Ill-Conditioned Nonlinear System Work-Precision Diagrams
Setup
Fetch required packages
using NonlinearSolve, LinearAlgebra, SparseArrays, DiffEqDevTools,
CairoMakie, Symbolics, BenchmarkTools, PolyesterForwardDiff, LinearSolve, Sundials,
Enzyme, SparseConnectivityTracer, DifferentiationInterface, SparseMatrixColorings
using SciMLLogging
import NLsolve, MINPACK, PETSc, RecursiveFactorization
const RUS = RadiusUpdateSchemes;
BenchmarkTools.DEFAULT_PARAMETERS.seconds = 0.2;Define a utility to timeout the benchmark after a certain time.
# Taken from ReTestItems.jl
function timeout(f, timeout)
cond = Threads.Condition()
timer = Timer(timeout) do tm
close(tm)
ex = ErrorException("timed out after $timeout seconds")
@lock cond notify(cond, ex; error = false)
end
Threads.@spawn begin
try
ret = $f()
isopen(timer) && @lock cond notify(cond, ret)
catch e
isopen(timer) &&
@lock cond notify(cond, CapturedException(e, catch_backtrace()); error = true)
finally
close(timer)
end
end
return @lock cond wait(cond) # will throw if we timeout
endtimeout (generic function with 1 method)Define the Brussletor problem.
brusselator_f(x, y) = (((x - 3 // 10) ^ 2 + (y - 6 // 10) ^ 2) ≤ 0.01) * 5
limit(a, N) = ifelse(a == N + 1, 1, ifelse(a == 0, N, a))
function init_brusselator_2d(xyd, N)
N = length(xyd)
u = zeros(N, N, 2)
for I in CartesianIndices((N, N))
x = xyd[I[1]]
y = xyd[I[2]]
u[I, 1] = 22 * (y * (1 - y))^(3 / 2)
u[I, 2] = 27 * (x * (1 - x))^(3 / 2)
end
return u
end
function generate_brusselator_problem(N::Int; sparsity = nothing, kwargs...)
xyd_brusselator = range(0; stop = 1, length = N)
function brusselator_2d_loop(du_, u_, p)
A, B, α, δx = p
α = α / δx ^ 2
du = reshape(du_, N, N, 2)
u = reshape(u_, N, N, 2)
@inbounds @simd for I in CartesianIndices((N, N))
i, j = Tuple(I)
x, y = xyd_brusselator[I[1]], xyd_brusselator[I[2]]
ip1, im1 = limit(i + 1, N), limit(i - 1, N)
jp1, jm1 = limit(j + 1, N), limit(j - 1, N)
du[i, j, 1] = α * (u[im1, j, 1] + u[ip1, j, 1] + u[i, jp1, 1] + u[i, jm1, 1] -
4u[i, j, 1]) +
B + u[i, j, 1] ^ 2 * u[i, j, 2] - (A + 1) * u[i, j, 1] +
brusselator_f(x, y)
du[i, j, 2] = α * (u[im1, j, 2] + u[ip1, j, 2] + u[i, jp1, 2] + u[i, jm1, 2] -
4u[i, j, 2]) +
A * u[i, j, 1] - u[i, j, 1] ^ 2 * u[i, j, 2]
end
return nothing
end
return NonlinearProblem(
NonlinearFunction(brusselator_2d_loop; sparsity),
vec(init_brusselator_2d(xyd_brusselator, N)),
(3.4, 1.0, 10.0, step(xyd_brusselator));
kwargs...
