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Dense GPU Offload — Where Does the GPU Start to Win?

Should you move your dense solve to the GPU? The answer is size-dependent, and the honest accounting has to separate transfer from compute: offload pays PCIe both ways (H2D for the matrix, D2H for the solution), and below some N that overhead swamps any factorization speedup. This benchmark measures the crossover on the folder's GPU runner, with transfer cost reported explicitly — a GPU benchmark that hides transfer is marketing, not evidence.

A crossover claim is only as strong as its CPU baseline, so the CPU side is not one algorithm but the best of several — including LinearSolve's default choice, which is the number a user gets by not specifying an algorithm at all. Compared, all through the same LinearProblem API:

  • CPU: LUFactorization (OpenBLAS), RFLUFactorization (RecursiveFactorization), the LinearSolve default algorithm choice, and MKLLUFactorization on machines where LinearSolve loads MKL at all — it deliberately does not on AMD EPYC, which is this folder's runner CPU, and the document prints a note instead of a broken column when that applies
  • CudaOffloadLUFactorization — full-precision GPU offload
  • CudaOffloadQRFactorization — QR variant
  • CUDAOffload32MixedLUFactorization — Float32 factorization with Float64 refinement of the result; its correctness gate is accordingly looser, and its win condition (bandwidth-bound GPUs) is precisely what this document tests
using BenchmarkTools, Random, Printf
using LinearAlgebra, LinearSolve, RecursiveFactorization, MKL_jll
using CUDA

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()))
println("CPU: ", Sys.cpu_info()[1].model, " — ", Sys.CPU_THREADS,
    " hardware threads, ", BLAS.get_num_threads(), " BLAS threads, ",
    Threads.nthreads(), " Julia threads")

# (name, algorithm, gate tolerance, device class); `nothing` is LinearSolve's
# default algorithm choice for the problem — the baseline a non-expert gets.
algs = Any[
    ("CPU OpenBLAS LU", LUFactorization(), 1e-10, :cpu),
    ("CPU RFLU", RFLUFactorization(), 1e-10, :cpu),
    ("CPU default choice", nothing, 1e-10, :cpu),
    ("GPU LU offload", CudaOffloadLUFactorization(), 1e-10, :gpu),
    ("GPU QR offload", CudaOffloadQRFactorization(), 1e-10, :gpu),
    ("GPU 32-mixed LU", CUDAOffload32MixedLUFactorization(), 1e-4, :gpu),
]

# LinearSolve deliberately does not load MKL on CPUs where it defaults away
# from it — notably AMD EPYC, this folder's runner CPU (LinearSolve.jl#518,
# the LoadMKL_JLL preference). Probe once and include the MKL row only where
# it actually runs, so the CPU baselines are exactly the options a user of
# this machine has; a NaN column with load errors is not a baseline.
mkl_works = try
    solve(LinearProblem(Matrix(8.0I, 8, 8), ones(8)), MKLLUFactorization())
    true
catch
    false
end
if mkl_works
    insert!(algs, 2, ("CPU MKL LU", MKLLUFactorization(), 1e-10, :cpu))
else
    println("MKL LU excluded: LinearSolve does not load MKL_jll on this CPU ",
        "by default (LoadMKL_JLL preference; see LinearSolve.jl#518).")
end
cpu_idx = findall(a -> a[4] == :cpu, algs)
gpu_idx = findall(a -> a[4] == :gpu, algs)

ns = [256, 512, 1024, 2048, 4096, 8192]
GPU: Tesla V100-PCIE-32GB
CPU: AMD EPYC 9354 32-Core Processor — 58 hardware threads, 29 BLAS threads
, 58 Julia threads
MKL LU excluded: LinearSolve does not load MKL_jll on this CPU by default (
LoadMKL_JLL preference; see LinearSolve.jl#518).
6-element Vector{Int64}:
  256
  512
 1024
 2048
 4096
 8192

Methodology

Per size: a correctness gate against a reference solve, then the end-to-end solve time (evals=1, fresh problem per sample — the full cost a user pays), and separately the pure round-trip transfer time for the same data (CuArray(A) up, Array(x) down). Transfer is measured with CUDA.@sync so asynchronous copies can't hide.

res_time = fill(NaN, length(ns), length(algs))
res_xfer = fill(NaN, length(ns))

for (i, n) in enumerate(ns)
    rng = MersenneTwister(123)
    A = rand(rng, n, n) + n * I
    b = rand(rng, n)
    ref = A \ b
    @info "n=$n"

    for (j, (name, alg, tol, _)) in enumerate(algs)
        try
            sol = solve(LinearProblem(A, b), alg)
            err = norm(sol.u - ref) / norm(ref)
            if !(err < tol)
                @warn "correctness gate failed — omitted" name n err
                continue
            end
            res_time[i, j] = @belapsed solve(LinearProblem($A, $b), $alg).u evals=1
        catch e
            @warn "$name failed at n=$n" exception=(e,)
        end
    end

    # Round-trip transfer for the same data, isolated.
    res_xfer[i] = @belapsed begin
        Ag = CuArray($A)
        bg = CuArray($b)
        CUDA.@sync Ag
        Array(bg)
    end evals=1
end

