Doyle-Fuller-Newman (DFN) Battery Model Initialization

These benchmarks compares the runtime and error for a range of nonlinear solvers. The solvers are implemented in NonlinearProblemLibrary.jl, where you can find the problem function declarations. We test the following solvers:

• NonlinearSolve.jl's Newton Raphson method (NewtonRaphson()).
• NonlinearSolve.jl's Newton trust region method (TrustRegion()).
• NonlinearSolve.jl's Levenberg-Marquardt method (LevenbergMarquardt()).
• NonlinearSolve.jl's Broyden method (Broyden()).
• MINPACK's Modified Powell method (CMINPACK(method=:hybr)).
• MINPACK's Levenberg-Marquardt method (CMINPACK(method=:lm)).
• NLsolveJL's Newton Raphson (NLsolveJL(method=:newton)).
• NLsolveJL's Newton trust region (NLsolveJL()).
• NLsolveJL's Anderson acceleration (NLsolveJL(method=:anderson)).
• Sundials's Newton-Krylov method (KINSOL()).

Setup

Fetch required packages.

using NonlinearSolve, LinearSolve, StaticArrays, Sundials, SpeedMapping,
      BenchmarkTools, LinearAlgebra, DiffEqDevTools, PolyesterForwardDiff, CairoMakie,
      RecursiveFactorization, Enzyme
using SciMLLogging
import MINPACK, NLsolve, PETSc
import LineSearches

const RUS = RadiusUpdateSchemes;

Declare the benchmarked solvers (and their names and plotting options).

HagerZhang() = LineSearchesJL(; method = LineSearches.HagerZhang())
MoreThuente() = LineSearchesJL(; method = LineSearches.MoreThuente())