)
endgenerate_brusselator_problem (generic function with 1 method)function get_ordering(x::AbstractMatrix)
idxs = Vector{Int}(undef, size(x, 1))
placed = zeros(Bool, size(x, 1))
idx = 1
for j in size(x, 2):-1:1
row = view(x, :, j)
idxs_row = sortperm(row; by = x -> isnan(x) ? Inf : (x == -1 ? Inf : x))
for i in idxs_row
if !placed[i] && !isnan(row[i]) && row[i] ≠ -1
idxs[idx] = i
placed[i] = true
idx += 1
idx > length(idxs) && break
end
end
idx > length(idxs) && break
end
return idxs
endget_ordering (generic function with 1 method)Scaling of Sparsity Detection Algorithm
We increase the problem size, and compute the jacobian 10 times similar to a real workload where the jacobian is computed several times and amortizes the cost for computing the sparsity pattern.
test_problem = generate_brusselator_problem(4)
bruss_f!, u0 = (du, u) -> test_problem.f(du, u, test_problem.p), test_problem.u0
y = similar(u0)
J = Float64.(ADTypes.jacobian_sparsity(bruss_f!, y, u0, TracerSparsityDetector()))
colors = fast_coloring(J, ColoringProblem(), GreedyColoringAlgorithm())
begin
J_ = similar(J)
rows = rowvals(J)
vals = nonzeros(J)
for j in 1:size(J, 2)
for i in nzrange(J, j)
row = rows[i]
J_[j, row] = colors[j] # spy does a ordering I can't figure out. so transposing it here
end
end
end
function cache_and_compute_10_jacobians(adtype, f!::F, y, x, p) where {F}
prep = DifferentiationInterface.prepare_jacobian(f!, y, adtype, x, Constant(p))
J = DifferentiationInterface.jacobian(f!, y, prep, adtype, x, Constant(p))
for _ in 1:9
DifferentiationInterface.jacobian!(f!, y, J, prep, adtype, x, Constant(p))
end
return J
end
# Cap at 2^6: DenseSparsityDetector is O(N²) and dominated CI time at N≥128.
Ns = [2^i for i in 1:6];
adtypes = [
(
AutoSparse(
AutoFiniteDiff();
sparsity_detector = TracerSparsityDetector(),
coloring_algorithm = GreedyColoringAlgorithm(LargestFirst())
),
[:finitediff, :exact_sparse]
),
(
AutoSparse(
AutoPolyesterForwardDiff();
sparsity_detector = TracerSparsityDetector(),
coloring_algorithm = GreedyColoringAlgorithm(LargestFirst())
),
[:polyester, :exact_sparse]
),
(
AutoSparse(
AutoEnzyme(; mode = Enzyme.Forward);
sparsity_detector = TracerSparsityDetector(),
coloring_algorithm = GreedyColoringAlgorithm(LargestFirst())
),
[:enzyme, :exact_sparse]
),
(
AutoSparse(
AutoFiniteDiff();
sparsity_detector = DenseSparsityDetector(AutoFiniteDiff(); atol = 1e-5),
coloring_algorithm = GreedyColoringAlgorithm(LargestFirst())
),
[:finitediff, :approx_sparse]
),
(
AutoSparse(
AutoPolyesterForwardDiff();
sparsity_detector = DenseSparsityDetector(
AutoPolyesterForwardDiff(); atol = 1e-5
),
coloring_algorithm = GreedyColoringAlgorithm(LargestFirst())
),
[:polyester, :approx_sparse]
),
(
AutoSparse(
AutoEnzyme(; mode = Enzyme.Forward);
sparsity_detector = DenseSparsityDetector(
AutoEnzyme(; mode = Enzyme.Forward); atol = 1e-5
),
coloring_algorithm = GreedyColoringAlgorithm(LargestFirst())
),
[:enzyme, :approx_sparse]
),
(
AutoPolyesterForwardDiff(),
[:polyester, :none]
)
];
times = Matrix{Float64}(undef, length(Ns), length(adtypes));
for (i, N) in enumerate(Ns)
str = "$(lpad(N, 10)) "
test_problem = generate_brusselator_problem(N)
bruss_f! = test_problem.f
u0 = test_problem.u0
y = similar(u0)
for (j, (adtype, tags)) in enumerate(adtypes)
