{"id":"637f90a4-c84b-4126-87d4-643c597dff19","arxiv_id":"2509.20937","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Lower-precision solves inside Schwarz methods converge for M-matrix problems when the rounding is sign-aware and two norm/componentwise conditions hold; experiments suggest single precision suffices.","lead":"This paper analyzes what happens when the small sub-problems inside Schwarz domain-decomposition solvers are solved with fewer digits of precision than the rest of the computation. It derives conditions under which this still converges and finds that, for the tested problems, about five accurate digits (single precision) are enough.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 guarantees convergence for exact inverses of the sign-rounded subdomain matrices, while the experiments use actual low-precision LU solves; the theory may not cover the implemented algorithm.","rationale":"The reader's conditional verdict is reasonable, but the most load-bearing gap is not primarily hardware fidelity or the unverified large-N condition (19). It is the mismatch between the object analyzed in Theorem 3 and the object executed in the experiments. The theorem's weak-regular splitting proof requires \\tilde A_i^{-1} >= 0 and \\tilde A_i^{-1}F_i >= 0, which is enforced by (19) after the sign-informed rounding. These are properties of the exact inverse of the rounded matrix. In a genuine u_l-precision LU solve, the computed factors solve (\\tilde A_i + \\Delta_i)v = b with unstructured backward error; there is no reason for \\Delta_i to preserve nonnegativity or the splitting inequalities. The paper nowhere bounds the contribution of \\Delta_i, so Theorem 3 does not logically cover the actual numerical method whose convergence is reported. This is a testable modeling gap rather than a mathematical error in the theorem, and it strengthens the case for a conditional verdict: the theoretical section should either explicitly restrict to exact solves of rounded matrices or add a backward-error condition absorbing the solve-time rounding. The proposed experiment cleanly separates these two algorithms and would settle whether the concern lands.","tokens_in":23361,"tokens_out":12155,"duration_ms":108823,"concrete_test":"Re-run the N=2500 cases of Figures 1 and 2 twice: (i) with the actual chop/advanpix low-precision solve exactly as in the paper, and (ii) with the local solver replaced by the exact inverse of the sign-rounded matrix \\tilde A_i, computed in high-precision or rational arithmetic from the same rounded entries, while all other parts of the Schwarz iteration remain in working precision. Compare the convergence factors \\rho_conv and the first iterations where conditions (16) and (19) are satisfied. If the convergence curves coincide within the precision-u_l tolerance, the idealized model is representative and this concern is resolved; if they differ measurably, or if case (ii) converges while case (i) does not in a regime satisfying (16)-(19), then Theorem 3 does not validate the implemented floating-point algorithm.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conditional guarantee is narrower than the experiments it is claimed to explain. Theorem 3 and the derivation (15)-(19) analyze the iteration obtained by replacing A_i^{-1} with the exact inverse of \\tilde A_i = A_i + F_i, where F_i is the sign-informed matrix-rounding error from (21). The numerical experiments in Figures 1 and 2 are run with the chop and advanpix toolboxes, which implement a full u_l-precision LU solve: every arithmetic operation in the factorization and in the triangular solves is rounded to u_l. The computed local solution therefore satisfies a perturbed system (\\tilde A_i + \\Delta_i)v = b with \\Delta_i of size O(u_l \\|\\tilde A_i\\|), and \\Delta_i has no sign structure. Equations (16) and (19) are conditions on F_i only; they do not constrain \\Delta_i. Consequently, even when all conditions of Theorem 3 hold for \\tilde A_i, the actual iteration operator differs from \\tilde T_⋆ by an unstructured perturbation not accounted for in the weak-regular splitting proof. The paper does not state or prove that the solve-time rounding is dominated by or absorbable into the matrix-rounding conditions. Thus the headline claim that the analysis identifies the minimum number of digits needed for multiprecision Schwarz local solves is, strictly, established only for the idealized exact-inverse-of-rounded-matrix method; its applicability to the actual computed solves rests on an unstated robustness assumption, separately from the large-N issue for condition (19).