Ill-Conditioned Nonlinear System Work-Precision Diagrams
Setup
Fetch required packages
using NonlinearSolve, SparseDiffTools, LinearAlgebra, SparseArrays, DiffEqDevTools,
CairoMakie, Symbolics, BenchmarkTools, PolyesterForwardDiff, LinearSolve, Sundials
import NLsolve, MINPACK
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
p = (3.4, 1.0, 10.0, step(xyd_brusselator))
u0 = init_brusselator_2d(xyd_brusselator, N)
nlfunc = NonlinearFunction(brusselator_2d_loop; sparsity)
return NonlinearProblem(nlfunc, vec(u0), p; kwargs...)
endgenerate_brusselator_problem (generic function with 1 method)function __ad_backend(sparsity, psize, ck)
if sparsity === nothing
if psize ≥ 16
return AutoPolyesterForwardDiff(; chunksize = ck)
else
return AutoForwardDiff(; chunksize = ck)
end
else
if psize ≥ 16
return AutoSparsePolyesterForwardDiff(; chunksize = ck)
else
return AutoSparseForwardDiff(; chunksize = ck)
end
end
end
function __set_ad_chunksize(solver::GeneralizedFirstOrderAlgorithm{CJ, N}, ck,
sparsity, psize) where {CJ, N}
ad = __ad_backend(sparsity, psize, ck)
return GeneralizedFirstOrderAlgorithm{CJ, N}(; solver.descent, solver.linesearch,
solver.trustregion, jacobian_ad = ad, solver.max_shrink_times, solver.forward_ad,
solver.reverse_ad)
end
function __set_ad_chunksize(solver::ApproximateJacobianSolveAlgorithm{CJ, N}, ck,
sparsity, psize) where {CJ, N}
ad = __ad_backend(sparsity, psize, ck)
initialization = solver.initialization isa NonlinearSolve.TrueJacobianInitialization ?
NonlinearSolve.TrueJacobianInitialization(NonlinearSolve.FullStructure(), ad) : solver.initialization
return ApproximateJacobianSolveAlgorithm{CJ, N}(; solver.descent, solver.linesearch,
solver.trustregion, solver.update_rule, solver.max_shrink_times, solver.max_resets,
initialization, solver.reinit_rule)
end
function __set_ad_chunksize(solver::SimpleNewtonRaphson, ck, sparsity, psize)
solver.autodiff === nothing || return solver
autodiff = __ad_backend(nothing, psize, ck)
return SimpleNewtonRaphson(; autodiff)
end
function __set_ad_chunksize(solver::SimpleTrustRegion, ck, sparsity, psize)
solver.autodiff === nothing || return solver
autodiff = __ad_backend(nothing, psize, ck)
return SimpleTrustRegion(; autodiff)
end
__set_ad_chunksize(solver, ck, sparsity, psize) = solver
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)
sd = SymbolicsSparsityDetection()
adtype = AutoSparseFiniteDiff()
J = sparse_jacobian(adtype, sd, bruss_f!, y, u0)
colors = matrix_colors(J)
begin
J_ = copy(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, sd, f!::F, y, x) where {F}
cache = sparse_jacobian_cache(adtype, sd, f!, y, x)
J = init_jacobian(cache)
for i in 1:10
sparse_jacobian!(J, adtype, cache, f!, y, x)
end
return J
end
Ns = [2^i for i in 1:8]
sd = [SymbolicsSparsityDetection(), ApproximateJacobianSparsity(; ntrials = 3)]
adtypes = [AutoSparseFiniteDiff(), AutoSparsePolyesterForwardDiff(; chunksize = 8)]
algs = vcat(vec(collect(Iterators.product(sd, adtypes))),
[(NoSparsityDetection(), AutoPolyesterForwardDiff(; chunksize = 8))])
times = Matrix{Float64}(undef, length(Ns), length(algs))
for (i, N) in enumerate(Ns)
@info N
test_problem = generate_brusselator_problem(N)
bruss_f!, u0 = (du, u) -> test_problem.f(du, u, test_problem.p), test_problem.u0
y = similar(u0)
for (j, (sd, adtype)) in enumerate(algs)
if N < 2^9 || (N ≥ 2^9 && sd isa SymbolicsSparsityDetection)
times[i, j] = @belapsed $cache_and_compute_10_jacobians($adtype, $sd, $bruss_f!, $y, $u0)
@info times[i, j]
else
@info "Skipping would have timed out"
times[i, j] = NaN
end
end
endPlotting the results.