GC.enable(false) # for PETSc

solvers_all = [
    (; pkg = :nonlinearsolve, name = "Default PolyAlgorithm",
        solver = Dict(:alg => FastShortcutNonlinearPolyalg(; autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve, name = "Newton Raphson",
        solver = Dict(:alg => NewtonRaphson(; autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Newton Raphson (HagerZhang LineSearch)",
        solver = Dict(:alg => NewtonRaphson(; linesearch = HagerZhang(), autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Newton Raphson (MoreThuente LineSearch)",
        solver = Dict(:alg => NewtonRaphson(; linesearch = MoreThuente(), autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Newton Raphson (BackTracking LineSearch)",
        solver = Dict(:alg => NewtonRaphson(; linesearch = BackTracking(), autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Newton Krylov with GMRES",
        solver = Dict(:alg => NewtonRaphson(; linsolve = KrylovJL_GMRES(), autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve, name = "DFSane", solver = Dict(:alg => DFSane())),
    (; pkg = :nonlinearsolve, name = "Trust Region",
        solver = Dict(:alg => TrustRegion(; autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Trust Region (NLsolve Update)",
        solver = Dict(:alg => TrustRegion(; radius_update_scheme = RUS.NLsolve, autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Trust Region (Nocedal Wright)",
        solver = Dict(:alg => TrustRegion(; radius_update_scheme = RUS.NocedalWright, autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Trust Region (Hei)",
        solver = Dict(:alg => TrustRegion(; radius_update_scheme = RUS.Hei, autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Trust Region (Yuan)",
        solver = Dict(:alg => TrustRegion(; radius_update_scheme = RUS.Yuan, autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Trust Region (Bastin)",
        solver = Dict(:alg => TrustRegion(; radius_update_scheme = RUS.Bastin, autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Trust Region (Fan)",
        solver = Dict(:alg => TrustRegion(; radius_update_scheme = RUS.Fan, autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve, name = "Levenberg-Marquardt",
        solver = Dict(:alg => LevenbergMarquardt(; autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Levenberg-Marquardt with Cholesky",
        solver = Dict(:alg => LevenbergMarquardt(;
            linsolve = CholeskyFactorization(), autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Levenberg-Marquardt (No Geodesic Accln.)",
        solver = Dict(:alg => LevenbergMarquardt(; disable_geodesic = Val(true), autodiff = AutoEnzyme()))),
    (; pkg = :nonlinearsolve,
        name = "Levenberg-Marquardt (No Geodesic Accln.) with Cholesky",
        solver = Dict(:alg => LevenbergMarquardt(; disable_geodesic = Val(true),
            linsolve = CholeskyFactorization(), autodiff = AutoEnzyme()))), (;
        pkg = :wrapper, name = "Newton Raphson [Sundials]",
        solver = Dict(:alg => KINSOL(; linear_solver = :LapackDense, maxsetupcalls = 1))),
    (; pkg = :wrapper,
        name = "Newton Krylov [Sundials]",
        solver = Dict(:alg => KINSOL(; linear_solver = :GMRES, maxsetupcalls = 1, krylov_dim = 1000))),
    (; pkg = :wrapper,
        name = "Newton Raphson with LineSearch [Sundials]",
        solver = Dict(:alg => KINSOL(; globalization_strategy = :LineSearch, maxsetupcalls = 1))), (;
        pkg = :wrapper, name = "Newton Raphson [NLsolve.jl]",
        solver = Dict(:alg => NLsolveJL(; method = :newton, autodiff = :forward))),
    (; pkg = :wrapper, name = "Trust Region [NLsolve.jl]",
        solver = Dict(:alg => NLsolveJL(; autodiff = :forward))), (;
        pkg = :wrapper, name = "Modified Powell [MINPACK]",
        solver = Dict(:alg => CMINPACK(; method = :hybr))),
    (; pkg = :wrapper, name = "Levenberg-Marquardt [MINPACK]",
        solver = Dict(:alg => CMINPACK(; method = :lm))), (;
        pkg = :wrapper, name = "Speed Mapping [SpeedMapping.jl]",
        solver = Dict(:alg => SpeedMappingJL())),
    (; pkg = :wrapper,
        name = "Newton Raphson [PETSc]",
        solver = Dict(:alg => PETScSNES(; snes_type = "newtonls", snes_linesearch_type = "basic"))),
    (; pkg = :wrapper,
        name = "Newton Raphson with QR [PETSc]",
        solver = Dict(:alg => PETScSNES(;
            snes_type = "newtonls", snes_linesearch_type = "basic", pc_type = "qr"))),
    (; pkg = :wrapper, name = "Newton Raphson with BackTracking [PETSc]",
        solver = Dict(:alg => PETScSNES(; snes_type = "newtonls"))),
    (; pkg = :wrapper, name = "Newton Raphson with BackTracking & QR [PETSc]",
        solver = Dict(:alg => PETScSNES(; snes_type = "newtonls", pc_type = "qr"))),
    (; pkg = :wrapper, name = "Trust Region [PETSc]",
        solver = Dict(:alg => PETScSNES(; snes_type = "newtontr"))),
    (; pkg = :wrapper,
        name = "Newton Krylov with GMRES [PETSc]",
        solver = Dict(:alg => PETScSNES(;
            snes_type = "newtonls", snes_linesearch_type = "basic",
            ksp_type = "gmres", ksp_gmres_restart = 1000)))
];

