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Cache Reuse — What the init/solve! API Buys You

LinearSolve.jl's headline API difference from A \ b is the cache: build a problem once with init, then call solve! repeatedly while updating b or A in place. Every downstream SciML solver (ODE Jacobians, Newton iterations) leans on this. This benchmark quantifies what it's worth, for each algorithm family:

  1. Naive — solve(LinearProblem(A, b), alg) from scratch every time. What you pay without the cache.
  2. Cached, new b — cache.b = b′; solve!(cache). Direct methods skip factorization entirely: this isolates the backsolve.
  3. Cached, new A (same sparsity) — cache.A = A′; solve!(cache). Forces numeric refactorization but reuses symbolic analysis, orderings, and workspaces (this path exercises the recent direct-BLAS workspace reuse and sparse symbolic-reuse work in LinearSolve v5).

Also reported: allocations per cached re-solve — for direct methods a warm cache should allocate (near) zero, which is what makes the API safe inside tight loops.

CI budget note: two fixed problem sizes per family, three modes, ~10 algorithms — minutes, not hours.

using BenchmarkTools, Random
using LinearAlgebra, SparseArrays, LinearSolve, Sparspak
using RecursiveFactorization, FastLapackInterface
import Pardiso
import ParU_jll
using Plots

BenchmarkTools.DEFAULT_PARAMETERS.seconds = 0.5
BenchmarkTools.DEFAULT_PARAMETERS.samples = 20

# Same FD generator as the folder's other sparse documents.
A ⊕ B = kron(I(size(B, 1)), A) + kron(B, I(size(A, 1)))
function lattice(n; Tv = Float64)
    d = fill(2 * one(Tv), n); d[1] = one(Tv); d[end] = one(Tv)
    spdiagm(1 => -ones(Tv, n - 1), 0 => d, -1 => -ones(Tv, n - 1))
end
lattice(L...; Tv = Float64) = lattice(L[1]; Tv) ⊕ lattice(L[2:end]...; Tv)
function fdmatrix(N; dim = 2, Tv = Float64, δ = 1.0e-2)
    n = N^(1 / dim) |> ceil |> Int
    lattice([n for i in 1:dim]...; Tv) + Tv(δ) * I
end

# (family, name, alg). Dense algs get a dense A; sparse algs the 2-D FD matrix.
algs = [
    (:dense,  "LU",           LUFactorization()),
    (:dense,  "RFLU",         RFLUFactorization()),
    (:dense,  "MKL LU",       MKLLUFactorization()),
    (:sparse, "UMFPACK",      UMFPACKFactorization()),
    (:sparse, "KLU",          KLUFactorization()),
    (:sparse, "SupernodalLU", SupernodalLUFactorization()),
    (:sparse, "Sparspak",     SparspakFactorization()),
    (:sparse, "ParU",         ParUFactorization()),
]

const DENSE_N = 500
const SPARSE_N = 40_000    # ~200×200 grid, 2-D

function make_problem(family)
    rng = MersenneTwister(123)
    A = family === :dense ? (rand(rng, DENSE_N, DENSE_N) + DENSE_N * I) :
        fdmatrix(SPARSE_N; dim = 2)
    n = size(A, 1)
    b = rand(rng, n)
    # Same-sparsity update: scale values, keep the pattern (dense: fresh matrix).
    A2 = family === :dense ? (rand(rng, n, n) + n * I) :
        SparseMatrixCSC(n, n, copy(A.colptr), copy(A.rowval), 1.1 .* A.nzval)
    b2 = rand(rng, n)
    return A, A2, b, b2
end
make_problem (generic function with 1 method)

Methodology

For each algorithm we measure four numbers:

function bench_reuse(family, alg)
    A, A2, b, b2 = make_problem(family)

    # Correctness gate on both the fresh solve and the A-update path — a cache
    # that silently returns the OLD factorization's answer after `cache.A = A2`
    # would be fast and wrong, which is the failure mode this guards against.
    # NOTE: `cache.A = X` hands the backend the array itself, and several
    # factorizations (RFLU, MKL) factorize it IN PLACE — the caller's matrix is
    # destroyed. Others (LU) copy into an internal workspace. Always assign a copy
    # if you still need the matrix; this benchmark does, so its residual checks
    # test the solve rather than the wreckage.
    cache = init(LinearProblem(copy(A), b), alg)
    u1 = copy(solve!(cache).u)
    cache.b = b2
    u2 = copy(solve!(cache).u)
    cache.A = copy(A2)
    u3 = copy(solve!(cache).u)
    r1 = norm(A * u1 - b) / norm(b)
    r2 = norm(A * u2 - b2) / norm(b2)
    r3 = norm(A2 * u3 - b2) / norm(b2)
    if !(r1 < 1e-8 && r2 < 1e-8 && r3 < 1e-8)
        @warn "correctness gate failed" alg r1 r2 r3
        return nothing
    end