fig = begin
cycle = Cycle([:marker], covary = true)
plot_theme = Theme(Lines = (; cycle), Scatter = (; cycle))
with_theme(plot_theme) do
fig = Figure(; size = (1400, 1400 * 0.5))
ax = Axis(fig[1, 1]; title = "Sparsity Pattern for 2D Brusselator Jacobian",
titlesize = 22, titlegap = 10,
xticksize = 20, yticksize = 20, xticklabelsize = 20, yticklabelsize = 20,
xtickwidth = 2.5, ytickwidth = 2.5, spinewidth = 2.5, yreversed = true)
spy!(ax, J_; markersize = 16, marker = :circle, framecolor = :lightgray,
colormap = :seaborn_bright, strokewidth = 3)
ax = Axis(fig[1, 2]; title = "Scaling of Sparse Jacobian Computation Algorithms",
titlesize = 22, titlegap = 10, xscale = log10, yscale = log10,
xticksize = 20, yticksize = 20, xticklabelsize = 20, yticklabelsize = 20,
xtickwidth = 2.5, ytickwidth = 2.5, spinewidth = 2.5,
xlabel = L"Input Dimension ($\mathbf{N}$)", ylabel = L"Time $\mathbf{(s)}$", xlabelsize = 22,
ylabelsize = 22, yaxisposition = :right)
colors = cgrad(:seaborn_bright, length(algs); categorical = true)
line_list = []
scatter_list = []
Ns_ = Ns .^ 2 .* 2
push!(line_list,
lines!(
ax, Ns_, times[:, 1], color = colors[1], linewidth = 5, linestyle = :dash))
push!(
scatter_list, scatter!(ax, Ns_, times[:, 1], color = colors[1], strokewidth = 3, markersize = 16))
push!(line_list, lines!(ax, Ns_, times[:, 2], color = colors[2], linewidth = 5))
push!(
scatter_list, scatter!(ax, Ns_, times[:, 2], color = colors[2], strokewidth = 3, markersize = 16))
push!(line_list,
lines!(
ax, Ns_, times[:, 3], color = colors[3], linewidth = 5, linestyle = :dash))
push!(
scatter_list, scatter!(ax, Ns_, times[:, 3], color = colors[3], strokewidth = 3, markersize = 16))
push!(line_list, lines!(ax, Ns_, times[:, 4], color = colors[4], linewidth = 5))
push!(
scatter_list, scatter!(ax, Ns_, times[:, 4], color = colors[4], strokewidth = 3, markersize = 16))
push!(line_list,
lines!(
ax, Ns_, times[:, 5], color = colors[5], linewidth = 5, linestyle = :dot))
push!(
scatter_list, scatter!(ax, Ns_, times[:, 5], color = colors[5], strokewidth = 3, markersize = 16))
group_symbolics = [
[
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 [1, 3]]
group_approx = [
[
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 [2, 4]]
legend_polyester = [
[
LineElement(; color = line_list[5].color,
linestyle = line_list[5].linestyle,
linewidth = line_list[5].linewidth),
MarkerElement(; color = scatter_list[5].color,
marker = scatter_list[5].marker,
strokewidth = scatter_list[5].strokewidth,
markersize = scatter_list[5].markersize)
]
]
axislegend(ax,
[group_symbolics, group_approx, legend_polyester],
[
["Finite Diff", "Forward Diff"],
["Finite Diff", "Threaded Forward Diff"],
["Threaded Forward Diff"]
],
["Symbolic Sparsity", "Approx. Sparsity", "No Sparsity"];
position = :rb, 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 (Dense)", alg = NewtonRaphson(; linsolve = nothing)),
(; pkg = :nonlinearsolve, sparsity = :approx, name = "NR (Approx. Sparse)", alg = NewtonRaphson(; linsolve = nothing)),
(; pkg = :nonlinearsolve, sparsity = :exact, name = "NR (Exact Sparse)", alg = NewtonRaphson(; linsolve = nothing)),
(; pkg = :wrapper, sparsity = :none, name = "NR [NLsolve.jl]", alg = NLsolveJL(; method = :newton, autodiff = :forward)),