# Dense sparsity probing is cubic in problem size; skip for N>32.
if tags[2] === :approx_sparse && N > 32
times[i, j] = NaN
str = str * lpad("NaN", 16)
continue
end
times[i, j] = @belapsed begin
$(cache_and_compute_10_jacobians)(
$(adtype), $(bruss_f!), $(y), $(u0), $(test_problem.p)
)
end
str = str * "$(lpad(times[i, j], 16))"
end
println(str)
end
nothing2 1.72e-5 1.612e-5 1.576e-5 1.063e-5
1.064e-5 1.019e-5 8.709e-6
4 7.283e-5 5.045e-5 6.2109e-5 6.3529e-5
3.822e-5 3.8499e-5 5.1679e-5
8 0.000233768 0.000163238 0.000172008 0.000298217
0.000202737 0.000201148 0.000229338
16 0.001060678 0.000730732 0.000708092 0.002775538
0.002153306 0.002073386 0.001169617
32 0.004863074 0.003326322 0.003931685 0.037416123
0.029633142 0.026266001 0.069032972
64 0.023852948 0.022124027 0.018766356 NaN
NaN NaN 0.555817775Plotting the results.
symbol_to_adname = Dict(
:finitediff => "Finite Diff",
:forwarddiff => "Forward Mode AD",
:polyester => "Threaded Forward Mode AD",
:enzyme => "Forward Mode AD (Enzyme)"
)
fig = begin
cycle = Cycle([:marker], covary = true)
plot_theme = Theme(Lines = (; cycle), Scatter = (; cycle))
with_theme(plot_theme) do
fig = Figure(; size = (1400, 1400 * 0.5))
ax = Axis(fig[1, 1]; title = "Sparsity Pattern for 2D Brusselator Jacobian",
titlesize = 22, titlegap = 10,
xticksize = 20, yticksize = 20, xticklabelsize = 20, yticklabelsize = 20,
xtickwidth = 2.5, ytickwidth = 2.5, spinewidth = 2.5, yreversed = true)
spy!(ax, J_; markersize = 1, marker = :circle, framecolor = :lightgray,
colormap = :tableau_20)
ax = Axis(fig[1, 2]; title = "Scaling of Sparse Jacobian Computation",
titlesize = 22, titlegap = 10, xscale = log2, yscale = log2,
xticksize = 20, yticksize = 20, xticklabelsize = 20, yticklabelsize = 20,
xtickwidth = 2.5, ytickwidth = 2.5, spinewidth = 2.5,
xlabel = L"Input Dimension ($\mathbf{N}$)",
ylabel = L"Time $\mathbf{(s)}$", xlabelsize = 22,
ylabelsize = 22, yaxisposition = :right)
colors = cgrad(:tableau_20, length(adtypes); categorical = true)
line_list = []
scatter_list = []
Ns_ = Ns .^ 2 .* 2
linestyles = [:solid, :solid, :solid, :dash, :dash, :dash, :dot, :dot]
for (i, times) in enumerate(eachcol(times))
l = lines!(
Ns_, times; linewidth = 5, color = colors[i], linestyle = linestyles[i])
push!(line_list, l)
sc = scatter!(Ns_, times; markersize = 16, strokewidth = 2, color = colors[i])
push!(scatter_list, sc)
end
tracer_idxs = [idx for idx in 1:length(adtypes) if :exact_sparse ∈ adtypes[idx][2]]
group_tracer = [[
LineElement(;
color = line_list[idx].color,
linestyle = line_list[idx].linestyle,
linewidth = line_list[idx].linewidth
),
MarkerElement(;
color = scatter_list[idx].color,
marker = scatter_list[idx].marker,
strokewidth = scatter_list[idx].strokewidth,
markersize = scatter_list[idx].markersize
)
] for idx in tracer_idxs]
local_sparse_idxs = [idx
for idx in 1:length(adtypes)
if :approx_sparse ∈ adtypes[idx][2]]
group_local_sparse = [[
LineElement(;
color = line_list[idx].color,
linestyle = line_list[idx].linestyle,
linewidth = line_list[idx].linewidth
),
MarkerElement(;
color = scatter_list[idx].color,
marker = scatter_list[idx].marker,
strokewidth = scatter_list[idx].strokewidth,
markersize = scatter_list[idx].markersize