Setup the problem

function f!(out, u, p = nothing)
    out[1] = -u[33]
    out[2] = -u[32]
    out[3] = -u[31]
    out[4] = 1.9876764062810574e10(u[1] + u[4]) - 1.9876764062810574e10u[23]
    out[5] = -u[2] + (-1.5546404484393263e-11exp(-19.460872248113507(-0.4 - u[10] + u[26])))
    out[6] = -1.9876764062810574e10u[14] + 1.9876764062810574e10(u[3] + u[6])
    out[7] = -1.9876764062810574e10u[4] +
             114676.11822324689(-exp(-19.460872248113507(-0.6608489145760508 + u[25])) +
                                exp(19.460872248113507(-0.6608489145760508 + u[25])))
    out[8] = -1.9876764062810574e10u[12] + 1.9876764062810574e10(u[2] + u[5])
    out[9] = -1.9876764062810574e10u[6] +
             114676.1182232469(-exp(-19.460872248113507(-0.6608489145760508 - u[11] +
                                                        u[27])) +
                               exp(19.460872248113507(-0.6608489145760508 - u[11] + u[27])))
    out[10] = -4.284490145672665e10u[19] + 4.284490145672665e10(u[31] + u[7])
    out[11] = -4.284490145672665e10u[21] + 4.284490145672665e10(u[32] + u[8])
    out[12] = -4.284490145672665e10u[22] + 4.284490145672665e10(u[33] + u[9])
    out[13] = 0.025692579121085843(7.680104664733624e7(u[10] - u[11]) +
                                   7.680104664733624e7u[10]) - 1.793773306620288e9u[12]
    out[14] = -u[3] +
              (-1.5546404484393263e-11exp(-19.460872248113507(-0.4 - u[11] + u[27])))
    out[15] = -1.9876764062810574e10u[5] +
              114676.1182232469(-exp(-19.460872248113507(-0.6608489145760506 - u[10] +
                                                         u[26])) +
                                exp(19.460872248113507(-0.6608489145760506 - u[10] +
                                                       u[26])))
    out[16] = 0.025692579121085843(7.680104664733624e7(-u[10] + u[11]) +
                                   1.4529008434739566e8(u[11] - u[13])) -
              1.793773306620288e9u[14]
    out[17] = -1.793773306620288e9u[14] - 1.4404300298567445e9(-u[26] + u[27])
    out[18] = 0.025692579121085843(5.1142109690283257e8(-u[11] + u[13]) +
                                   4.7254130462088e9(u[13] - u[15]))
    out[19] = 0.025692579121085843(4.7254130462088e9(-u[13] + u[15]) +
                                   4.7254130462088e9(u[15] - u[16]))
    out[20] = 0.025692579121085843(4.7254130462088e9(-u[15] + u[16]) +
                                   2.3040372207628164e8(u[16] - u[17]))
    out[21] = 0.025692579121085843(7.200116314883803e7(-u[16] + u[17]) +
                                   3.6900178974461965e7(u[17] - u[18])) -
              2.193876971198113e9u[19]
    out[22] = -4.284490145672665e10u[7] +
              147554.10828979727(-exp(-19.460872248113507(-3.3618450059739535 - u[17] +
                                                          u[28])) +
                                 exp(19.460872248113507(-3.3618450059739535 - u[17] +
                                                        u[28])))
    out[23] = 0.025692579121085843(3.6900178974461965e7(-u[17] + u[18]) +
                                   3.6900178974461965e7(u[18] - u[20])) -
              2.193876971198113e9u[21]
    out[24] = -4.284490145672665e10u[8] +
              147554.10828979727(-exp(-19.460872248113507(-3.3618450059739535 - u[18] +
                                                          u[29])) +
                                 exp(19.460872248113507(-3.3618450059739535 - u[18] +
                                                        u[29])))
    out[25] = 948060.7678835923(-u[18] + u[20]) - 2.193876971198113e9u[22]
    out[26] = -u[1] + (-1.5546404484393263e-11exp(-19.460872248113507(-0.4 + u[25])))
    out[27] = -2.193876971198113e9u[22] +
              0.025692579121085843(-37499.99999999999u[24] -
                                   8.296874999999998e10(-u[29] + u[30]))
    out[28] = -1.793773306620288e9u[23] +
              0.025692579121085843(34090.90909090909u[24] -
                                   5.6064049586776855e10(u[25] - u[26]))
    out[29] = -1.793773306620288e9u[12] +
              0.025692579121085843(-5.6064049586776855e10(-u[25] + u[26]) -
                                   5.6064049586776855e10(u[26] - u[27]))
    out[30] = -2.193876971198113e9u[19] - 2.1316811739525905e9(u[28] - u[29])
    out[31] = -2.193876971198113e9u[21] +
              0.025692579121085843(-8.296874999999998e10(-u[28] + u[29]) -
                                   8.296874999999998e10(u[29] - u[30]))
    out[32] = -4.284490145672665e10u[9] +
              147554.10828979727(-exp(-19.460872248113507(-3.3618450059739535 - u[20] +
                                                          u[30])) +
                                 exp(19.460872248113507(-3.3618450059739535 - u[20] +
                                                        u[30])))
    out[33] = 292.3000724036127 + u[24]
    nothing
end