(; pkg = :wrapper, sparsity = :none, name = "Mod. NR [Sundials]", alg = KINSOL()),
(; pkg = :nonlinearsolve, sparsity = :none, name = "TR (Dense)", alg = TrustRegion(; radius_update_scheme = RUS.NLsolve, linsolve = nothing)),
(; pkg = :nonlinearsolve, sparsity = :approx, name = "TR (Approx. Sparse)", alg = TrustRegion(; radius_update_scheme = RUS.NLsolve, linsolve = nothing)),
(; pkg = :nonlinearsolve, sparsity = :exact, name = "TR (Exact Sparse)", alg = TrustRegion(; radius_update_scheme = RUS.NLsolve, linsolve = nothing)),
(; pkg = :wrapper, sparsity = :none, name = "TR [NLsolve.jl]", alg = NLsolveJL(; autodiff = :forward)),
(; pkg = :wrapper, sparsity = :none, name = "Mod. Powell [MINPACK]", alg = CMINPACK()),
]
runtimes_scaling = zeros(length(solvers_scaling), length(Ns)) .- 1
for (i, N) in enumerate(Ns)
prob_dense = generate_brusselator_problem(N)
prob_approx_sparse = generate_brusselator_problem(N;
sparsity = ApproximateJacobianSparsity())
prob_exact_sparse = generate_brusselator_problem(N;
sparsity = SymbolicsSparsityDetection())
@info "Benchmarking N = $N"
for (j, solver) in enumerate(solvers_scaling)
ptype = solver.sparsity
alg = solver.alg
name = solver.name
if ptype == :none
prob = prob_dense
elseif ptype == :approx
prob = prob_approx_sparse
elseif ptype == :exact
prob = prob_exact_sparse
end
if (j > 1 && runtimes_scaling[j - 1, i] == -1) || (alg isa CMINPACK && N > 32) ||
(alg isa KINSOL && N > 64) ||
(alg isa NLsolveJL && N > 64 && alg.method == :trust_region) ||
(alg isa GeneralizedFirstOrderAlgorithm{nothing, :TrustRegion} && N > 64) ||
(alg isa NLsolveJL && N > 128 && alg.method == :newton) ||
(alg isa GeneralizedFirstOrderAlgorithm{nothing, :NewtonRaphson} && N > 128 && ptype == :none) ||
(alg isa GeneralizedFirstOrderAlgorithm{nothing, :NewtonRaphson} && N > 150 && ptype == :approx)
# The last benchmark failed so skip this too
runtimes_scaling[j, i] = NaN
@warn "$(name): Would Have Timed out"
else
alg_concrete = __set_ad_chunksize(alg, NonlinearSolve.pickchunksize(prob.u0),
prob.f.sparsity, length(prob.u0))
function __benchmark_function()
sol = solve(prob, alg_concrete; abstol=1e-6, reltol=1e-6,
termination_condition=AbsNormTerminationMode())
runtimes_scaling[j, i] = @belapsed solve($prob, $alg_concrete; abstol=1e-6,
reltol=1e-6, termination_condition=$AbsNormTerminationMode())
@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(:seaborn_bright, length(solvers_scaling); categorical = true)
theme = Theme(Lines = (cycle = cycle,), Scatter = (cycle = cycle,))
LINESTYLES = Dict(
(:nonlinearsolve, :none) => :solid,
(:nonlinearsolve, :approx) => :dash,
(:nonlinearsolve, :exact) => :dashdot,
# (:simplenonlinearsolve, :none) => :solid,
(: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 = log10, yscale = log10, 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\n(Fastest to Slowest)";
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}Jacobian-Free Newton / TR Krylov Methods
In this section, we will benchmark jacobian-free nonlinear solvers with Krylov methods. We will use preconditioning from AlgebraicMultigrid.jl and IncompleteLU.jl. Unfortunately, our ability to use 3rd party software is limited here, since only Sundials.jl supports jacobian-free methods via :GMRES.
