Finite Difference Sparse PDE Jacobian Factorization Benchmarks

using BenchmarkTools, Random, VectorizationBase
using LinearAlgebra, SparseArrays, LinearSolve, Sparspak
# PureUMFPACK backs PureUMFPACKFactorization via LinearSolvePureUMFPACKExt.
# Use `import` (not `using`): PureUMFPACK ≤0.1 exports `solve`, which collides
# with LinearSolve/CommonSolve. PureKLU / SupernodalLU need no extra load.
import PureUMFPACK
import Pardiso
import ParU_jll
using Plots

BenchmarkTools.DEFAULT_PARAMETERS.seconds = 0.5

# Why do I need to set this ?
BenchmarkTools.DEFAULT_PARAMETERS.samples = 10

# Sparse matrix generation on  a n-dimensional rectangular grid. After
# https://discourse.julialang.org/t/seven-lines-of-julia-examples-sought/50416/135
# by A. Braunstein.

A ⊕ B = kron(I(size(B, 1)), A) + kron(B, I(size(A, 1)))

function lattice(n; Tv = Float64)
    d = fill(2 * one(Tv), n)
    d[1] = one(Tv)
    d[end] = one(Tv)
    spdiagm(1 => -ones(Tv, n - 1), 0 => d, -1 => -ones(Tv, n - 1))
end

lattice(L...; Tv = Float64) = lattice(L[1]; Tv) ⊕ lattice(L[2:end]...; Tv)

#
# Create a matrix similar to that of a finite difference discretization in a `dim`-dimensional
# unit cube of  ``-Δu + δu`` with approximately N unknowns. It is strictly diagonally dominant.
#
function fdmatrix(N; dim = 2, Tv = Float64, δ = 1.0e-2)
    n = N^(1 / dim) |> ceil |> Int
    lattice([n for i in 1:dim]...; Tv) + Tv(δ) * I
end

algs = [
    UMFPACKFactorization(),
    KLUFactorization(),
    PureKLUFactorization(),
    PureUMFPACKFactorization(),
    SupernodalLUFactorization(),
    MKLPardisoFactorize(),
    SparspakFactorization()
    # ParUFactorization() is EXCLUDED: repeated ParU factorizations in one
    # process ratchet Julia's GC allocation accounting irreversibly (ParU's
    # METIS ordering allocates through the counted-malloc path; GC.gc(true)
    # does not reset the pressure). This sweep's ~400 cumulative ParU
    # factorizations at up to ~40k unknowns are far past the ~140 where a
    # minimal reproducer wedges — sub-second solves become hours, and the
    # slowdown also lands on other solvers' analysis phases as collateral.
    # Two benchmark-runner CI runs of this folder wedged this way (9h and 23h)
    # before diagnosis. Standalone ParU solves are unaffected. See the
    # LinearSolve.jl issue for the reproducer and status.
]
cols = [:red, :blue, :green, :magenta, :turquoise, :orange, :purple] # one color per alg

__parameterless_type(T) = Base.typename(T).wrapper
parameterless_type(x) = __parameterless_type(typeof(x))
parameterless_type(::Type{T}) where {T} = __parameterless_type(T)

#
# kmax=12 gives ≈ 40_000 unknowns max, can be watched in real time
# kmax=15 gives ≈ 328_000 unknows, you can go make a coffee.
# Main culprit is KLU factorization in 3D.
#
function run_and_plot(dim; kmax = 12)
    ns = [10 * 2^k for k in 0:kmax]

    res = [Float64[] for i in 1:length(algs)]

    for i in 1:length(ns)
        rng = MersenneTwister(123)
        A = fdmatrix(ns[i]; dim)
        n = size(A, 1)
        @info "dim=$(dim): $n × $n"
        b = rand(rng, n)
        u0 = rand(rng, n)

        for j in 1:length(algs)
            bt = @belapsed solve(prob, $(algs[j])).u setup=(prob = LinearProblem(copy($A),
                copy($b);
                u0 = copy($u0),
                alias = LinearAliasSpecifier(alias_A = true, alias_b = true)))
            push!(res[j], bt)
        end
    end

    p = plot(;
        ylabel = "Time/s",
        xlabel = "N",
        yscale = :log10,
        xscale = :log10,
        title = "Time for NxN  sparse LU Factorization $(dim)D",
        label = string(Symbol(parameterless_type(algs[1]))),
        legend = :outertopright)

    for i in 1:length(algs)
        plot!(p, ns, res[i];
            linecolor = cols[i],
            label = "$(string(Symbol(parameterless_type(algs[i]))))")
    end
    p
end

run_and_plot (generic function with 1 method)

run_and_plot(1)

run_and_plot(2)

run_and_plot(3)

Appendix

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/LinearSolve","SparsePDE.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 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-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolve/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
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⌃ [7ed4a6bd] LinearSolve v5.5.0
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⌃ [91a5bcdd] Plots v1.41.6
⌃ [b7e1f0a2] PureUMFPACK v0.1.4
⌃ [f2c3362d] RecursiveFactorization v0.2.26
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.1.5]
  [e56a9233] Sparspak v0.3.15
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⌃ [9e0b026c] ParU_jll v1.0.0+0
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  [2f01184e] SparseArrays v1.12.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolve/Manifest.toml`
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  [cddc5d3d] s2n_tls_jll v1.7.3+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [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
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [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 v1.3.0+1
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.4+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+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 -m`