)
] for idx in local_sparse_idxs]
non_sparse_idxs = [idx for idx in 1:length(adtypes) if :none ∈ adtypes[idx][2]]
group_nonsparse = [[
LineElement(;
color = line_list[idx].color,
linestyle = line_list[idx].linestyle,
linewidth = line_list[idx].linewidth
),
MarkerElement(;
color = scatter_list[idx].color,
marker = scatter_list[idx].marker,
strokewidth = scatter_list[idx].strokewidth,
markersize = scatter_list[idx].markersize
)
] for idx in non_sparse_idxs]
axislegend(
ax,
[group_tracer, group_local_sparse, group_nonsparse],
[
[symbol_to_adname[adtypes[idx][2][1]] for idx in tracer_idxs],
[symbol_to_adname[adtypes[idx][2][1]] for idx in local_sparse_idxs],
[symbol_to_adname[adtypes[idx][2][1]] for idx in non_sparse_idxs]
],
["Exact Sparsity", "Approx. Local Sparsity", "Dense"];
position = :lt, framevisible = true, framewidth = 2.5, titlesize = 18,
labelsize = 16, patchsize = (40.0f0, 20.0f0)
)
fig
end
end
save("brusselator_sparse_jacobian_scaling.svg", fig)CairoMakie.Screen{SVG}Scaling with Problem Size
First, let us experiment the scaling of each algorithm with the problem size.
Ns = vcat(collect(2 .^ (2:7)), [150, 175, 200])
solvers_scaling = [
(; pkg = :nonlinearsolve, sparsity = :none,
name = "NR (No Sparsity)", alg = NewtonRaphson()),
(; pkg = :nonlinearsolve, sparsity = :exact,
name = "NR (Exact Sparsity)", alg = NewtonRaphson()),
(; pkg = :wrapper, sparsity = :none, name = "NR [NLsolve.jl]",
alg = NLsolveJL(; method = :newton, autodiff = :forward)),
(; pkg = :wrapper, sparsity = :none, name = "NR [Sundials]",
alg = KINSOL(; linear_solver = :LapackDense, maxsetupcalls = 1)),
(; pkg = :wrapper,
sparsity = :none,
name = "NR [PETSc] (No Sparsity)",
alg = PETScSNES(; snes_type = "newtonls", snes_linesearch_type = "basic", autodiff = missing)),
(; pkg = :wrapper, sparsity = :exact, name = "NR [PETSc] (Exact Sparsity)",
alg = PETScSNES(; snes_type = "newtonls", snes_linesearch_type = "basic")), (;
pkg = :nonlinearsolve, sparsity = :none, name = "TR (No Sparsity)",
alg = TrustRegion(; radius_update_scheme = RUS.NLsolve)),
(; pkg = :nonlinearsolve, sparsity = :exact, name = "TR (Exact Sparsity)",
alg = TrustRegion(; radius_update_scheme = RUS.NLsolve)),
(; pkg = :wrapper, sparsity = :none, name = "TR [NLsolve.jl]",
alg = NLsolveJL(; autodiff = :forward)),
(; pkg = :wrapper, sparsity = :none, name = "TR [PETSc] (No Sparsity)",
alg = PETScSNES(; snes_type = "newtontr", autodiff = missing)),
(; pkg = :wrapper, sparsity = :exact, name = "TR [PETSc] (Exact Sparsity)",
alg = PETScSNES(; snes_type = "newtontr")), (; pkg = :wrapper, sparsity = :none,
name = "Mod. Powell [MINPACK]", alg = CMINPACK())
]
GC.enable(false) # for PETSc
runtimes_scaling = fill(-1.0, length(solvers_scaling), length(Ns))
for (i, N) in enumerate(Ns)
prob_dense = generate_brusselator_problem(N)
prob_exact_sparse = generate_brusselator_problem(N;
sparsity = TracerSparsityDetector()
)
@info "Benchmarking N = $N"
for (j, solver) in enumerate(solvers_scaling)
ptype = solver.sparsity
alg = solver.alg
name = solver.name
prob = if ptype == :none
prob_dense
elseif ptype == :approx
# With Tracing based sparsity detection, we dont need this any more
error("Approximate Sparsity not implemented")