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an algebraic convergence analysis for Schwarz methods in which the local subdomain solves are performed at a lower precision than the working precision. The authors propose sign-informed rounding of the rescaled subdomain matrices so that the rounding error matrix F_i is nonnegative, and they derive sufficient conditions, namely ||A_i^{-1}F_i||<1 in (16) and A_i^{-1} >= A_i^{-1}F_i A_i^{-1} in (19), under which the multiprecision damped additive, restricted additive, and multiplicative Schwarz iterations converge (Theorem 3); a Weyl-type eigenvalue condition (29) is added for the Stieltjes case (Theorem 4). The numerical experiments with the chop and advanpix toolboxes suggest that about four or five decimal digits suffice for the tested model problems, and the paper also contains perturbation bounds and pseudospectral statements for the use of these methods as preconditioners. The central theoretical scaffolding is coherent, but there is a gap between the idealized solver analyzed in the theorems and the actual low-precision LU solves used in the experiments, and for larger N only condition (16) is verified.","tokens_in":23850,"tokens_out":9079,"duration_ms":77666,"significance":"If the gap between theory and implementation is closed, this would be a useful contribution: it provides the first algebraic convergence conditions explicitly designed for multiprecision Schwarz methods, the conditions are derived rather than fitted, and the paper makes its code available. The sign-informed rounding idea is simple and effective, and the experiments are careful in distinguishing precisions for which the sufficient conditions hold from those for which they do not. The numerical evidence that single precision suffices for the tested M-matrix problems is interesting and reproducible. At present, however, the main theorem covers only exact inverses of the rounded matrices, which weakens the stated headline that the analysis identifies the minimum number of digits for the computed local solves.","major_comments":[{"comment":"Theorem 3 analyzes the iteration obtained by replacing A_i^{-1} with the exact inverse of the sign-rounded matrix \\tilde A_i = A_i + F_i, but the numerical experiments are run with chop and advanpix, which round every operation in the LU factorization and in the triangular solves. The computed local solution therefore satisfies a perturbed system with an additional unstructured backward error \\Delta_i of size O(u_l ||\\tilde A_i||), and conditions (16) and (19) give no control on \\Delta_i because they constrain only the sign-structured entry-rounding error F_i. As stated, the theorem does not cover the implemented algorithm; the paper should either prove that the solve-time rounding is dominated by or absorbable into F_i, or explicitly restrict the convergence claims and the interpretation of Figures 1-2 to the idealized exact-inverse-of-rounded-matrix method plus an unproved robustness assumption.","section":"Section 4.1, Theorem 3; Figures 1-2"},{"comment":"For N > 2500 the paper verifies only condition (16) and not condition (19); the text itself says this tests whether the observation in Remark 3 holds for larger N. Consequently the large-N claim that 'four or five digits suffice' rests on the unproven assertion that (16) is generally the binding condition. The paper should provide support for this assertion, either by verifying (19) with a cheaper componentwise estimate, by proving a relevant implication from (16) to (19) under additional assumptions, or by explicitly labeling the large-N statements as heuristic and outside the scope of Theorem 3.","section":"Section 4.1, Remark 3; Figure 4"},{"comment":"The statement of Proposition 1 says that for any \\delta \\in (0,\\epsilon) the bound (41) holds, but the right-hand side of (41) contains the factor 1/(\\delta - \\epsilon), which is negative or undefined on that interval. The correct condition should be \\delta > \\epsilon; please fix the interval and re-check the subsequent use of the bound in the discussion of pseudospectral stability.","section":"Section 5, Proposition 1, Eq. (41)"},{"comment":"Equation (13) and the expression for E_i use (D_r^i)^{-1} on both sides of \\tilde A_i, but the rescaling in (8) is A_i := \\mu D_r^i A_i D_c^i. In the general non-symmetric case the right factor must be (D_c^i)^{-1}; as printed, the formula for E_i and the equivalence in (14) are valid only when the scaling is symmetric, and the general derivation should be corrected.","section":"Section 4.1, Eq. (13)"}],"minor_comments":[{"comment":"In the abstract, 'proecision' should be 'precision'.","section":"Abstract"},{"comment":"The sentence 'taking \\theta=1/2 still guarantees convergence convergence' contains a duplicated