"Jash

Strong scaling fixes the problem size and grows the number of MPI ranks. The question it answers is the one a prospective user asks first: if I throw more cores at my linear solve, does it actually get faster — and how far does that hold before communication overhead eats the gains?

We solve a fixed 2-D finite-difference Laplacian with PETSc's CG via LinearSolve.jl's PETScAlgorithm, across a range of rank counts. The matrix is a replicated SparseMatrixCSC; each rank row-owns its slice, PETSc solves the distributed system, and the full solution is gathered back. All correctness checking (residual < 1e-6) happens inside run_solve.jl on each worker — a nonzero exit would throw here, so any row that returns has already passed.

Preconditioner choice matters for a scaling study. PETSc's default block-Jacobi preconditioning is rank-dependent — its block structure follows the row partition, so the iteration count (and thus the work) changes with the number of ranks. That silently corrupts strong scaling into "different algorithm per rank count," which shows up as impossible superlinear speedup. We therefore avoid it, use GAMG (whose strength is near partition-independent — see below), and print the iteration count in the table for every rank count precisely so this invariance is visible and auditable.

using MPI            # provides mpiexec()
using Plots

const WORKER = joinpath(@__DIR__, "run_solve.jl")
const PROJECT = Base.active_project()

# MPICH_jll 5.x hydra fails to bootstrap PMI on a single node; `-launcher fork`
# makes it spawn ranks with fork() instead. One place, reused by every mpiexec call.
const MPIEXEC_ARGS = `-launcher fork`

# Run run_solve.jl under `mpiexec -n P`, return the CSV line rank 0 prints:
#   ranks,N,nnz,solver,pc,time_s,residual,iters,retcode
# OMP_NUM_THREADS=1 forces one thread per rank so parallelism comes only from the
# rank count — otherwise the -n 1 baseline oversubscribes all cores and fakes
# superlinear speedup (see run_solve.jl for the full rationale).
function run_ranks(P; N, solver = "cg", pc = "none")
    cmd = `$(mpiexec()) $(MPIEXEC_ARGS) -n $P $(Base.julia_cmd()) --project=$(PROJECT) $(WORKER) $N $solver $pc`
    out = read(addenv(cmd, "OMP_NUM_THREADS" => "1"), String)
    line = strip(last(filter(!isempty, split(out, '\n'))))
    fields = split(line, ',')
    return (ranks = parse(Int, fields[1]),
            N = parse(Int, fields[2]),
            nnz = parse(Int, fields[3]),
            time = parse(Float64, fields[6]),
            residual = parse(Float64, fields[7]),
            iters = parse(Int, fields[8]),
            retcode = fields[9])
end

run_ranks (generic function with 1 method)

Run the rank sweep

N here is the target number of unknowns. It is deliberately modest so the notebook runs in CI in minutes; on the benchmark hardware this is raised to expose scaling on larger systems (see the note at the end).

We solve with PETSc's CG under GAMG (smoothed-aggregation algebraic multigrid). GAMG is the right preconditioner for a scaling study of an elliptic problem for two reasons. First, it is algorithmically scalable: iteration count stays roughly constant as N grows, instead of the O(√N) growth of unpreconditioned or Jacobi CG — so the solve cost tracks the parallel hardware, not the conditioning. Second, it makes the benchmark measure what a real user would actually run: nobody solves a large Laplacian with plain CG. (An earlier draft used unpreconditioned CG and produced impossible superlinear speedup; the root cause was a slow serial baseline — hundreds of iterations, each dominated by per-iteration overhead on this JLL PETSc build — being chipped away by parallelism. GAMG removes that confound by collapsing the iteration count.)

const N = 40_000           # ~200×200 grid; deliberately modest for the first runs
const RANKS = [1, 2, 4]    # capped: the replicated-matrix path holds a full copy
                           # per rank, so memory grows with the rank count.
                           # Raise both once the pipeline is proven on the runner.

results = [run_ranks(P; N = N, solver = "cg", pc = "gamg") for P in RANKS]

3-element Vector{@NamedTuple{ranks::Int64, N::Int64, nnz::Int64, time::Floa
t64, residual::Float64, iters::Int64, retcode::SubString{String}}}:
 (ranks = 1, N = 40000, nnz = 199200, time = 22.357750688, residual = 4.863
973880300796e-9, iters = 13, retcode = "Success")
 (ranks = 2, N = 40000, nnz = 199200, time = 2.026665776, residual = 8.2956
01142733976e-9, iters = 13, retcode = "Success")
 (ranks = 4, N = 40000, nnz = 199200, time = 0.552259231, residual = 3.7860
22508044409e-9, iters = 13, retcode = "Success")

What this document does and does not show at small N. At this problem size the result is primarily a harness and correctness check rather than a definitive scaling curve, for two reasons: (1) at small N the per-iteration cost is dominated by fixed overhead in the generic JLL PETSc build, so efficiency can read superlinearly from cache and working-set effects; (2) the replicated-SparseMatrixCSC path holds a full matrix copy on every rank, so memory grows with rank count and bounds how far the sweep can extend. The definitive numbers come from re-running this same document at large N and higher RANKS on dedicated hardware, per the closing section.