Results

using Plots
p = plot(; xlabel = "N", ylabel = "time / s", xscale = :log2, yscale = :log10,
    title = "Dense solve: CPU vs GPU offload", legend = :topleft)
for (j, (name, _, _, class)) in enumerate(algs)
    mask = .!isnan.(res_time[:, j])
    any(mask) && plot!(p, ns[mask], res_time[mask, j];
        marker = class == :cpu ? :circle : :diamond, label = name)
end
plot!(p, ns, res_xfer; linestyle = :dash, color = :gray,
    label = "transfer round-trip only")
p

println("   N   | " * join([rpad(a[1], 18) for a in algs], "| ") * "| transfer (s)")
println("-"^(9 + 20 * length(algs) + 13))
for (i, n) in enumerate(ns)
    vals = join([@sprintf("%17.4g ", res_time[i, j]) for j in 1:length(algs)], "| ")
    @printf("%6d | %s| %12.4g\n", n, vals, res_xfer[i])
end

# Crossover: first size where the best GPU variant beats the BEST CPU option
# (not just one CPU algorithm — a crossover against a slow baseline is not a
# crossover). The best-CPU column names which algorithm set the bar.
best_cpu = [minimum(filter(!isnan, res_time[i, cpu_idx]); init = Inf) for i in 1:length(ns)]
best_gpu = [minimum(filter(!isnan, res_time[i, gpu_idx]); init = Inf) for i in 1:length(ns)]
println()
for (i, n) in enumerate(ns)
    j = cpu_idx[argmin(replace(res_time[i, cpu_idx], NaN => Inf))]
    @printf("%6d | best CPU: %-18s %.4g s | best GPU: %.4g s | GPU/CPU: %.2fx\n",
        n, algs[j][1], best_cpu[i], best_gpu[i], best_cpu[i] / best_gpu[i])
end
cross = findfirst(i -> best_gpu[i] < best_cpu[i], 1:length(ns))
println()
println(cross === nothing ?
    "No crossover in the measured range — the best CPU option wins throughout." :
    "Crossover: GPU offload first beats the best CPU option at N = $(ns[cross]).")
N   | CPU OpenBLAS LU   | CPU RFLU          | CPU default choice| GPU LU
 offload    | GPU QR offload    | GPU 32-mixed LU   | transfer (s)
---------------------------------------------------------------------------
-------------------------------------------------------------------
   256 |          0.001182 |         0.0002462 |         0.0002652 |       
  0.0009261 |          0.003256 |         0.0007535 |    8.475e-05
   512 |             1.133 |           0.01199 |          0.007805 |       
   0.002215 |           0.00809 |          0.002074 |    0.0002329
  1024 |           0.02306 |          0.008911 |           0.02603 |       
   0.005602 |           0.01842 |          0.004234 |    0.0007239
  2048 |           0.07961 |           0.03984 |           0.09721 |       
    0.03332 |           0.06457 |           0.02972 |     0.002685
  4096 |            0.2803 |            0.1672 |            0.3506 |       
     0.1027 |            0.2186 |            0.1236 |      0.01053
  8192 |             1.184 |            0.8811 |             1.464 |       
     0.4114 |            0.9035 |            0.4871 |      0.04187

   256 | best CPU: CPU RFLU           0.0002462 s | best GPU: 0.0007535 s |
 GPU/CPU: 0.33x
   512 | best CPU: CPU default choice 0.007805 s | best GPU: 0.002074 s | G
PU/CPU: 3.76x
  1024 | best CPU: CPU RFLU           0.008911 s | best GPU: 0.004234 s | G
PU/CPU: 2.10x
  2048 | best CPU: CPU RFLU           0.03984 s | best GPU: 0.02972 s | GPU
/CPU: 1.34x
  4096 | best CPU: CPU RFLU           0.1672 s | best GPU: 0.1027 s | GPU/C
PU: 1.63x
  8192 | best CPU: CPU RFLU           0.8811 s | best GPU: 0.4114 s | GPU/C
PU: 2.14x

Crossover: GPU offload first beats the best CPU option at N = 512.

Reading the result

The dashed transfer line is the floor no offload algorithm can beat: where a solver's curve approaches it, the algorithm is transfer-bound and further GPU speedup is irrelevant at that size. The stated crossover N is the actionable number — below it, stay on the CPU; above it, offload pays. It is computed against the best CPU option at each size, and the best-CPU column shows which algorithm sets that bar — on the runner's EPYC, where LinearSolve deliberately does not load MKL, that is RFLU across this sweep; a GPU "win" over plain OpenBLAS alone would say more about BLAS libraries than about the GPU. The spread among the CPU rows is itself a result — the gap between the slowest CPU row and the default choice is what hand-picking the wrong algorithm costs, and how close the default sits to the best row is the value of LinearSolve's automatic selection. The 32-mixed variant's gap to full-precision GPU LU shows what halving the factorization's memory traffic buys; its residual (gated at 1e-4) is the accuracy price.

Caveats: one GPU model per run (the runner's), Float64 inputs, well-conditioned random matrices (no pivoting stress). The size dependence of the crossover on GPU generation is deliberately out of scope — this document publishes from a fixed runner precisely so the number is stable.

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","DenseGPUOffload.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`
  [6e4b80f9] BenchmarkTools v1.8.0
⌃ [052768ef] CUDA v6.2.0
⌅ [45b445bb] CUDSS v0.7.0
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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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  [e7412a2a] Ogg_jll v1.3.6+0
⌃ [9bd350c2] OpenSSH_jll v10.4.1+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.0+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.3+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+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
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+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
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [8f1865be] ZeroMQ_jll v4.3.6+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [1e29f10c] demumble_jll v1.3.0+0
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.13.3+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [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`