word and should read 'still guarantees convergence'.","section":"Section 5 (near Eq. (38)-(39))"},{"comment":"Both captions contain 'are are analogous'; the duplicated word should be removed.","section":"Captions of Figures 3 and 8"},{"comment":"The notation \\tilde A_i is used both for the rounded rescaled subdomain matrix and for the rounding of the original matrix; please introduce distinct symbols or state the convention explicitly to avoid ambiguity in the derivation of (13)-(19).","section":"Section 4.1, notation around Eq. (13)"}],"recommendation":"major_revision","confidential_remarks":"This is a serious and timely contribution that fits the journal. The main requested revision is not to the experiments but to making the theorem statement match the implemented algorithm, or to adding an explicit robustness lemma for solve-time rounding; the large-N verification gap and the Proposition 1 interval error are also fixable. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — this paper is worth a look if you work on mixed-precision iterative solvers. It analyzes the Schwarz iteration itself with lower-precision subdomain solves, which is new: earlier mixed-precision domain decomposition papers mostly studied preconditioned CG empirically. The main contribution is Theorem 3: for M-matrices, if the rounded local matrices A~i = Ai + Fi have the right sign structure (via the new sign-aware round-up routine), and conditions (16) and (19) hold, then the multiprecision damped AS, RAS, and MS iterations converge. Theorem 4 adds a Stieltjes-matrix variant using Weyl's inequality. The conditions are genuinely sufficient, not fitted to the numerical results, and the experiments verify them independently. They also test preconditioned GMRES and note that four to five digits suffice on their model problems, which aligns with fp32. Code is available. If I have one concern, it's this: the theorem analyzes the exact inverse of the rounded matrix A~i^{-1}, but the experiments are run with chop and advanpix, which round every arithmetic operation inside the LU factorization and triangular solves. The computed solve satisfies (A~i + Δ_i)v = b with an unstructured Δ_i of size O(u_l). Conditions (16) and (19) say nothing about Δ_i, so the rigorous guarantee stops short of the implemented algorithm. I don't think this sinks the paper—the experiments stand on their own—but the authors should either prove a perturbation result for unstructured Δ_i, or state explicitly that the theory is for an idealized model and the numerics are evidence for a plausible robustness assumption. Also note: for N > 2500 they only verify (16), not (19), because the latter becomes too expensive; Remark 3 admits that the claim that (16) is the binding condition is not proved. The \"twice as fast\" comments are speculative—there are no timings. There are minor slips: equation (13) has a scaling typo (the second D^r_i should be D^c_i). The Section 5 pseudospectral bound is a reasonable application of existing GMRES perturbation theory, not a new tool. All told: a solid extension of inexact-Schwarz analysis, not a breakthrough, and the gap above is the place to push. It deserves a serious referee; I'd accept it with revisions that close or at least acknowledge the idealization.","headline":"First convergence analysis of Schwarz local solves in lower precision; the theory is clean but covers an idealized exact-inverse model, and the practical claim rests on an unstated robustness assumption.","tokens_in":24176,"tokens_out":5841,"would_cite":true,"duration_ms":48989,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F10","65F08","65N55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing exact subdomain solves in Schwarz methods with sign-informed lower-precision solves preserves convergence for M-matrices when two local rounding-error conditions hold; in tested model problems four to five digits of local…","keywords":["multiprecision arithmetic","mixed precision","Schwarz methods","domain decomposition","M-matrix","additive Schwarz","multiplicative Schwarz","low-precision local solves"],"falsifier":"Take one of the paper's model problems with $N=108900$ and two overlapping subdomains, implement the low-precision local solves in actual IEEE fp16 hardware or a simulator that models underflow and overflow, and check condition (19) explicitly; if there is a precision at which $\\|A_i^{-1}F_i\\|<1$ holds but $A_i^{-1}\\ge A_i^{-1}F_iA_i^{-1}$ fails and the multiprecision multiplicative Schwarz iteration has spectral radius at least 1 or fails to converge in the error norm, then the sufficiency claim and the Remark 3 assumption that (16) is generally binding would be refuted for that