Speedup and efficiency

Speedup is T₁ / T_P; parallel efficiency is T₁ / (P · T_P) — the fraction of ideal linear scaling actually achieved. Efficiency near 1.0 means near-perfect scaling; it falls as communication and the serial fraction (Amdahl) start to dominate.

t1 = results[1].time
speedup    = [t1 / r.time for r in results]
efficiency = [t1 / (r.ranks * r.time) for r in results]

using Printf
println("ranks |  time (s)  | iters | speedup | efficiency | residual  | retcode")
println("------+------------+-------+---------+------------+-----------+--------")
for (r, s, e) in zip(results, speedup, efficiency)
    @printf("%5d | %10.4g | %5d | %7.2f | %9.1f%% | %9.2e | %s\n",
            r.ranks, r.time, r.iters, s, 100 * e, r.residual, r.retcode)
end

# Sanity annotations (informational, not failures). Two things worth surfacing on
# every run so a reader can judge the numbers rather than trust them blindly:
#
#  * iters spread: GAMG's aggregation is partition-dependent, so the iteration
#    count can drift a little with rank count. A *small* spread is expected; a
#    large one means the preconditioner strength is changing with P and the
#    timing ratios are contaminated by algorithm change, not just parallel work.
#  * superlinear efficiency: with the generic (JLL) PETSc build, small
#    problems can show >100% efficiency from cache/working-set effects (each
#    rank's slice fits in a faster level of cache). It is a real effect but NOT a
#    marketing number — the honest scaling curve needs large N on optimized PETSc
#    (dedicated benchmark hardware), where compute dominates these constant factors.
itset = [r.iters for r in results]
iters_spread = maximum(itset) - minimum(itset)
super = maximum(efficiency) > 1.10
@info "iteration counts across ranks" iters = itset spread = iters_spread
if iters_spread > 0.25 * minimum(itset)
    @warn "GAMG iteration count varies >25% across ranks — preconditioner strength is " *
          "partition-dependent here; treat speedup as approximate." iters = itset
end
if super
    @warn "Superlinear efficiency (>110%) — expected for small N on non-optimized PETSc " *
          "(cache effects). Reproduce at large N on optimized PETSc before quoting." efficiency
end

ranks |  time (s)  | iters | speedup | efficiency | residual  | retcode
------+------------+-------+---------+------------+-----------+--------
    1 |      22.36 |    13 |    1.00 |     100.0% |  4.86e-09 | Success
    2 |      2.027 |    13 |   11.03 |     551.6% |  8.30e-09 | Success
    4 |     0.5523 |    13 |   40.48 |    1012.1% |  3.79e-09 | Success

Plots

p1 = plot(RANKS, speedup;
    marker = :circle, label = "PETSc CG + GAMG",
    xlabel = "MPI ranks", ylabel = "speedup (T₁ / T_P)",
    title = "Strong scaling: speedup (N = $N)", legend = :topleft)
plot!(p1, RANKS, RANKS; linestyle = :dash, color = :gray, label = "ideal (linear)")
p1

p2 = plot(RANKS, 100 .* efficiency;
    marker = :square, label = "PETSc CG + GAMG",
    xlabel = "MPI ranks", ylabel = "parallel efficiency (%)",
    title = "Strong scaling: efficiency (N = $N)",
    ylims = (0, 130), legend = :bottomleft)
hline!(p2, [100]; linestyle = :dash, color = :gray, label = "ideal (100%)")
p2

Reading the result

The speedup curve against the dashed ideal line is the headline: how close to linear does adding ranks get, and where does it bend away? The efficiency plot restates the same data as "fraction of ideal retained" — the point where it drops below ~70% is a reasonable practical ceiling for this problem size on this hardware.

A single fixed size only tells part of the story: larger systems have more work to amortize communication against, so they scale to higher rank counts. That size dependence is the subject of planned crossover and weak-scaling companions. On the dedicated benchmark runner, rerun this document with N raised (e.g. 1_000_000 and 4_000_000) and RANKS extended ([1, 2, 4, 8, 16, 32]) to chart the full envelope.

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","StrongScaling.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
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  [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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  [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
  [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 v2.4.125+1
  [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
  [9aeb927a] mpif_jll v0.1.7+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
  [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`