using AlgebraicMultigrid, IncompleteLU
function incompletelu(W, du, u, p, t, newW, Plprev, Prprev, solverdata)
if newW === nothing || newW
Pl = ilu(W, τ = 50.0)
else
Pl = Plprev
end
Pl, nothing
end
function algebraicmultigrid(W, du, u, p, t, newW, Plprev, Prprev, solverdata)
if newW === nothing || newW
Pl = aspreconditioner(ruge_stuben(convert(AbstractMatrix, W)))
else
Pl = Plprev
end
Pl, nothing
end
function algebraicmultigrid_jacobi(W, du, u, p, t, newW, Plprev, Prprev, solverdata)
if newW === nothing || newW
A = convert(AbstractMatrix, W)
Pl = AlgebraicMultigrid.aspreconditioner(AlgebraicMultigrid.ruge_stuben(
A, presmoother = AlgebraicMultigrid.Jacobi(rand(size(A, 1))),
postsmoother = AlgebraicMultigrid.Jacobi(rand(size(A, 1)))))
else
Pl = Plprev
end
Pl, nothing
end
Ns = 2 .^ (2:7)
solvers_scaling_jacobian_free = [
(; pkg = :nonlinearsolve, name = "Newton Krylov", alg = NewtonRaphson(; linsolve = KrylovJL_GMRES())),
(; pkg = :nonlinearsolve, name = "Newton Krylov (ILU)", alg = NewtonRaphson(; linsolve = KrylovJL_GMRES(), precs = incompletelu, concrete_jac = true)),
(; pkg = :nonlinearsolve, name = "Newton Krylov (AMG)", alg = NewtonRaphson(; linsolve = KrylovJL_GMRES(), precs = algebraicmultigrid, concrete_jac = true)),
(; pkg = :nonlinearsolve, name = "Newton Krylov (AMG Jacobi)", alg = NewtonRaphson(; linsolve = KrylovJL_GMRES(), precs = algebraicmultigrid_jacobi, concrete_jac = true)),
(; pkg = :nonlinearsolve, name = "TR Krylov", alg = TrustRegion(; linsolve = KrylovJL_GMRES())),
(; pkg = :nonlinearsolve, name = "TR Krylov (ILU)", alg = TrustRegion(; linsolve = KrylovJL_GMRES(), precs = incompletelu, concrete_jac = true)),
(; pkg = :nonlinearsolve, name = "TR Krylov (AMG)", alg = TrustRegion(; linsolve = KrylovJL_GMRES(), precs = algebraicmultigrid, concrete_jac = true)),
(; pkg = :nonlinearsolve, name = "TR Krylov (AMG Jacobi)", alg = TrustRegion(; linsolve = KrylovJL_GMRES(), precs = algebraicmultigrid_jacobi, concrete_jac = true)),
(; pkg = :wrapper, name = "Newton Krylov [Sundials]", alg = KINSOL(; linear_solver = :GMRES)),
]
runtimes_scaling = zeros(length(solvers_scaling_jacobian_free), length(Ns)) .- 1
for (i, N) in enumerate(Ns)
prob = generate_brusselator_problem(N;
sparsity = ifelse(N ≥ 6, SymbolicsSparsityDetection(), ApproximateJacobianSparsity()))
@info "Benchmarking N = $N"
for (j, solver) in enumerate(solvers_scaling_jacobian_free)
alg = solver.alg
name = solver.name
if (j > 1 && runtimes_scaling[j - 1, i] == -1)
# The last benchmark failed so skip this too
runtimes_scaling[j, i] = NaN
@warn "$(name): Would Have Timed out"
else
alg_concrete = __set_ad_chunksize(alg, NonlinearSolve.pickchunksize(prob.u0),
prob.f.sparsity, length(prob.u0))
function __benchmark_function()
sol = solve(prob, alg_concrete; abstol=1e-6, reltol=1e-6,
linsolve_kwargs = (; abstol = 1e-9, reltol = 1e-9),
termination_condition=AbsNormTerminationMode())
if SciMLBase.successful_retcode(sol) || norm(sol.resid, Inf) ≤ 1e-5
runtimes_scaling[j, i] = @belapsed solve($prob, $alg_concrete; abstol=1e-6,
reltol=1e-6, termination_condition=$AbsNormTerminationMode())
else
runtimes_scaling[j, i] = NaN
end
@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(:seaborn_bright, length(solvers_scaling_jacobian_free); categorical = true)
theme = Theme(Lines = (cycle = cycle,), Scatter = (cycle = cycle,))
LINESTYLES = Dict(