case.","tokens_in":23161,"feed_emoji":"🧮","tokens_out":10975,"duration_ms":89298,"temperature":0.7,"pith_summary":"Schwarz domain-decomposition methods split a large linear system into overlapping subdomain problems; this paper asks how few digits the subdomain solves can use before the outer iteration stops converging. The authors prove that for M-matrices—nonsingular matrices with nonpositive off-diagonal entries and nonnegative inverse—the answer is controlled by two local conditions. If each subdomain matrix is rounded with a sign-aware rule (positive entries up, negative entries toward zero) and if $\\|A_i^{-1}F_i\\|<1$ together with $A_i^{-1}\\ge A_i^{-1}F_iA_i^{-1}$ hold, then the multiprecision damped additive, restricted additive, and multiplicative Schwarz iterations converge. In the numerical experiments on discretized reaction-advection-diffusion equations, the conditions are met with about four to five accurate decimal digits, making single precision sufficient for the local solves and half precision viable for smaller problems. The practical payoff is that the expensive local solves can be run at reduced precision, with the required precision chosen from computable algebraic conditions rather than by trial and error.","feed_headline":"Five digits of local accuracy keep Schwarz solvers convergent","feed_subtitle":"Low-precision local solves save cost; four to five digits match double precision on tested M-matrix PDEs.","key_machinery":"The load-bearing object is the sign-informed rounding rule $\\operatorname{roundMmtrx}(X,u_\\ell)$: positive entries are rounded upward and negative entries toward zero, so that $\\tilde A_i$ differs from $A_i$ by an entrywise nonnegative error matrix $F_i$. This makes the splitting $A_i=\\tilde A_i-(\\tilde A_i-A_i)$ weak regular once the inverse perturbation is controlled. The analysis expands $\\tilde A_i^{-1}=A_i^{-1}(I+F_iA_i^{-1})^{-1}$ as a Neumann series; condition (16) guarantees the series converges and condition (19) guarantees every term keeps the inverse nonnegative. The row/column rescaling of the subdomain systems into the target precision's exponent range, taken from the cited half-precision rescaling algorithm, is what allows the conditions to be met in practice and prevents overflow.","core_discovery":"The central claim is Theorem 3: for an M-matrix $A$ partitioned into overlapping subdomains, replacing each exact local solve $A_i^{-1}$ by $\\tilde A_i^{-1}$, where $\\tilde A_i$ is the sign-informed low-precision rounding of the rescaled subdomain matrix, yields convergent multiprecision Schwarz methods provided (16) and (19) hold for every $i$: $\\|A_i^{-1}F_i\\|<1$ in some norm and $A_i^{-1}\\ge A_i^{-1}F_iA_i^{-1}$ elementwise, with $F_i=\\tilde A_i-A_i$. The convergence is $\\rho(\\tilde T_{\\mathrm{MS}})<1$, $\\rho(\\tilde T_{\\mathrm{RAS}})<1$, and $\\rho(\\tilde T_{\\mathrm{AS},\\theta})<1$ for damping $\\theta<1/q$, where $q$ is the coloring number of the subdomain overlap graph. The rounding rule is engineered so that $F_i\\ge0$, which turns the splitting into a weak regular splitting; the two conditions make the Neumann expansion of $\\tilde A_i^{-1}$ valid and nonnegative. In the finite-difference experiments on three nonsymmetric and three symmetric M-matrix problems, the first precision at which the conditions hold is $d_\\ell=4$ or 5 digits, the observed convergence factor is then essentially identical to double precision, and the same threshold stabilizes the iteration count of preconditioned GMRES. Theorem 4 gives the analogous SPD/Stieltjes statement with an additional eigenvalue condition $\\lambda_{\\min}(A_i)\\ge 2|\\lambda_{-\\infty}(F_i)|$.","pith_inferences":["Editorial inference: if condition (16) is the binding one, as the paper's Remark 3 suggests but does not prove, then for large problems the precision threshold could be checked cheaply from $\\|A_i^{-1}\\|\\,\\|F_i\\|$ or a norm estimate, avoiding the dense-matrix verification of (19) that becomes untenable as $N$ grows.","Editorial inference: the sign-informed rounding recipe is not tied to Schwarz specifically; it should transfer to other splittings or multisplittings whose convergence rests on weak regular splittings, such as block Jacobi or overlapping block smoothers, giving them a similarly principled mixed-precision mode.","Editorial inference: the experiments suggest a testable rule of thumb: choose the local precision so that its unit roundoff is near $10^{-5}$, essentially independent of the subdomain condition numbers ($10^3$ to $10^{11}$ in the tests); checking this on three-dimensional problems with many subdomains and actual fp16/fp32 hardware would show whether the rule survives outside