elseif ptype == :exact
prob_exact_sparse
end
# Cascade on -1 (timeout): leave -1 so later solvers at this N also skip.
# Size-limit skips use NaN and do not cascade.
if (j > 1 && runtimes_scaling[j - 1, i] == -1) ||
(alg isa CMINPACK && N > 32) ||
(alg isa KINSOL && N > 64) ||
(alg isa NLsolveJL && N > 64 && alg.method == :trust_region) ||
(alg isa GeneralizedFirstOrderAlgorithm && alg.name == :TrustRegion && N > 64) ||
(alg isa NLsolveJL && N > 64 && alg.method == :newton) ||
(alg isa GeneralizedFirstOrderAlgorithm && alg.name == :NewtonRaphson &&
N > 64 && ptype == :none) ||
(alg isa PETScSNES && N > 64)
if j > 1 && runtimes_scaling[j - 1, i] == -1
runtimes_scaling[j, i] = -1
else
runtimes_scaling[j, i] = NaN
end
@warn "$(name): Would Have Timed out"
else
function benchmark_function()
termination_condition = (alg isa PETScSNES || alg isa KINSOL) ?
nothing :
AbsNormTerminationMode(Base.Fix1(maximum, abs))
sol = solve(prob, alg; abstol = 1e-6, reltol = 1e-6, termination_condition)
runtimes_scaling[j, i] = @belapsed solve($prob, $alg; abstol = 1e-6,
reltol = 1e-6, termination_condition = $termination_condition)
@info "$(name): $(runtimes_scaling[j, i]) | $(norm(sol.resid, Inf)) | $(sol.retcode)"
end
timeout(benchmark_function, 600)
# Keep -1 on timeout so subsequent solvers at this N cascade-skip.
if runtimes_scaling[j, i] == -1
@warn "$(name): Timed out"
end
end
end
println()
end
# Normalize timeout sentinels for plotting (log-scale).
runtimes_scaling = map(x -> x == -1 ? NaN : x, runtimes_scaling)12×9 Matrix{Float64}:
0.000113398 0.0008585 0.0559755 … NaN NaN NaN
0.000211058 0.000745371 0.00350314 1.70608 2.60908 3.98496
8.4919e-5 0.000982199 0.023241 NaN NaN NaN
9.2899e-5 0.000698552 0.013871 NaN NaN NaN
0.00102924 0.003772 0.0285505 NaN NaN NaN
0.000658253 0.00147389 0.00477854 … NaN NaN NaN
0.000134939 0.000816571 0.0250265 NaN NaN NaN
0.000212028 0.000745701 0.00356366 NaN NaN NaN
9.5899e-5 0.00102779 0.0237114 NaN NaN NaN
0.00235851 0.00932486 0.0794934 NaN NaN NaN
0.000972429 0.00340767 0.0180865 … NaN NaN NaN
8.0389e-5 0.00230232 0.119524 NaN NaN NaNPlot the results.