    t_naive = @belapsed solve(LinearProblem($A, $b), $alg).u

    cache_newb = init(LinearProblem(A, b), alg)
    solve!(cache_newb)
    t_newb = @belapsed solve!(c).u setup=(c = $cache_newb; c.b = $b2) evals=1

    cache_newA = init(LinearProblem(A, b), alg)
    solve!(cache_newA)
    t_newA = @belapsed solve!(c).u setup=(c = $cache_newA; c.A = copy($A2)) evals=1

    # Steady-state allocations: repeated solve! on a warm cache, same b.
    warm = init(LinearProblem(A, b), alg)
    solve!(warm); solve!(warm)
    allocs = @allocated solve!(warm)

    return (; t_naive, t_newb, t_newA, allocs)
end

results = []
for (family, name, alg) in algs
    r = try
        bench_reuse(family, alg)
    catch e
        @warn "$name failed" exception=(e,)
        nothing
    end
    r === nothing || push!(results, (; family, name, r...))
    r === nothing || @info name t_naive=r.t_naive t_newb=r.t_newb t_newA=r.t_newA allocs=r.allocs
end

Results

using Printf
println("alg           | family |  naive (s) | new-b (s) | new-A (s) | b-speedup | A-speedup | allocs/solve")
println("--------------+--------+------------+-----------+-----------+-----------+-----------+-------------")
for r in results
    @printf("%-13s | %-6s | %10.3g | %9.3g | %9.3g | %8.1fx | %8.2fx | %d\n",
        r.name, r.family, r.t_naive, r.t_newb, r.t_newA,
        r.t_naive / r.t_newb, r.t_naive / r.t_newA, r.allocs)
end
alg           | family |  naive (s) | new-b (s) | new-A (s) | b-speedup | A
-speedup | allocs/solve
--------------+--------+------------+-----------+-----------+-----------+--
---------+-------------
LU            | dense  |    0.00551 |  5.69e-05 |   0.00483 |     96.8x |  
   1.14x | 0
RFLU          | dense  |    0.00311 |  5.68e-05 |   0.00249 |     54.8x |  
   1.25x | 0
MKL LU        | dense  |    0.00401 |  6.92e-05 |   0.00377 |     57.9x |  
   1.06x | 0
UMFPACK       | sparse |      0.207 |   0.00593 |     0.152 |     34.8x |  
   1.36x | 0
KLU           | sparse |      0.241 |    0.0106 |     0.182 |     22.8x |  
   1.33x | 0
SupernodalLU  | sparse |      0.143 |    0.0112 |    0.0555 |     12.8x |  
   2.57x | 0
Sparspak      | sparse |      0.132 |    0.0206 |     0.108 |      6.4x |  
   1.22x | 965928
ParU          | sparse |      0.533 |    0.0164 |     0.281 |     32.5x |  
   1.89x | 968360
sparse_res = filter(r -> r.family === :sparse, results)
vals = hcat([ [r.t_naive for r in sparse_res],
              [r.t_newA  for r in sparse_res],
              [r.t_newb  for r in sparse_res] ]...)
p = plot()   # grouped bars via repeated bar! calls keep deps minimal
xs = 1:length(sparse_res)
w = 0.25
for (k, (lab, col)) in enumerate(zip(("naive", "new A (refactor)", "new b (backsolve)"),
                                     (:gray, :steelblue, :seagreen)))
    bar!(p, xs .+ (k - 2) * w, vals[:, k]; bar_width = w, label = lab, color = col)
end
plot!(p; xticks = (xs, [r.name for r in sparse_res]), yscale = :log10,
    ylabel = "time / s (log)", title = "Sparse matrix reuse cost (N = $(SPARSE_N))",
    legend = :topright)
p

Krylov warm start

For iterative methods the cache carries a different asset: the previous solution. With warm_start, repeated solves against slowly-varying right-hand sides start from the last u instead of zero — the payoff is iterations, which we report directly (time follows iterations for a fixed operator).

