"Jash

The most actionable question about a distributed linear solver is not how it scales but when to reach for it at all: below some problem size, a good serial sparse direct factorization wins outright, because the distributed solve pays MPI setup and communication that a single process never does. This document measures that crossover for 2-D finite-difference Laplacians: the best serial direct solvers available through LinearSolve.jl against the distributed PETSc GAMG-CG path at a fixed rank count, across a size sweep.

The output is a rule of thumb of the form "below N, use a serial factorization; above it, the distributed solve pays." It is a property of this problem class and this hardware, not a universal constant, and the document says so.

using MPI            # provides mpiexec()
using LinearAlgebra, SparseArrays, LinearSolve
using BenchmarkTools
using Plots
using Printf

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

const WORKER = joinpath(@__DIR__, "run_solve.jl")
const PROJECT = Base.active_project()
const MPIEXEC_ARGS = `-launcher fork`   # MPICH_jll hydra PMI workaround; see run_solve.jl
const P_DIST = 4                        # fixed rank count for the distributed line

# Serial solves run in-process; one BLAS thread to match the one-thread-per-rank
# discipline of the distributed worker, so the comparison is core-for-core fair:
# 1 serial core against P_DIST distributed cores, which is exactly the tradeoff
# a user weighs.
BLAS.set_num_threads(1)

# Same 2-D 5-point Laplacian the worker builds (kept in sync with run_solve.jl).
function laplacian_2d(m::Int)
    n = m * m
    I_ = Int[]; J_ = Int[]; V = Float64[]
    lin(i, j) = (j - 1) * m + i
    @inbounds for j in 1:m, i in 1:m
        k = lin(i, j)
        push!(I_, k); push!(J_, k); push!(V, 4.0)
        if i > 1
            push!(I_, k); push!(J_, lin(i - 1, j)); push!(V, -1.0)
        end
        if i < m
            push!(I_, k); push!(J_, lin(i + 1, j)); push!(V, -1.0)
        end
        if j > 1
            push!(I_, k); push!(J_, lin(i, j - 1)); push!(V, -1.0)
        end
        if j < m
            push!(I_, k); push!(J_, lin(i, j + 1)); push!(V, -1.0)
        end
    end
    return sparse(I_, J_, V, n, n)
end

serial_algs = [
    ("UMFPACK", UMFPACKFactorization()),
    ("KLU", KLUFactorization()),
    ("SupernodalLU", SupernodalLUFactorization()),
]

# Target unknown counts (the worker rounds to a square grid). Modest for CI;
# the closing section describes raising the range on dedicated hardware.
const SIZES = [10_000, 22_500, 40_000, 90_000]

4-element Vector{Int64}:
 10000
 22500
 40000
 90000

Methodology

Serial entries time a full solve (setup + factorization + solve, fresh per sample) behind a correctness gate. The distributed entry launches the same worker the scaling documents use, which itself gates on the true residual and reports the end-to-end assemble/solve/gather time along with the iteration count. Both sides therefore pay their honest setup costs: factorization for the direct solvers, MPI launch + PETSc setup + communication for the distributed solve.

function bench_serial(A, b, alg)
    ref = A \ b
    sol = solve(LinearProblem(A, b), alg)
    err = norm(sol.u - ref) / norm(ref)
    err < 1e-8 || return NaN
    return @belapsed solve(LinearProblem($A, $b), $alg).u evals=1
end

function bench_dist(N)
    cmd = `$(mpiexec()) $(MPIEXEC_ARGS) -n $P_DIST $(Base.julia_cmd()) --project=$(PROJECT) $(WORKER) $N cg gamg`
    out = read(addenv(cmd, "OMP_NUM_THREADS" => "1"), String)
    f = split(strip(last(filter(!isempty, split(out, '\n')))), ',')
    return (time = parse(Float64, f[6]), iters = parse(Int, f[8]),
        n = parse(Int, f[2]), retcode = f[9])
end

rows = []
for N in SIZES
    m = round(Int, sqrt(N))
    A = laplacian_2d(m)
    n = size(A, 1)
    rng_b = ones(n)   # deterministic RHS, matching the worker
    @info "n=$n"
    serial = [(name, bench_serial(A, rng_b, alg)) for (name, alg) in serial_algs]
    dist = bench_dist(N)
    push!(rows, (; n, serial, dist))
end