fig = begin
ASPECT_RATIO = 0.7
WIDTH = 1200
HEIGHT = round(Int, WIDTH * ASPECT_RATIO)
STROKEWIDTH = 2.5
cycle = Cycle([:marker], covary = true)
colors = cgrad(:tableau_20, length(solvers_scaling); categorical = true)
theme = Theme(Lines = (cycle = cycle,), Scatter = (cycle = cycle,))
LINESTYLES = Dict(
(:nonlinearsolve, :none) => :solid,
(:nonlinearsolve, :exact) => :dashdot,
# (:simplenonlinearsolve, :none) => :solid,
(:wrapper, :exact) => :dash,
(:wrapper, :none) => :dot
)
Ns_ = Ns .^ 2 .* 2
with_theme(theme) do
fig = Figure(; size = (WIDTH, HEIGHT))
ax = Axis(fig[1, 1:3], ylabel = L"Time ($s$)", xlabel = L"Problem Size ($N$)",
xscale = log2, yscale = log2, xlabelsize = 22, ylabelsize = 22,
xticklabelsize = 20, yticklabelsize = 20, xtickwidth = STROKEWIDTH,
ytickwidth = STROKEWIDTH, spinewidth = STROKEWIDTH)
idxs = get_ordering(runtimes_scaling)
ls, scs = [], []
for (i, solver) in zip(idxs, solvers_scaling[idxs])
linestyle = LINESTYLES[(solver.pkg, solver.sparsity)]
l = lines!(Ns_, runtimes_scaling[i, :]; linewidth = 5, color = colors[i],
linestyle)
sc = scatter!(Ns_, runtimes_scaling[i, :]; markersize = 16, strokewidth = 2,
color = colors[i])
push!(ls, l)
push!(scs, sc)
end
main_legend = [[
LineElement(;
color = ls[idx].color, linestyle = ls[idx].linestyle,
linewidth = ls[idx].linewidth),
MarkerElement(;
color = scs[idx].color, marker = scs[idx].marker,
markersize = scs[idx].markersize, strokewidth = scs[idx].strokewidth)
]
for idx in 1:length(solvers_scaling)]
sparsity_legend = [
LineElement(; linestyle = :solid, linewidth = 5),
# LineElement(; linestyle = :dash, linewidth = 5),
LineElement(; linestyle = :dashdot, linewidth = 5)
]
axislegend(ax, main_legend, [s.name for s in solvers_scaling[idxs]],
"Successful Solvers";
framevisible = true, framewidth = STROKEWIDTH, orientation = :vertical,
titlesize = 20, nbanks = 1, labelsize = 16,
tellheight = true, tellwidth = false, patchsize = (60.0f0, 20.0f0),
position = :rb)
axislegend(ax, sparsity_legend,
[
"No Sparsity Detection",
# "Approx. Sparsity",
"Exact Sparsity"
],
"Sparsity Detection"; framevisible = true, framewidth = STROKEWIDTH,
orientation = :vertical, titlesize = 20, nbanks = 1, labelsize = 16,
tellheight = true, tellwidth = false, patchsize = (60.0f0, 20.0f0),
position = :lt)
fig[0, :] = Label(fig,
"Brusselator 2D: Scaling of First-Order Nonlinear Solvers with Problem Size",
fontsize = 24, tellwidth = false, font = :bold)
return fig
end
end
save("brusselator_scaling.svg", fig)CairoMakie.Screen{SVG}Work-Precision Diagram
In this section, we will generate the work-precision of the solvers. All solvers that can exploit sparsity will automatically do so.