n = 1
x_sol = [
    -3.889310081682032e-13,
    -5.690845522092043e-13,
    -1.4900105367898274e-12,
    -2.1680981422696e-5,
    -3.284624075480569e-5,
    -8.820027287447222e-5,
    9.53999632159426e-5,
    2.1138249693289567e-5,
    1.1829446876191545e-5,
    0.019709320908045884,
    0.06927785744111935,
    -3.2846241323890243e-5,
    0.13786323434647954,
    -8.820027436448276e-5,
    0.14528607936456214,
    0.15270892438264475,
    0.3049460860584471,
    0.3812355737657502,
    9.53999632159426e-5,
    0.40860971681949443,
    2.1138249693289567e-5,
    1.1829446876191545e-5,
    -2.1680981811627007e-5,
    -292.3000724036127,
    0.5895178515117894,
    0.5896685912243755,
    0.5897784273806014,
    3.837532182598256,
    3.8376303660343676,
    3.837750304468262,
    0.0,
    0.0,
    0.0
]
x_start = zeros(length(x_sol))
x_start[25:27] .= 0.6608489145760508
x_start[28:30] .= 3.3618450059739433

dict = Dict("n" => n, "start" => x_start, "sol" => x_sol,
    "title" => "Doyle-Fuller-Newman (DFN) Battery Model Initialization")

testcase = (; prob = NonlinearProblem(f!, dict["start"]), true_sol = dict["sol"])

(prob = SciMLBase.NonlinearProblem{Vector{Float64}, true, SciMLBase.NullPar
ameters, SciMLBase.NonlinearFunction{true, SciMLBase.AutoSpecialize, typeof
(Main.var"##WeaveSandBox#232".f!), LinearAlgebra.UniformScaling{Bool}, Noth
ing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing
, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, No
thing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciML
Base.StandardNonlinearProblem, Nothing, Nothing}(SciMLBase.NonlinearFunctio
n{true, SciMLBase.AutoSpecialize, typeof(Main.var"##WeaveSandBox#232".f!), 
LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Not
hing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT
_OBSERVED_NO_TIME), Nothing, Nothing, Nothing, Nothing}(Main.var"##WeaveSan
dBox#232".f!, LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, n
othing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciM
LBase.DEFAULT_OBSERVED_NO_TIME, nothing, nothing, nothing, nothing), [0.0, 
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.6608489145760508, 0.
6608489145760508, 0.6608489145760508, 3.3618450059739433, 3.361845005973943
3, 3.3618450059739433, 0.0, 0.0, 0.0], SciMLBase.NullParameters(), SciMLBas
e.StandardNonlinearProblem(), nothing, nothing, Base.Pairs{Symbol, Union{},
 Tuple{}, @NamedTuple{}}()), true_sol = [-3.889310081682032e-13, -5.6908455
22092043e-13, -1.4900105367898274e-12, -2.1680981422696e-5, -3.284624075480
569e-5, -8.820027287447222e-5, 9.53999632159426e-5, 2.1138249693289567e-5, 
1.1829446876191545e-5, 0.019709320908045884  …  -292.3000724036127, 0.58951
78515117894, 0.5896685912243755, 0.5897784273806014, 3.837532182598256, 3.8
376303660343676, 3.837750304468262, 0.0, 0.0, 0.0])

Sets tolerances.

abstols = 1.0 ./ 10.0 .^ (3:0.5:6)
reltols = 1.0 ./ 10.0 .^ (3:0.5:6);

Prepares various helper functions for benchmarking a specific problem.

