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
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
Ns = [2^i for i in 1:8];
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, _)) in enumerate(adtypes)
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 1.974e-5 1.468e-5 1.609e-5 1.653e-5
1.022e-5 1.167e-5 1.229e-5
4 7.208e-5 5.1469e-5 5.451e-5 6.5759e-5
4.002e-5 4.252e-5 5.0959e-5
8 0.000271307 0.000211768 0.000204228 0.000333017
0.000241027 0.000229338 0.000660294
16 0.001048481 0.000876253 0.000823662 0.002903635
0.00234182 0.002141462 0.010793776
32 0.004439542 0.003891686 0.003448811 0.036322325
0.030441757 0.027064646 0.179627387
64 0.021031659 0.019135955 0.01737604 0.56334806
0.464029669 0.417120924 2.896505444
128 0.233382383 0.212826621 0.202984315 12.376377912
7.18849567 6.669337492 47.365001664
256 2.805276774 2.798079127 2.730241304 190.48951247
114.533257835 104.573494343 1110.189320069Plotting 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 = $N"
for (j, solver) in enumerate(solvers_scaling)
ptype = solver.sparsity
alg = solver.alg
name = solver.name
prob = if ptype == :none
prob_dense
elseif ptype == :approx
# With Tracing based sparsity detection, we dont need this any more
error("Approximate Sparsity not implemented")
elseif ptype == :exact
prob_exact_sparse
end
if (j > 1 && runtimes_scaling[j - 1, i] == -1) ||
(alg isa CMINPACK && N > 32) ||
(alg isa KINSOL && N > 64) ||
(alg isa NLsolveJL && N > 64 && alg.method == :trust_region) ||
(alg isa GeneralizedFirstOrderAlgorithm && alg.name == :TrustRegion && N > 64) ||
(alg isa NLsolveJL && N > 128 && alg.method == :newton) ||
(alg isa GeneralizedFirstOrderAlgorithm && alg.name == :NewtonRaphson && N > 128 && ptype == :none) ||
(alg isa PETScSNES && N > 64)
# The last benchmark failed so skip this too
runtimes_scaling[j, i] = NaN
@warn "$(name): Would Have Timed out"
else
function benchmark_function()
termination_condition = (alg isa PETScSNES || alg isa KINSOL) ?
nothing :
NonlinearSolveBase.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)) | $(sol.retcode)"
end
timeout(benchmark_function, 600)
if runtimes_scaling[j, i] == -1
@warn "$(name): Timed out"
runtimes_scaling[j, i] = NaN
end
end
end
println()
endPlot 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 = true), 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 = 2.647
6053818757766e-9).
[Info] Solver RobustMultiNewton (GMRES) successfully solved the problem (no
rm = 1.1664069101655303e-8).
[Info] Solver Newton Raphson successfully solved the problem (norm = 2.6476
053818757766e-9).
[Info] Solver Newton Krylov successfully solved the problem (norm = 1.16640
69101655303e-8).
[Info] Solver Trust Region successfully solved the problem (norm = 2.647605
3818757766e-9).
[Info] Solver TR Krylov successfully solved the problem (norm = 1.166406910
1655303e-8).
[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 = 1.24839
3048765494e-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 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 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
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
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/NonlinearProblem/Project.toml`
[2169fc97] AlgebraicMultigrid v1.0.0
[6e4b80f9] BenchmarkTools v1.6.0
[13f3f980] CairoMakie v0.13.4
[2b5f629d] DiffEqBase v6.167.2
[f3b72e0c] DiffEqDevTools v2.48.0
[a0c0ee7d] DifferentiationInterface v0.6.52
⌃ [7da242da] Enzyme v0.13.37
[40713840] IncompleteLU v0.2.1
[b964fa9f] LaTeXStrings v1.4.0
[d3d80556] LineSearches v7.3.0
[7ed4a6bd] LinearSolve v3.8.0
[4854310b] MINPACK v1.3.0
[2774e3e8] NLsolve v4.5.1
[b7050fa9] NonlinearProblemLibrary v0.1.3
[8913a72c] NonlinearSolve v4.6.0
[ace2c81b] PETSc v0.3.1
[98d1487c] PolyesterForwardDiff v0.1.3
[08abe8d2] PrettyTables v2.4.0
[f2c3362d] RecursiveFactorization v0.2.23
[31c91b34] SciMLBenchmarks v0.1.3
[efcf1570] Setfield v1.1.2
[727e6d20] SimpleNonlinearSolve v2.2.2
[9f842d2f] SparseConnectivityTracer v0.6.17
[0a514795] SparseMatrixColorings v0.4.18
[f1835b91] SpeedMapping v0.3.0
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[90137ffa] StaticArrays v1.9.13
[c3572dad] Sundials v4.28.0
[0c5d862f] Symbolics v6.38.0
Info Packages marked with ⌃ have new versions available and may be upgradable.
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonlinearProblem/Manifest.toml`
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[d1d4a3ce] BitFlags v0.1.9
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
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⌅ [8fc22ac5] FilePaths v0.8.3
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[a2bd30eb] Graphics v1.1.3
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[708ec375] Gumbo v0.8.3
[cd3eb016] HTTP v1.10.16
[eafb193a] Highlights v0.5.3
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[34004b35] HypergeometricFunctions v0.3.28
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[b5f81e59] IOCapture v0.2.5
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[91d4177d] Opus_jll v1.3.3+0
[8fa3689e] PETSc_jll v3.22.0+0
[36c8627f] Pango_jll v1.56.1+0
[30392449] Pixman_jll v0.44.2+0
[f50d1b31] Rmath_jll v0.5.1+0
[aabda75e] SCALAPACK32_jll v2.2.1+1
⌅ [fb77eaff] Sundials_jll v5.2.2+0
⌅ [02c8fc9c] XML2_jll v2.13.6+1
[ffd25f8a] XZ_jll v5.8.1+0
[4f6342f7] Xorg_libX11_jll v1.8.12+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.7+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+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 v3.11.0+0
[0ac62f75] libass_jll v0.15.2+0
[f638f0a6] libfdk_aac_jll v2.0.3+0
[b53b4c65] libpng_jll v1.6.47+0
[47bcb7c8] libsass_jll v3.6.6+0
[075b6546] libsixel_jll v1.10.5+0
[a9144af2] libsodium_jll v1.0.21+0
[f27f6e37] libvorbis_jll v1.3.7+2
[c5f90fcd] libwebp_jll v1.5.0+0
[1317d2d5] oneTBB_jll v2022.0.0+0
[1270edf5] x264_jll v10164.0.1+0
⌅ [dfaa095f] x265_jll v3.6.0+0
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[8bf52ea8] CRC32c
[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 v1.1.1+0
[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+4
[05823500] OpenLibm_jll v0.8.1+4
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[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`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.