob, abstols, reltols, ref_setups;
    names = ref_labels,
    appxsol = test_sol,
    dense = true,
    save_everystep = true,
    maxiters = Int(1e7),
)

plot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)

Discrete time-series errors (l2):

Code:

DENSE = true;
SAVE_EVERYSTEP = true;

_setups = [
  "EK1($order)" => Dict(:alg => EK1(order=order, smooth=DENSE))
  for order in 3:7
]

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

abstols = 1.0 ./ 10.0 .^ (6:11)
reltols = 1.0 ./ 10.0 .^ (3:8)

wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = labels,
    appxsol = test_sol,
    dense = DENSE,
    save_everystep = SAVE_EVERYSTEP,
    maxiters = Int(1e7),
    numruns = 5,
)

plot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))
plot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)

Interpolation errors (L2):

Code:

plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))
plot!(ref_wp_dense, x=:L2, 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!(2)

Comparison of the different initialization schemes

Code:

DENSE = false;
SAVE_EVERYSTEP = false;

abstols = 1.0 ./ 10.0 .^ (6:11)
reltols = 1.0 ./ 10.0 .^ (3:8)

orders = (3, 5, 8)
ps = []
for o in orders
    _setups = [
      "EK1($o) TaylorInit" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=TaylorModeInit(o)))
      "EK1($o) ForwardDiffInit" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ForwardDiffInit(o)))
      "EK1($o) SimpleInit" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=SimpleInit()))
      # "EK1($o) ClassicSolverInit" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ClassicSolverInit())) # unstable
    ]

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

    wp = WorkPrecisionSet(
        prob, abstols, reltols, setups;
        names = labels,
        appxsol = test_sol,
        dense = DENSE,
        save_everystep = SAVE_EVERYSTEP,
        maxiters = Int(1e7),
        numruns = 5,
    )

    p = plot(wp, color=[2 4 5 6], xticks = 10.0 .^ (-16:1:5))
    push!(ps, p)
end
plot(
    ps...,
    layout=(length(orders), 1),
    size = (1000, length(orders)*300),
    xlabel=["" "" "" "Error"],
)

Code:

DENSE = false;
SAVE_EVERYSTEP = false;

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (2:5)

_setups = [
    "EK1(8) TaylorInit(8)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(8)))
    "EK1(8) TaylorInit(7)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(7)))
    "EK1(8) TaylorInit(6)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(6)))
    "EK1(8) TaylorInit(5)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(5)))
    "EK1(8) TaylorInit(4)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(4)))
    "EK1(8) TaylorInit(3)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(3)))
    "EK1(8) TaylorInit(2)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(2)))
    # "EK1(8) TaylorInit(1)" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(1))) # fails, see above
]

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

wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = labels,
    appxsol = test_sol,
    dense = DENSE,
    save_everystep = SAVE_EVERYSTEP,
    maxiters = Int(1e7),
    numruns = 5,
)

plot(wp, palette=Plots.palette([:blue, :red], length(_setups)), xticks = 10.0 .^ (-16:1:5))

Solving the first- vs second-order ODE

Code:

function vanderpol2!(ddu, du, u, p, t)
    ddu[1] = p[1] * ((1 - u[1]^2) * du[1] - u[1])
end
p = [1e5]
tspan = (0.0, 2.0)
u0 = [2.0]
du0 = [0.0]
prob2 = SecondOrderODEProblem(vanderpol2!, du0, u0, tspan, p)

test_sol2 = solve(prob2, RadauIIA5(), abstol=1/10^14, reltol=1/10^14)
# plot(test_sol2, title="Van der Pol Solution (2nd order)", legend=false, ylims=(-5, 5), xticks=:auto)
nothing

Code:

DENSE = true;
SAVE_EVERYSTEP = true;

_setups = [
  "EK1(3) 1st order" => Dict(:alg => EK1(order=3, smooth=DENSE))
  "EK1(4) 1st order" => Dict(:alg => EK1(order=4, smooth=DENSE))
  "EK1(5) 1st order" => Dict(:alg => EK1(order=5, smooth=DENSE))
  "EK1(6) 1st order" => Dict(:alg => EK1(order=6, smooth=DENSE))
  "EK1(4) 2nd order" => Dict(:prob_choice => 2, :alg => EK1(order=4, smooth=DENSE))
  "EK1(5) 2nd order" => Dict(:prob_choice => 2, :alg => EK1(order=5, smooth=DENSE))
  "EK1(6) 2nd order" => Dict(:prob_choice => 2, :alg => EK1(order=6, smooth=DENSE))
  "EK1(7) 2nd order" => Dict(:prob_choice => 2, :alg => EK1(order=7, smooth=DENSE))
]

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

abstols = 1.0 ./ 10.0 .^ (6:11)
reltols = 1.0 ./ 10.0 .^ (3:8)

wp = WorkPrecisionSet(
    [prob, prob2], abstols, reltols, setups;
    names = labels,
    appxsol = [test_sol, test_sol2],
    dense = DENSE,
    save_everystep = SAVE_EVERYSTEP,
    maxiters = Int(1e7),
    numruns = 5,
)

color = [1 1 1 1 2 2 2 2]
plot(wp; x=:final, color)
plot!(ref_wp_dense, x=:final, color=:gray, alpha=0.7, linestyle=:dash)

Interpolation errors (L2):

Code:

plot(wp; x=:L2, color)
plot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)

Calibration

Code:

plot(wp; x=:final, y=:chi2_final, yguide="Chi-squared (final)", color)
plot_chisq_interval!(2)

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`
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  [91a5bcdd] Plots v1.39.0
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Info Packages marked with ⌃ have new versions available and may be upgradable.

Full manifest:

Pkg.status(mode=Pkg.PKGMODE_MANIFEST)

Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`
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⌅ [2efcf032] SymbolicIndexingInterface v0.2.2
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⌅ [89763e89] Libtiff_jll v4.5.1+1
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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`

Van der Pol benchmark

EK1 across orders

Calibration

Comparison of the different initialization schemes

Solving the first- vs second-order ODE

Calibration

Appendix

`EK1` across orders