function log_msg(msg; kwargs...)
    if startswith(msg, "[Info]")
        @info msg
    elseif startswith(msg, "[Warn]")
        @warn msg
    elseif startswith(msg, "[Error]")
        @error msg
    else
        @info msg
    end
    Base.printstyled(msg; kwargs...)
    return
end

function check_solver(prob, solver)
    try
        sol = solve(prob.prob, solver.solver[:alg]; abstol = 1e-4, reltol = 1e-4,
            maxiters = 10000)
        err = norm(sol.resid, Inf)
        if !SciMLBase.successful_retcode(sol.retcode)
            log_msg(
                "[Warn] Solver $(solver.name) returned retcode $(sol.retcode) with an residual norm = $(norm(sol.resid)).\n";
                color = :red)
            return false
        elseif err > 1e3
            log_msg(
                "[Warn] Solver $(solver.name) had a very large residual (norm = $(norm(sol.resid))).\n";
                color = :red)
            return false
        elseif isinf(err) || isnan(err)
            log_msg("[Warn] Solver $(solver.name) had a residual of $(err).\n";
                color = :red)
            return false
        end
        log_msg(
            "[Info] Solver $(solver.name) successfully solved the problem (norm = $(norm(sol.resid))).\n";
            color = :green)
    catch e
        log_msg("[Warn] Solver $(solver.name) threw an error: $e.\n"; color = :red)
        return false
    end
    return true
end

function generate_wpset(prob, solvers)
    # Finds the solvers that can solve the problem
    successful_solvers = filter(solver -> check_solver(prob, solver), solvers)

    return WorkPrecisionSet(prob.prob, abstols, reltols,
        getfield.(successful_solvers, :solver);
        names = getfield.(successful_solvers, :name), numruns = 50, error_estimate = :l∞,
        maxiters = 10000, verbose = SciMLLogging.Standard()),
    successful_solvers
end

generate_wpset (generic function with 1 method)

Benchmarks

We here run benchmarks for each of the solvers.

Problem

wp_set, successful_solvers = generate_wpset(testcase, solvers_all);

[Info] Solver Default PolyAlgorithm successfully solved the problem (norm =
 1.5941038377860503e-6).
[Info] Solver Newton Raphson successfully solved the problem (norm = 1.5941
038377860503e-6).
[Warn] Solver Newton Raphson (HagerZhang LineSearch) returned retcode MaxIt
ers with an residual norm = 196.3434604448566.
[Warn] Solver Newton Raphson (MoreThuente LineSearch) returned retcode MaxI
ters with an residual norm = 208.56523082587998.
[Warn] Solver Newton Raphson (BackTracking LineSearch) returned retcode Max
Iters with an residual norm = 205.06578109867118.
[Warn] Solver Newton Krylov with GMRES returned retcode Unstable with an re
sidual norm = 292.3000724036127.
[Warn] Solver DFSane returned retcode Stalled with an residual norm = 292.3
000724036127.
[Info] Solver Trust Region successfully solved the problem (norm = 6.873726
32864865e-7).
[Info] Solver Trust Region (NLsolve Update) successfully solved the problem
 (norm = 6.873420050791921e-7).
[Info] Solver Trust Region (Nocedal Wright) successfully solved the problem
 (norm = 1.593960440044342e-6).
[Info] Solver Trust Region (Hei) successfully solved the problem (norm = 7.
52082243191495e-7).
[Warn] Solver Trust Region (Yuan) returned retcode MaxIters with an residua
l norm = 116.21621694755135.
[Info] Solver Trust Region (Bastin) successfully solved the problem (norm =
 7.519362935748752e-7).
[Info] Solver Trust Region (Fan) successfully solved the problem (norm = 1.
5941128511089634e-6).
[Info] Solver Levenberg-Marquardt successfully solved the problem (norm = 7
.38426369554924e-5).
[Warn] Solver Levenberg-Marquardt with Cholesky returned retcode Stalled wi
th an residual norm = 0.0003179815605994228.
[Info] Solver Levenberg-Marquardt (No Geodesic Accln.) successfully solved 
the problem (norm = 5.246607252366464e-7).
[Warn] Solver Levenberg-Marquardt (No Geodesic Accln.) with Cholesky return
ed retcode InternalLinearSolveFailed with an residual norm = 0.000301547349
8789062.
[Info] Solver Newton Raphson [Sundials] successfully solved the problem (no
rm = 1.6205733409240167e-6).
[Warn] Solver Newton Krylov [Sundials] had a very large residual (norm = In
f).
[Warn] Solver Newton Raphson with LineSearch [Sundials] returned retcode Fa
ilure with an residual norm = 203.8024932082229.
[Info] Solver Newton Raphson [NLsolve.jl] successfully solved the problem (
norm = 1.5941104186169045e-6).
[Info] Solver Trust Region [NLsolve.jl] successfully solved the problem (no
rm = 6.940612093440701e-7).
[Warn] Solver Modified Powell [MINPACK] returned retcode Failure with an re
sidual norm = 292.3000724036127.
[Warn] Solver Levenberg-Marquardt [MINPACK] returned retcode Failure with a
n residual norm = 292.3000724036127.
[Warn] Solver Speed Mapping [SpeedMapping.jl] returned retcode Failure with
 an residual norm = 773.9600302958914.
[Warn] Solver Newton Raphson [PETSc] returned retcode Failure with an resid
ual norm = 8.918598151838181e8.
[Warn] Solver Newton Raphson with QR [PETSc] returned retcode Failure with 
an residual norm = 8.918620951328129e8.
[Warn] Solver Newton Raphson with BackTracking [PETSc] returned retcode Fai
lure with an residual norm = 280.83522493021263.
[Warn] Solver Newton Raphson with BackTracking & QR [PETSc] returned retcod
e Failure with an residual norm = 280.82134250907274.
[Warn] Solver Trust Region [PETSc] returned retcode Failure with an residua
l norm = 202.00730475517162.
[Warn] Solver Newton Krylov with GMRES [PETSc] returned retcode Failure wit
h an residual norm = 8.918598151838181e8.