solvers_all = [
(; pkg = :nonlinearsolve, name = "Default PolyAlg",
solver = Dict(:alg => FastShortcutNonlinearPolyalg())),
(; pkg = :nonlinearsolve, name = "RobustMultiNewton (GMRES)",
solver = Dict(:alg => RobustMultiNewton(; linsolve = KrylovJL_GMRES()))), (;
pkg = :nonlinearsolve, name = "Newton Raphson",
solver = Dict(:alg => NewtonRaphson(; linsolve = nothing))),
(; pkg = :nonlinearsolve, name = "Newton Krylov",
solver = Dict(:alg => NewtonRaphson(; linsolve = KrylovJL_GMRES()))),
(; pkg = :nonlinearsolve, name = "Trust Region", solver = Dict(:alg => TrustRegion())),
(; pkg = :nonlinearsolve, name = "TR Krylov",
solver = Dict(:alg => TrustRegion(; linsolve = KrylovJL_GMRES()))), (;
pkg = :wrapper, name = "NR [NLsolve.jl]",
solver = Dict(:alg => NLsolveJL(; method = :newton, autodiff = :forward))),
(; pkg = :wrapper, name = "TR [NLsolve.jl]",
solver = Dict(:alg => NLsolveJL(; autodiff = :forward))), (;
pkg = :wrapper, name = "NR [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 = "Mod. Powell [MINPACK]", solver = Dict(:alg => CMINPACK())),
(; pkg = :wrapper,
name = "NR [PETSc]",
solver = Dict(:alg => PETScSNES(;
snes_type = "newtonls", snes_linesearch_type = "basic", autodiff = missing))),
(; pkg = :wrapper, name = "TR [PETSc]",
solver = Dict(:alg => PETScSNES(; snes_type = "newtontr", autodiff = missing))),
(; pkg = :wrapper,
name = "Newton Krylov [PETSc]",
solver = Dict(:alg => PETScSNES(;
snes_type = "newtonls", snes_linesearch_type = "basic", ksp_type = "gmres",
autodiff = missing, snes_mf = true, ksp_gmres_restart = 1000)))
];prob_wpd = generate_brusselator_problem(32; sparsity = TracerSparsityDetector())
abstols = 1.0 ./ 10 .^ (2:10)
reltols = 1.0 ./ 10 .^ (2:10)
function check_solver(prob, solver)
try
sol = solve(prob, solver.solver[:alg]; abstol = 1e-4, reltol = 1e-4,
maxiters = 10000)
err = norm(sol.resid, Inf)
if !SciMLBase.successful_retcode(sol.retcode)
Base.printstyled(
"[Warn] Solver $(solver.name) returned retcode $(sol.retcode) with an residual norm = $(norm(sol.resid)).\n";
color = :red)
return false
elseif err > 1e3
Base.printstyled(
"[Warn] Solver $(solver.name) had a very large residual (norm = $(norm(sol.resid))).\n";
color = :red)
return false
elseif isinf(err) || isnan(err)
Base.printstyled("[Warn] Solver $(solver.name) had a residual of $(err).\n";
color = :red)
return false
end
Base.printstyled(
"[Info] Solver $(solver.name) successfully solved the problem (norm = $(norm(sol.resid))).\n";
color = :green)
catch e
Base.printstyled("[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, abstols, reltols,
getfield.(successful_solvers, :solver);
names = getfield.(successful_solvers, :name), numruns = 10, error_estimate = :l∞,
maxiters = 1000, verbose = SciMLLogging.Standard()),
successful_solvers
endgenerate_wpset (generic function with 1 method)wp_set, successful_solvers = generate_wpset(prob_wpd, solvers_all);[Info] Solver Default PolyAlg successfully solved the problem (norm = 2.639
1901755098182e-9).
[Info] Solver RobustMultiNewton (GMRES) successfully solved the problem (no
rm = 9.061694653242178e-5).
[Info] Solver Newton Raphson successfully solved the problem (norm = 2.6391
901755098182e-9).
[Info] Solver Newton Krylov successfully solved the problem (norm = 9.06169
4653242178e-5).
[Info] Solver Trust Region successfully solved the problem (norm = 2.639190
1755098182e-9).
[Info] Solver TR Krylov successfully solved the problem (norm = 9.061694653
242178e-5).
[Info] Solver NR [NLsolve.jl] successfully solved the problem (norm = 2.629
767216137896e-9).
[Info] Solver TR [NLsolve.jl] successfully solved the problem (norm = 2.629
767216137896e-9).
[Info] Solver NR [Sundials] successfully solved the problem (norm = 1.22227
32529298485e-6).