A = fdmatrix(SPARSE_N; dim = 2)
n = size(A, 1)
rng = MersenneTwister(42)
b = rand(rng, n)
db = 0.01 .* rand(rng, n)   # small perturbation: the "time-stepping" regime

for (label, ws) in (("WarmStart.Previous", KrylovJL_GMRES(warm_start = WarmStart.Previous)),
                    ("WarmStart.None", KrylovJL_GMRES(warm_start = WarmStart.None)))
    cache = init(LinearProblem(A, b), ws; reltol = 1e-8)
    iters = Int[]
    for step in 1:5
        sol = solve!(cache)
        push!(iters, sol.iters)
        cache.b = cache.b .+ db
    end
    println(rpad(label, 18), " iters per step: ", iters)
end
WarmStart.Previous iters per step: [224, 159, 159, 159, 159]
WarmStart.None     iters per step: [224, 224, 224, 224, 224]

Reading the result

  • new-b speedup is the headline. For direct methods this is factor-time / backsolve-time — typically one to two orders of magnitude. If your workload solves against many right-hand sides (adjoints, multiple loads), the cache API is the difference.
  • new-A speedup measures symbolic reuse. Same-pattern refactorization skips ordering/analysis; how much that's worth varies strongly by algorithm — this column is effectively a benchmark of each backend's refactorization path.
  • allocs/solve near zero is what makes solve! safe inside hot loops. Nonzero values here are worth filing as issues.
  • cache.A = X aliasing differs by backend. Some factorizations (RFLU, MKL LU) factorize the assigned array in place, destroying it; others (LU) copy into a reused workspace. If you need your matrix afterwards, assign a copy. This benchmark's correctness gate is what surfaced the difference.
  • Warm start converts cache reuse into iteration savings for Krylov methods in the time-stepping regime; with a fixed matrix and slowly-moving b, the iteration counts tell the story without timing noise.

What this does not show: changed-sparsity-pattern updates (a rebuild, by design), and multi-threaded interactions. Problem sizes are fixed per family; the size dependence of these ratios is covered by the SparseDirect document's sweep.

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/LinearSolve","CacheReuse.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 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/LinearSolve/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
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⌃ [7ed4a6bd] LinearSolve v5.5.0
  [b51810bb] MatrixDepot v1.1.0
  [46dd5b70] Pardiso v1.1.2
⌃ [91a5bcdd] Plots v1.41.6
⌃ [b7e1f0a2] PureUMFPACK v0.1.4
⌃ [f2c3362d] RecursiveFactorization v0.2.26
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.2.0]
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⌃ [10745b16] Statistics v1.11.1
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  [2f01184e] SparseArrays v1.12.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolve/Manifest.toml`
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  [4b2f31a3] Libmount_jll v2.42.0+0
  [89763e89] Libtiff_jll v4.7.3+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
⌅ [b5ada748] MPIABI_jll v0.1.5+0
  [7cb0a576] MPICH_jll v5.0.1+0
  [f1f71cc9] MPItrampoline_jll v5.5.6+0
  [c8ffd9c3] MbedTLS_jll v2.28.1010+0
  [9237b28f] MicrosoftMPI_jll v10.1.4+3
  [e7412a2a] Ogg_jll v1.3.6+0
  [fe0851c0] OpenMPI_jll v5.0.11+0
⌃ [9bd350c2] OpenSSH_jll v10.4.1+0
  [91d4177d] Opus_jll v1.6.1+0
⌃ [36c8627f] Pango_jll v1.57.1+0
⌃ [9e0b026c] ParU_jll v1.0.0+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
⌅ [02c8fc9c] XML2_jll v2.13.9+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 v1.8.3+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
⌅ [2b3700d1] aws_c_auth_jll v0.9.6+0
⌅ [70f11efc] aws_c_cal_jll v0.9.13+0
⌅ [73048d1d] aws_c_common_jll v0.12.6+0
⌅ [73a04cd5] aws_c_compression_jll v0.3.2+0
⌅ [3254fc65] aws_c_http_jll v0.10.13+0
⌅ [13c41daa] aws_c_io_jll v0.26.3+0
⌅ [bd1f34fb] aws_c_s3_jll v0.11.5+0
⌅ [1282aa60] aws_c_sdkutils_jll v0.2.4+1
⌅ [b2a88e68] aws_checksums_jll v0.2.10+0
  [c4b69c83] dlfcn_win32_jll v1.4.2+0
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [477f73a3] libaec_jll v1.1.7+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
⌅ [9aeb927a] mpif_jll v0.1.7+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌃ [cddc5d3d] s2n_tls_jll v1.7.3+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 v2025.11.4
  [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`