(:nonlinearsolve, :none) => :solid,
(:nonlinearsolve, :amg) => :dot,
(:nonlinearsolve, :amg_jacobi) => :dash,
(:nonlinearsolve, :ilu) => :dashdot,
)
Ns_ = Ns .^ 2 .* 2
with_theme(theme) do
fig = Figure(; size = (WIDTH, HEIGHT))
ax = Axis(fig[1, 1:2], ylabel = L"Time ($s$)", xlabel = L"Problem Size ($N$)",
xscale = log10, yscale = log10, xlabelsize = 22, ylabelsize = 22,
xticklabelsize = 20, yticklabelsize = 20, xtickwidth = STROKEWIDTH,
ytickwidth = STROKEWIDTH, spinewidth = STROKEWIDTH)
idxs = get_ordering(runtimes_scaling)
ls, scs, labels = [], [], []
for (i, solver) in zip(idxs, solvers_scaling_jacobian_free[idxs])
all(isnan, runtimes_scaling[i, :]) && continue
precon = occursin("AMG Jacobi", solver.name) ? :amg_jacobi : occursin("AMG", solver.name) ? :amg : occursin("ILU", solver.name) ? :ilu : :none
linestyle = LINESTYLES[(solver.pkg, precon)]
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)
push!(labels, solver.name)
end
axislegend(ax, [[l, sc] for (l, sc) in zip(ls, scs)], labels,
"Successful Solvers\n(Fastest to Slowest)";
framevisible=true, framewidth = STROKEWIDTH, orientation = :vertical,
titlesize = 20, labelsize = 16, position = :rb,
tellheight = true, tellwidth = false, patchsize = (40.0f0, 20.0f0))
axislegend(ax, [
LineElement(; linestyle = :solid, linewidth = 5),
LineElement(; linestyle = :dot, linewidth = 5),
LineElement(; linestyle = :dash, linewidth = 5),
LineElement(; linestyle = :dashdot, linewidth = 5),
], ["No Preconditioning", "AMG", "AMG Jacobi", "Incomplete LU"],
"Preconditioning"; framevisible=true, framewidth = STROKEWIDTH,
orientation = :vertical, titlesize = 20, labelsize = 16,
tellheight = true, tellwidth = true, patchsize = (40.0f0, 20.0f0),
position = :lt)
fig[0, :] = Label(fig,
"Brusselator 2D: Scaling of Jacobian-Free Nonlinear Solvers with Problem Size",
fontsize = 24, tellwidth = false, font = :bold)
return fig
end
end
save("brusselator_krylov_methods_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.
DEFAULT_FORWARD_AD = AutoSparsePolyesterForwardDiff(; chunksize = 12)
solvers_all = [
(; pkg = :nonlinearsolve, name = "Default PolyAlg", solver = Dict(:alg => FastShortcutNonlinearPolyalg(; autodiff = DEFAULT_FORWARD_AD))),
(; pkg = :nonlinearsolve, name = "RobustMultiNewton (GMRES)", solver = Dict(:alg => RobustMultiNewton(; linsolve = KrylovJL_GMRES(), autodiff = DEFAULT_FORWARD_AD))),
(; pkg = :nonlinearsolve, name = "Newton Raphson", solver = Dict(:alg => NewtonRaphson(; linsolve = nothing, autodiff = DEFAULT_FORWARD_AD))),
(; pkg = :nonlinearsolve, name = "Newton Krylov", solver = Dict(:alg => NewtonRaphson(; linsolve = KrylovJL_GMRES(), autodiff = DEFAULT_FORWARD_AD))),
(; pkg = :nonlinearsolve, name = "Trust Region", solver = Dict(:alg => TrustRegion(; autodiff = DEFAULT_FORWARD_AD))),
(; pkg = :nonlinearsolve, name = "TR Krylov", solver = Dict(:alg => TrustRegion(; linsolve = KrylovJL_GMRES(), autodiff = DEFAULT_FORWARD_AD))),
(; 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())),
(; pkg = :wrapper, name = "Newton Krylov [Sundials]", solver = Dict(:alg => KINSOL(; linear_solver = :GMRES))),
(; pkg = :wrapper, name = "Mod. Powell [MINPACK]", solver = Dict(:alg => CMINPACK())),
];prob_wpd = generate_brusselator_problem(32; sparsity = ApproximateJacobianSparsity())
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 = 0.000
20409942583029592).
