Pith Number

pith:MTV2REHB

pith:2026:MTV2REHBQCXBXFOAV5WD52WAMC

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Robust Matrix-Free Newton-Krylov Solvers via Automatic Differentiation

Marco Pasquale, Stefano Markidis

Automatic differentiation for Jacobian-vector products makes matrix-free Newton-Krylov solvers orders of magnitude faster and far more robust than finite differences.

arxiv:2605.13378 v1 · 2026-05-13 · cs.CE · physics.comp-ph

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Claims

C1strongest claim

By preventing degradation of the Krylov operator, AD accelerates computation by 2-3 orders of magnitude across both CPU and GPU architectures. More importantly, it drastically improves global solver robustness, achieving a minimum completion rate of 95%, compared to just 42% for FD.

C2weakest assumption

That the observed gains are caused solely by the accuracy of the Gateaux derivatives and will persist when the automatic-differentiation implementation, problem size, or floating-point precision changes.

C3one line summary

Forward-mode automatic differentiation replaces finite-difference approximations for Jacobian-vector products in JFNK solvers, delivering 2-3 orders of magnitude speedup and lifting minimum solver completion from 42% to 95% across Burgers, radiation diffusion, reaction-diffusion, and nonlinear time-

References

31 extracted · 31 resolved · 1 Pith anchors

[1] D. Knoll, D. Keyes, Jacobian-free Newton–Krylov methods: a survey of approaches and applications, Journal of Computational Physics 193 (2) 26 (2004) 357–397.doi:10.1016/j.jcp.2003.08.010. URLhttps://l 2004 · doi:10.1016/j.jcp.2003.08.010

[2] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics, 1995.doi:10.1137/1.9781611970944. URLhttps://epubs 1995 · doi:10.1137/1.9781611970944

[3] GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems 1986 · doi:10.1137/0907058

[4] M. R. Hestenes, E. Stiefel, Methods of Conjugate Gradients for Solving LinearSystems, JournalofResearchoftheNationalBureauofStandards 49 (6) (1952) 409–436 1952

[5] H. A. Van Der Vorst, Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems, SIAM Journal on Scientific and Statistical Computing 13 (2) (1992) 631– 1992 · doi:10.1137/0913035

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Receipt and verification

First computed	2026-05-18T02:44:47.873733Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

64eba890e180ae1b95c0af6c3eeac060b2c692adf076a2af7267d51c48ee6fd8

Aliases

arxiv: 2605.13378 · arxiv_version: 2605.13378v1 · doi: 10.48550/arxiv.2605.13378 · pith_short_12: MTV2REHBQCXB · pith_short_16: MTV2REHBQCXBXFOA · pith_short_8: MTV2REHB

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MTV2REHBQCXBXFOAV5WD52WAMC \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 64eba890e180ae1b95c0af6c3eeac060b2c692adf076a2af7267d51c48ee6fd8

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a93c34770cedaecb9472883ae67572009cbfa07af587427f27afef18ebf37f0a",
    "cross_cats_sorted": [
      "physics.comp-ph"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.CE",
    "submitted_at": "2026-05-13T11:34:32Z",
    "title_canon_sha256": "708eb2ce55b461502af8ba7190e9fdffa4398860a895dbe0334c4038d4e7dcde"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13378",
    "kind": "arxiv",
    "version": 1
  }
}