Plot and Save the Plot

fig = begin
    LINESTYLES = Dict(:nonlinearsolve => :solid, :simplenonlinearsolve => :dash,
        :wrapper => :dot)
    ASPECT_RATIO = 0.7
    WIDTH = 1200
    HEIGHT = round(Int, WIDTH * ASPECT_RATIO)
    STROKEWIDTH = 2.5

    colors = cgrad(:tableau_20, length(successful_solvers); categorical = true)
    cycle = Cycle([:marker], covary = true)
    plot_theme = Theme(Lines = (; cycle), Scatter = (; cycle))

    idxs = sortperm(median.(getfield.(wp_set.wps, :times)))

    with_theme(plot_theme) do
        fig = Figure(; size = (WIDTH, HEIGHT))
        # `textbf` doesn't work
        ax = Axis(fig[1, 1], ylabel = L"Time $\mathbf{(s)}$",
            xlabelsize = 22, ylabelsize = 22,
            xlabel = L"Error: $\mathbf{||f(u^\ast)||_\infty}$",
            xscale = log10, yscale = log10, xtickwidth = STROKEWIDTH,
            ytickwidth = STROKEWIDTH, spinewidth = STROKEWIDTH,
            xticklabelsize = 20, yticklabelsize = 20)

        ls, scs = [], []

        for (i, (wp, solver)) in enumerate(zip(wp_set.wps[idxs], successful_solvers[idxs]))
            (; name, times, errors) = wp
            errors = [err.l∞ for err in errors]
            l = lines!(ax, errors, times; linestyle = LINESTYLES[solver.pkg], label = name,
                linewidth = 5, color = colors[i])
            sc = scatter!(
                ax, errors, times; label = name, markersize = 16, strokewidth = 3,
                color = colors[i])
            push!(ls, l)
            push!(scs, sc)
        end

        axislegend(ax, [[l, sc] for (l, sc) in zip(ls, scs)],
            [solver.name for solver in successful_solvers[idxs]], "Successful Solvers";
            framevisible = true, framewidth = STROKEWIDTH, orientation = :vertical,
            titlesize = 20, nbanks = 1, labelsize = 16,#  margin = (0.0, 80.0, 0.0, 0.0),
            tellheight = false, tellwidth = true, patchsize = (40.0f0, 20.0f0),
            position = :rb)

        fig[0, :] = Label(fig,
            "Doyle-Fuller-Newman (DFN) Battery Model Initialization: Work Precision Diagram",
            fontsize = 24, tellwidth = false, font = :bold)

        fig
    end
end

save("battery_problem_work_precision.svg", fig)

