"Jash

Weak scaling grows the problem with the number of ranks, holding the work per rank fixed. It answers the question distributed solvers actually exist for: can I solve a proportionally bigger problem in the same wall-clock time by adding hardware? Strong scaling (the sibling document) fixes the problem and asks for speedup; weak scaling is usually the more honest test for memory-bound sparse solves, because per-rank working set stays constant by construction.

We solve 2-D finite-difference Laplacians with PETSc's CG under GAMG via LinearSolve.jl's PETScAlgorithm, with the target unknown count proportional to the rank count. Two properties are on trial:

• Algorithmic scalability: GAMG's iteration count should stay near-constant as N grows. The iteration column makes this auditable on every run.
• Parallel weak efficiencyT_1 / T_P: with perfect weak scaling every rank count solves its (proportionally larger) system in the baseline's time.

using MPI            # provides mpiexec()
using Plots
using Printf

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

# MPICH_jll 5.x hydra can fail PMI bootstrap on a single node; `-launcher fork`
# spawns ranks with fork() instead and is harmless where hydra is healthy.
const MPIEXEC_ARGS = `-launcher fork`

# Run run_solve.jl under `mpiexec -n P`; parse the CSV line rank 0 prints:
#   ranks,N,nnz,solver,pc,time_s,residual,iters,retcode
# One thread per rank so parallelism comes only from the rank count.
function run_ranks(P; N, solver = "cg", pc = "gamg")
    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'))))
    f = split(line, ',')
    return (ranks = parse(Int, f[1]), N = parse(Int, f[2]), nnz = parse(Int, f[3]),
        time = parse(Float64, f[6]), residual = parse(Float64, f[7]),
        iters = parse(Int, f[8]), retcode = f[9])
end

run_ranks (generic function with 1 method)

Run the sweep

Work per rank is deliberately modest so the document proves the pipeline inside a small CI budget; raise N_PER_RANK (and extend RANKS) on dedicated hardware to chart the full envelope. Note the replicated-SparseMatrixCSC input path stores a full matrix copy per rank, so total memory grows with P × N here even though the solver's per-rank work is constant; the closing section discusses this bound.

const N_PER_RANK = 10_000
const RANKS = [1, 2, 4]

results = [run_ranks(P; N = N_PER_RANK * P) 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 = 10000, nnz = 49600, time = 0.811717794, residual = 2.81871
56057856e-9, iters = 12, retcode = "Success")
 (ranks = 2, N = 19881, nnz = 98841, time = 0.895897876, residual = 6.96859
3402376254e-9, iters = 12, retcode = "Success")
 (ranks = 4, N = 40000, nnz = 199200, time = 0.548885054, residual = 3.7860
22508044409e-9, iters = 13, retcode = "Success")

Weak efficiency

t1 = results[1].time
weak_eff = [t1 / r.time for r in results]

println("ranks |    N    |  time (s)  | iters | weak eff. | residual  | retcode")
println("------+---------+------------+-------+-----------+-----------+--------")
for (r, e) in zip(results, weak_eff)
    @printf("%5d | %7d | %10.4g | %5d | %8.1f%% | %9.2e | %s\n",
        r.ranks, r.N, r.time, r.iters, 100 * e, r.residual, r.retcode)
end

# Auditability annotations, mirroring the strong-scaling document: iteration
# growth means the preconditioner is not algorithmically scaling (the time
# column then conflates solver work with parallel overhead), and efficiency
# above ~110% at these sizes usually reflects cache effects on a generic
# (JLL) PETSc build rather than real scaling.
itset = [r.iters for r in results]
if maximum(itset) - minimum(itset) > 0.25 * minimum(itset)
    @warn "GAMG iteration count grows >25% across the sweep — algorithmic " *
          "scalability is not holding at these sizes." iters = itset
end
if maximum(weak_eff) > 1.10
    @warn "Weak efficiency exceeds 110% — treat as a cache/working-set artifact " *
          "at small N, not a real result." weak_eff
end

ranks |    N    |  time (s)  | iters | weak eff. | residual  | retcode
------+---------+------------+-------+-----------+-----------+--------
    1 |   10000 |     0.8117 |    12 |    100.0% |  2.82e-09 | Success
    2 |   19881 |     0.8959 |    12 |     90.6% |  6.97e-09 | Success
    4 |   40000 |     0.5489 |    13 |    147.9% |  3.79e-09 | Success

Plot

p = plot(RANKS, 100 .* weak_eff;
    marker = :square, label = "PETSc CG + GAMG",
    xlabel = "MPI ranks (N = $(N_PER_RANK) × ranks)",
    ylabel = "weak efficiency (%)",
    title = "Weak scaling: constant work per rank",
    ylims = (0, 130), legend = :bottomleft)
hline!(p, [100]; linestyle = :dash, color = :gray, label = "ideal (100%)")
p

Reading the result

Flat iteration counts with efficiency near 100% is the headline weak-scaling claim: the solver does proportionally more work in the same time as ranks are added. Efficiency decay isolates parallel overhead (communication, setup), since constant per-rank work removes the serial-fraction confound that complicates strong scaling.

Three caveats bound what this document can show. First, at the committed sizes the per-iteration cost is small enough that fixed overheads matter; the definitive curve needs larger N_PER_RANK on dedicated hardware. Second, the replicated-input path makes every rank assemble the full matrix, so the constant-work-per-rank premise holds for the solve phase but not for assembly: assembly cost grows with total N on every rank, which drags measured weak efficiency down even when the solver itself scales. Third, the same replication caps how far RANKS can extend before total memory becomes the binding constraint. A partitioned-input variant removes the second and third limits and is the natural follow-up for large-rank weak scaling.

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","WeakScaling.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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  [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`