[Info] Solver Newton Krylov [Sundials] successfully solved the problem (nor
m = 0.0005045549665406284).
[Info] Solver Mod. Powell [MINPACK] successfully solved the problem (norm =
1.9629370283177898e-6).
[Warn] Solver NR [PETSc] returned retcode Failure with an residual norm = 0
.008277151682674046.
[Info] Solver TR [PETSc] successfully solved the problem (norm = 0.00113188
002920334).
[Warn] Solver Newton Krylov [PETSc] returned retcode Failure with an residu
al norm = 0.022043824067284137.Plotting the Work-Precision Diagram.
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))
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 = log2, yscale = log2, xtickwidth = STROKEWIDTH,
ytickwidth = STROKEWIDTH, spinewidth = STROKEWIDTH,
xticklabelsize = 20, yticklabelsize = 20)
idxs = sortperm(median.(getfield.(wp_set.wps, :times)))
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 = 2,
color = colors[i])
push!(ls, l)
push!(scs, sc)
end
xlims!(ax; high = 1)
ylims!(ax; low = 5e-3)
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, position = :rb,
titlesize = 20, labelsize = 16, patchsize = (40.0f0, 20.0f0))
fig[0, :] = Label(fig, "Brusselator Steady State PDE: Work Precision Diagram",
fontsize = 24, tellwidth = false, font = :bold)
fig
end
end
save("brusselator_wpd.svg", fig)CairoMakie.Screen{SVG}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","bruss.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
[6e4b80f9] BenchmarkTools v1.8.0
[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
[b7050fa9] NonlinearProblemLibrary v0.1.7
⌃ [8913a72c] NonlinearSolve v4.21.0
[ace2c81b] PETSc v0.4.10
[98d1487c] PolyesterForwardDiff v0.1.4
⌃ [08abe8d2] PrettyTables v3.4.5
⌃ [f2c3362d] RecursiveFactorization v0.2.26
⌃ [31c91b34] SciMLBenchmarks v0.1.3
[a6db7da4] SciMLLogging v2.0.4
[efcf1570] Setfield v1.1.2
⌃ [727e6d20] SimpleNonlinearSolve v2.13.1
[9f842d2f] SparseConnectivityTracer v1.2.2
[0a514795] SparseMatrixColorings v0.4.27
[f1835b91] SpeedMapping v0.4.1
[860ef19b] StableRNGs v1.0.4
⌃ [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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[e2ed5e7c] Bijections v0.2.2
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⌃ [70df07ce] BracketingNonlinearSolve v1.12.4
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[c3611d14] ColorVectorSpace v0.11.0
[5ae59095] Colors v0.13.1
⌅ [861a8166] Combinatorics v1.0.2
[38540f10] CommonSolve v0.2.13
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.1.2
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[163ba53b] DiffResults v1.1.0
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⌃ [77dc65aa] FunctionWrappersWrappers v1.12.1
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⌅ [61eb1bfa] GPUCompiler v1.23.0
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[5c1252a2] GeometryBasics v0.5.11
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[a2bd30eb] Graphics v1.1.3
[86223c79] Graphs v1.14.0
[3955a311] GridLayoutBase v0.11.2
⌅ [eafb193a] Highlights v0.5.3
[3e5b6fbb] HostCPUFeatures v0.1.18
[34004b35] HypergeometricFunctions v0.3.30
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⌅ [682c06a0] JSON v0.21.4
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[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.1.0+0
⌅ [dad2f222] LLVMExtra_jll v0.0.44+0
[1d63c593] LLVMOpenMP_jll v22.1.7+0
[ad6e5548] LibTracyClient_jll v0.13.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
⌅ [b5ada748] MPIABI_jll v0.1.5+0
[7cb0a576] MPICH_jll v5.0.1+0
[f1f71cc9] MPItrampoline_jll v5.5.6+0
[9237b28f] MicrosoftMPI_jll v10.1.4+3
[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`