CairoMakie.Screen{SVG}

Summary of successful solvers

Finally, we print a summary of which solvers successfully solved which problems.

solver_successes = [(solver in successful_solvers) ? "✔" : "✖" for solver in solvers_all];

using PrettyTables
io = IOBuffer()
println(io, "```@raw html")
pretty_table(io, reshape(solver_successes, 1, :); backend = :html,
    column_labels = getfield.(solvers_all, :name), alignment = :c)
println(io, "```")
Docs.Text(String(take!(io)))

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/NonlinearProblem","nonlinear_battery_problem.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NonlinearProblem/Project.toml`
⌅ [2169fc97] AlgebraicMultigrid v1.2.0
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  [13f3f980] CairoMakie v0.15.13
⌃ [2b5f629d] DiffEqBase v7.12.0
⌃ [f3b72e0c] DiffEqDevTools v3.2.0
⌃ [a0c0ee7d] DifferentiationInterface v0.7.20
⌃ [7da242da] Enzyme v0.13.198
  [40713840] IncompleteLU v0.2.1
⌃ [b964fa9f] LaTeXStrings v1.4.0
⌃ [d3d80556] LineSearches v7.5.1
⌅ [7ed4a6bd] LinearSolve v3.87.0
  [4854310b] MINPACK v1.3.0
⌅ [2774e3e8] NLsolve v4.5.1
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⌃ [8913a72c] NonlinearSolve v4.21.0
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  [a6db7da4] SciMLLogging v2.0.4
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⌃ [727e6d20] SimpleNonlinearSolve v2.13.1
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⌃ [90137ffa] StaticArrays v1.9.18
⌃ [c3572dad] Sundials v6.4.2
  [0c5d862f] Symbolics v7.36.0
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`

And the full manifest:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NonlinearProblem/Manifest.toml`
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  [e7412a2a] Ogg_jll v1.3.6+0
  [656ef2d0] OpenBLAS32_jll v0.3.34+0
  [6cdc7f73] OpenBLASConsistentFPCSR_jll v0.3.34+0
⌃ [18a262bb] OpenEXR_jll v3.4.13+0
  [fe0851c0] OpenMPI_jll v5.0.11+0
⌃ [9bd350c2] OpenSSH_jll v10.4.1+0
  [458c3c95] OpenSSL_jll v3.5.7+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
⌃ [8fa3689e] PETSc_jll v3.22.1+0
  [36c8627f] Pango_jll v1.58.0+0
  [30392449] Pixman_jll v0.46.4+0
  [f50d1b31] Rmath_jll v0.5.2+0
⌃ [aabda75e] SCALAPACK32_jll v2.2.300+0
  [ca45d3f4] SuiteSparse32_jll v7.12.1+0
  [fb77eaff] Sundials_jll v7.5.0+0
⌅ [02c8fc9c] XML2_jll v2.13.9+0
  [ffd25f8a] XZ_jll v5.8.3+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [8f1865be] ZeroMQ_jll v4.3.6+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [b792d7bf] cminpack_jll v1.3.12+0
  [9a68df92] isoband_jll v0.2.3+0
⌃ [a4ae2306] libaom_jll v3.13.3+0
  [0ac62f75] libass_jll v0.17.4+0
  [8e53e030] libdrm_jll v2.4.134+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [b53b4c65] libpng_jll v1.6.58+0
  [075b6546] libsixel_jll v1.10.5+0
  [a9144af2] libsodium_jll v1.0.21+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [c5f90fcd] libwebp_jll v1.6.0+0
⌅ [9aeb927a] mpif_jll v0.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [8bf52ea8] CRC32c v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [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`