Results

println("    N    | " * join([rpad(name, 10) for (name, _) in serial_algs], "| ") *
        "| dist P=$(P_DIST) | dist iters")
println("-"^78)
for r in rows
    svals = join([@sprintf("%9.3g ", t) for (_, t) in r.serial], "| ")
    @printf("%8d | %s| %9.3g | %d (%s)\n", r.n, svals, r.dist.time,
        r.dist.iters, r.dist.retcode)
end

best_serial = [minimum(filter(!isnan, [t for (_, t) in r.serial]); init = Inf) for r in rows]
dist_times = [r.dist.time for r in rows]
ns = [r.n for r in rows]
cross = findfirst(i -> dist_times[i] < best_serial[i], 1:length(rows))
println()
println(cross === nothing ?
    "No crossover in the measured range: the best serial factorization wins throughout." :
    "Crossover: the distributed solve first beats the best serial factorization at N = $(ns[cross]).")

N    | UMFPACK   | KLU       | SupernodalLU| dist P=4 | dist iters
---------------------------------------------------------------------------
---
   10000 |     0.027 |    0.0271 |     0.023 |     0.058 | 12 (Success)
   22500 |    0.0798 |    0.0955 |    0.0545 |     0.195 | 12 (Success)
   40000 |     0.154 |     0.231 |     0.128 |     0.552 | 13 (Success)
   90000 |     0.419 |     0.838 |     0.337 |      2.77 | 14 (Success)

No crossover in the measured range: the best serial factorization wins thro
ughout.

p = plot(; xlabel = "N", ylabel = "time / s", xscale = :log10, yscale = :log10,
    title = "Serial direct vs distributed (P = $(P_DIST))", legend = :topleft)
for (j, (name, _)) in enumerate(serial_algs)
    ys = [r.serial[j][2] for r in rows]
    mask = .!isnan.(ys)
    any(mask) && plot!(p, ns[mask], ys[mask]; marker = :circle, label = name)
end
plot!(p, ns, dist_times; marker = :diamond, linewidth = 2,
    label = "PETSc GAMG-CG, $(P_DIST) ranks")
p

Reading the result

Where the diamond line crosses below the best serial curve is the size at which distribution starts paying on this problem class. Below it, the winning move is a serial factorization (and, for repeated solves, its cached reuse — see the CacheReuse document in the LinearSolve folder). The distributed iteration counts are printed so a reader can check that GAMG stays algorithmically flat across the sweep; if iterations grew, the crossover would be an artifact of solver strength rather than parallelism.

Scope: 2-D FD Laplacians (SPD, well-conditioned, planar sparsity), one fixed rank count, modest sizes chosen to fit a CI budget. The conclusions transfer to problems of similar structure, not to arbitrary sparse systems. On dedicated hardware, extending SIZES upward and adding rank counts turns the single crossover point into a crossover frontier; that is the intended follow-up once this baseline is established.

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/LinearSolveDistributed","SerialCrossover.jmd")

Computer Information:

Julia Version 1.12.6
Commit 15346901f00 (2026-04-09 19:20 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 `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolveDistributed/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
⌃ [7ed4a6bd] LinearSolve v5.5.0
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  [91a5bcdd] Plots v1.41.6
⌃ [0bca4576] SciMLBase v3.36.0
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.1.5]
  [a0a7dd2c] SparseMatricesCSR v0.6.12
  [37e2e46d] LinearAlgebra v1.12.0
  [de0858da] Printf v1.11.0
  [2f01184e] SparseArrays v1.12.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

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⌅ [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
  [4607b0f0] SuiteSparse
  [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`