[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 = 5.94616
2345262808e-6).
[Warn] Solver Newton Krylov [Sundials] returned retcode Failure with an res
idual norm = 34.06162309153361.
[Info] Solver Mod. Powell [MINPACK] successfully solved the problem (norm =
1.962972484440553e-6).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(:seaborn_bright, 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 = log10, yscale = log10, 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 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 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: 128 default, 0 interactive, 64 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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonlinearProblem/Project.toml`
[2169fc97] AlgebraicMultigrid v0.6.0
[6e4b80f9] BenchmarkTools v1.5.0
[13f3f980] CairoMakie v0.11.9
[2b5f629d] DiffEqBase v6.148.0
[f3b72e0c] DiffEqDevTools v2.44.2
[40713840] IncompleteLU v0.2.1
[b964fa9f] LaTeXStrings v1.3.1
[7ed4a6bd] LinearSolve v2.27.0
[4854310b] MINPACK v1.2.0
[2774e3e8] NLsolve v4.5.1
[b7050fa9] NonlinearProblemLibrary v0.1.2
[8913a72c] NonlinearSolve v3.8.0
[98d1487c] PolyesterForwardDiff v0.1.1
[08abe8d2] PrettyTables v2.3.1
[31c91b34] SciMLBenchmarks v0.1.3
[efcf1570] Setfield v1.1.1
[727e6d20] SimpleNonlinearSolve v1.6.0
[47a9eef4] SparseDiffTools v2.17.0
[f1835b91] SpeedMapping v0.3.0
[860ef19b] StableRNGs v1.0.1
[90137ffa] StaticArrays v1.9.3
[c3572dad] Sundials v4.24.0
⌃ [0c5d862f] Symbolics v5.23.0
Info Packages marked with ⌃ have new versions available and may be upgradable.And the full manifest:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NonlinearProblem/Manifest.toml`
⌃ [47edcb42] ADTypes v0.2.6
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⌅ [ec485272] ArnoldiMethod v0.2.0
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[861a8166] Combinatorics v1.0.2
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[bbf7d656] CommonSubexpressions v0.3.0
[34da2185] Compat v4.14.0
[b152e2b5] CompositeTypes v0.1.3
[2569d6c7] ConcreteStructs v0.2.3
⌃ [f0e56b4a] ConcurrentUtilities v2.3.1
[8f4d0f93] Conda v1.10.0
[187b0558] ConstructionBase v1.5.4
[d38c429a] Contour v0.6.2
[a2441757] Coverage v1.6.0
[c36e975a] CoverageTools v1.3.0
[adafc99b] CpuId v0.3.1
[a8cc5b0e] Crayons v4.1.1
[9a962f9c] DataAPI v1.16.0
[864edb3b] DataStructures v0.18.18
[e2d170a0] DataValueInterfaces v1.0.0
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[163ba53b] DiffResults v1.1.0
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[b4f34e82] Distances v0.10.11
[31c24e10] Distributions v0.25.107
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⌃ [5b8099bc] DomainSets v0.7.9
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⌃ [f151be2c] EnzymeCore v0.6.5
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⌃ [5789e2e9] FileIO v1.16.2
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⌃ [cd3eb016] HTTP v1.10.3
[eafb193a] Highlights v0.5.2
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[90137ffa] StaticArrays v1.9.3
[1e83bf80] StaticArraysCore v1.4.2
[82ae8749] StatsAPI v1.7.0
[2913bbd2] StatsBase v0.34.2
[4c63d2b9] StatsFuns v1.3.1
[7792a7ef] StrideArraysCore v0.5.2
