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Sparse Direct on GPU — CUDSS vs CPU Factorizations

Does GPU sparse direct pay? CUDSS provides LU on CuSparseMatrixCSR, reached through LinearSolve's ordinary LUFactorization once the matrix lives on the device. This benchmark compares it against the strong CPU baselines (UMFPACK, KLU) on 2-D finite-difference matrices, with the two costs a real workload cares about kept separate: factor + first solve and cached re-solve — plus the H2D transfer of the CSR structure, reported explicitly, since a matrix that lives on the CPU has to pay it.

using BenchmarkTools, Random, Printf
using LinearAlgebra, SparseArrays, LinearSolve
using CUDA, CUDSS

BenchmarkTools.DEFAULT_PARAMETERS.seconds = 0.5
BenchmarkTools.DEFAULT_PARAMETERS.samples = 5

@assert CUDA.functional() "This benchmark requires a functional CUDA GPU"
println("GPU: ", CUDA.name(CUDA.device()))

# 2-D FD Laplacian (same generator family as the CPU sparse documents).
lap1d(k) = spdiagm(1 => -ones(k - 1), 0 => fill(2.0, k), -1 => -ones(k - 1))
function fd2d(m)
    kron(I(m), lap1d(m)) + kron(lap1d(m), I(m)) + 0.01I
end

ms = [50, 100, 200, 320, 450]          # grid sides → N = 2.5e3 … 2.0e5
GPU: Tesla V100-PCIE-32GB
5-element Vector{Int64}:
  50
 100
 200
 320
 450

Methodology

CPU entries get the SparseMatrixCSC problem; the GPU entry gets the same matrix as CuSparseMatrixCSR with a CuVector right-hand side. Every timing is behind a correctness gate against a CPU reference solve. Re-solve updates b on the cache and calls solve! — for CUDSS this isolates its numeric backsolve exactly as UMFPACK/KLU's is isolated on the CPU.

function bench_cpu(A, b, alg, ref)
    cache = init(LinearProblem(A, b), alg)
    sol = solve!(cache)
    err = norm(sol.u - ref) / norm(ref)
    err < 1e-8 || return nothing
    t_first = @belapsed solve!(c) setup=(
        c = init(LinearProblem($A, $b), $alg)) evals=1
    t_re = @belapsed solve!($cache)
    return (; t_first, t_re)
end

function bench_cudss(A, b, ref)
    Ag = CUDA.CUSPARSE.CuSparseMatrixCSR(A)
    bg = CuArray(b)
    cache = init(LinearProblem(Ag, bg), LUFactorization())
    sol = solve!(cache)
    err = norm(Array(sol.u) - ref) / norm(ref)
    err < 1e-8 || return nothing
    t_first = @belapsed CUDA.@sync(solve!(c)) setup=(
        c = init(LinearProblem($Ag, $bg), LUFactorization())) evals=1
    t_re = @belapsed CUDA.@sync(solve!($cache))
    t_h2d = @belapsed begin
        G = CUDA.CUSPARSE.CuSparseMatrixCSR($A)
        CUDA.@sync G
    end evals=1
    return (; t_first, t_re, t_h2d)
end

rows = []
for m in ms
    A = fd2d(m); n = size(A, 1)
    rng = MersenneTwister(123)
    b = rand(rng, n)
    ref = A \ b
    @info "grid $m×$m → n=$n, nnz=$(nnz(A))"
    umf = bench_cpu(A, b, UMFPACKFactorization(), ref)
    klu = bench_cpu(A, b, KLUFactorization(), ref)
    gpu = bench_cudss(A, b, ref)
    push!(rows, (; n, umf, klu, gpu))
end

Results

using Plots
p = plot(; xlabel = "N", ylabel = "time / s", xscale = :log10, yscale = :log10,
    title = "Sparse factor+solve: CPU vs CUDSS", legend = :topleft)
for (label, f) in (("UMFPACK", r -> r.umf), ("KLU", r -> r.klu), ("CUDSS (GPU)", r -> r.gpu))
    xs = [r.n for r in rows if f(r) !== nothing]
    ys = [f(r).t_first for r in rows if f(r) !== nothing]
    isempty(xs) || plot!(p, xs, ys; marker = :circle, label = label)
end
gx = [r.n for r in rows if r.gpu !== nothing]
plot!(p, gx, [r.gpu.t_h2d for r in rows if r.gpu !== nothing];
    linestyle = :dash, color = :gray, label = "CSR H2D transfer only")
p

println("    N   | UMF first | KLU first | GPU first | UMF re   | KLU re   | GPU re   | H2D")
println("--------+-----------+-----------+-----------+----------+----------+----------+--------")
for r in rows
    f(x, fld) = x === nothing ? NaN : getfield(x, fld)
    @printf("%7d | %9.3g | %9.3g | %9.3g | %8.3g | %8.3g | %8.3g | %.3g\n",
        r.n, f(r.umf, :t_first), f(r.klu, :t_first), f(r.gpu, :t_first),
        f(r.umf, :t_re), f(r.klu, :t_re), f(r.gpu, :t_re), f(r.gpu, :t_h2d))
end
N   | UMF first | KLU first | GPU first | UMF re   | KLU re   | GPU re 
  | H2D
--------+-----------+-----------+-----------+----------+----------+--------
--+--------
   2500 |   0.00737 |   0.00201 |   0.00128 | 6.72e-05 | 5.69e-05 | 0.00021
7 | 0.000153
  10000 |    0.0169 |    0.0158 |   0.00247 | 0.000341 | 0.000335 | 0.00036
1 | 0.000209
  40000 |    0.0888 |     0.127 |   0.00674 |  0.00184 |  0.00299 | 0.00081
3 | 0.000469
 102400 |     0.301 |     0.525 |    0.0156 |  0.00895 |   0.0128 |  0.0014
8 | 0.000985
 202500 |     0.695 |      1.34 |    0.0259 |     0.02 |   0.0286 |  0.0021
2 | 0.00182

Reading the result

Two questions decide GPU sparse direct, and the table answers both. First, does CUDSS's factor+solve beat the best CPU factorization at your size — and if it does, does the margin survive the H2D transfer when your matrix starts on the CPU? Second, in re-solve-dominated workloads (the common case), compare the re-solve columns alone: the matrix transfers once, so the H2D cost amortizes away and the backsolve throughput is the whole story.

Not shown: structures beyond 2-D FD grids, CUDSS's symbolic/numeric split across same-pattern refactorizations, and multi-GPU. Those belong to a follow-up once this baseline is published.

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/LinearSolveGPU","SparseGPU.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 9354 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 58 default, 1 interactive, 58 GC (on 58 virtual cores)
Environment:
  JULIA_CPU_THREADS = 58
  JULIA_NUM_PRECOMPILE_TASKS = 58
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolveGPU/Project.toml`
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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:

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  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [a9144af2] libsodium_jll v1.0.21+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+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 v2025.5.20
  [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`