[69024149] StringEncodings v0.3.7
[892a3eda] StringManipulation v0.3.4
[09ab397b] StructArrays v0.6.18
[c3572dad] Sundials v4.24.0
[2efcf032] SymbolicIndexingInterface v0.3.11
[d1185830] SymbolicUtils v1.5.1
⌃ [0c5d862f] Symbolics v5.23.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.11.1
[62fd8b95] TensorCore v0.1.1
[8290d209] ThreadingUtilities v0.5.2
⌅ [731e570b] TiffImages v0.6.8
[a759f4b9] TimerOutputs v0.5.23
[3bb67fe8] TranscodingStreams v0.10.4
[d5829a12] TriangularSolve v0.1.20
[410a4b4d] Tricks v0.1.8
[981d1d27] TriplotBase v0.1.0
[781d530d] TruncatedStacktraces v1.4.0
[9d95972d] TupleTools v1.5.0
[5c2747f8] URIs v1.5.1
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[a7c27f48] Unityper v0.1.6
[3d5dd08c] VectorizationBase v0.21.65
[81def892] VersionParsing v1.3.0
[19fa3120] VertexSafeGraphs v0.2.0
[44d3d7a6] Weave v0.10.12
[efce3f68] WoodburyMatrices v1.0.0
[ddb6d928] YAML v0.4.9
[c2297ded] ZMQ v1.2.2
[6e34b625] Bzip2_jll v1.0.8+1
[4e9b3aee] CRlibm_jll v1.0.1+0
[83423d85] Cairo_jll v1.18.0+1
[5ae413db] EarCut_jll v2.2.4+0
[2e619515] Expat_jll v2.5.0+0
⌃ [b22a6f82] FFMPEG_jll v4.4.2+2
[f5851436] FFTW_jll v3.3.10+0
[a3f928ae] Fontconfig_jll v2.13.93+0
[d7e528f0] FreeType2_jll v2.13.1+0
[559328eb] FriBidi_jll v1.0.10+0
[78b55507] Gettext_jll v0.21.0+0
⌃ [f8c6e375] Git_jll v2.36.1+2
[7746bdde] Glib_jll v2.80.0+0
[3b182d85] Graphite2_jll v1.3.14+0
[528830af] Gumbo_jll v0.10.2+0
[2e76f6c2] HarfBuzz_jll v2.8.1+1
[905a6f67] Imath_jll v3.1.7+0
[1d5cc7b8] IntelOpenMP_jll v2024.0.2+0
[aacddb02] JpegTurbo_jll v3.0.2+0
[c1c5ebd0] LAME_jll v3.100.1+0
[1d63c593] LLVMOpenMP_jll v15.0.7+0
[dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
[d4300ac3] Libgcrypt_jll v1.8.7+0
[7add5ba3] Libgpg_error_jll v1.42.0+0
[94ce4f54] Libiconv_jll v1.17.0+0
[4b2f31a3] Libmount_jll v2.39.3+0
[38a345b3] Libuuid_jll v2.39.3+1
[856f044c] MKL_jll v2024.0.0+0
[e7412a2a] Ogg_jll v1.3.5+1
[18a262bb] OpenEXR_jll v3.1.4+0
⌅ [9bd350c2] OpenSSH_jll v8.9.0+1
⌅ [458c3c95] OpenSSL_jll v1.1.23+0
[efe28fd5] OpenSpecFun_jll v0.5.5+0
[91d4177d] Opus_jll v1.3.2+0
[36c8627f] Pango_jll v1.52.1+0
[30392449] Pixman_jll v0.42.2+0
[f50d1b31] Rmath_jll v0.4.0+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
[02c8fc9c] XML2_jll v2.12.5+0
[aed1982a] XSLT_jll v1.1.34+0
[4f6342f7] Xorg_libX11_jll v1.8.6+0
[0c0b7dd1] Xorg_libXau_jll v1.0.11+0
[a3789734] Xorg_libXdmcp_jll v1.1.4+0
[1082639a] Xorg_libXext_jll v1.3.4+4
[ea2f1a96] Xorg_libXrender_jll v0.9.10+4
[14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
[c7cfdc94] Xorg_libxcb_jll v1.15.0+0
[c5fb5394] Xorg_xtrans_jll v1.5.0+0
[8f1865be] ZeroMQ_jll v4.3.5+0
[9a68df92] isoband_jll v0.2.3+0
[a4ae2306] libaom_jll v3.4.0+0
[0ac62f75] libass_jll v0.15.1+0
[f638f0a6] libfdk_aac_jll v2.0.2+0
[b53b4c65] libpng_jll v1.6.43+1
[47bcb7c8] libsass_jll v3.6.4+0
[075b6546] libsixel_jll v1.10.3+0
[a9144af2] libsodium_jll v1.0.20+0
[f27f6e37] libvorbis_jll v1.3.7+1
[1270edf5] x264_jll v2021.5.5+0
[dfaa095f] x265_jll v3.5.0+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 v1.1.0+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+2
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.8.0+1
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`