two-subdomain model pr"],"forward_implications":["For the tested M-matrix model problems, four to five accurate decimal digits in the local solves satisfy the theoretical conditions, and further precision buys almost no visible improvement in the first 60 iterations.","Single precision is therefore sufficient for the subdomain solves in these problems, and half precision is viable for small $N$; on hardware with native low-precision arithmetic this can reduce the cost of the dominant part of the algorithm without meaningful convergence loss.","The multiprecision Schwarz methods remain effective as left preconditioners for GMRES: iteration counts to reach a relative residual of $10^{-12}$ are stable once the same precision thresholds are met.","Increasing the local precision cannot worsen the spectral radius of the multiprecision additive, restricted additive, or multiplicative Schwarz iteration matrices, by the monotonicity result (22).","The convergence guarantees hold for the methods as stand-alone stationary iterations, with separate sufficient conditions for nonsymmetric M-matrices (Theorem 3) and for symmetric Stieltjes matrices (Theorem 4)."],"supporting_citations":[{"why":"Supplies the weak regular splitting convergence theory for overlapping additive Schwarz iterations that the multiprecision analysis extends.","marker":"[17]"},{"why":"Supplies the algebraic convergence theory for restricted additive Schwarz and the inexact-solve framework used for RAS and multiplicative methods.","marker":"[18]"},{"why":"Supplies the algebraic convergence theory for multiplicative Schwarz and the coloring/damping conditions reprised in Theorems 3 and 4.","marker":"[3]"},{"why":"Provides the complementary condition-number-based adaptive precision block-Jacobi approach that motivates choosing a per-subdomain precision.","marker":"[2]"},{"why":"Provides the row/column rescaling procedure that fits the subdomain systems into the low precision's exponent range before rounding.","marker":"[32]"},{"why":"Supplies the low-precision simulation toolbox used in the numerical experiments, including underflow/overflow modeling.","marker":"[31]"},{"why":"Supplies the adjustable-digit simulation environment used for the precision scans and larger-N experiments.","marker":"[37]"},{"why":"Provides the pseudospectral GMRES perturbation bound used to analyze multiprecision Schwarz preconditioners.","marker":"[45]"}],"fun_headline_variants":["Schwarz solvers need just 5 digits locally","Four to five digits enough for Schwarz local solves","Low-precision local solves keep Schwarz convergent","Save digits, keep convergence in Schwarz methods","Multiprecision Schwarz: 5 digits do the job"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the simulated low precisions used in the experiments faithfully reproduce real half/single precision arithmetic (rounding directions, exponent range, underflow/overflow), and that for large problems the unverified condition (19) is indeed implied by the verified condition (16), an assertion the paper leaves as a remark.","fun_headline_variants_meta":{"raw":{"variants":["Schwarz solvers need just 5 digits locally","Four to five digits enough for Schwarz local solves","Low-precision local solves keep Schwarz convergent","Save digits, keep convergence in Schwarz methods","Multiprecision Schwarz: 5 digits do the job"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000685,"raw_usage":{"total_tokens":3109,"prompt_tokens":951,"completion_tokens":2158,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":2086}},"tokens_in":567,"tokens_out":2158,"duration_ms":13579,"temperature":1.0,"reasoning_tokens":2086,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:47:31.565214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take one of the paper's model problems with $N=108900$ and two overlapping subdomains, implement the low-precision local solves in actual IEEE fp16 hardware or a simulator that models underflow and overflow, and check condition (19) explicitly; if there is a precision at which $\\|A_i^{-1}F_i\\|<1$ holds but $A_i^{-1}\\ge A_i^{-1}F_iA_i^{-1}$ fails and the multiprecision multiplicative Schwarz iteration has spectral radius at least 1 or fails to converge in the error norm, then the sufficiency claim and the Remark 3 assumption that (16) is generally binding would be refuted for that case.","supporting_citations":[{"cited_title":"Frommer and D","cited_arxiv_id":null,"evidence_quote":"Supplies the weak regular splitting convergence theory for overlapping additive Schwarz iterations that the multiprecision analysis extends."},{"cited_title":"Frommer and D","cited_arxiv_id":null,"evidence_quote":"Supplies the algebraic convergence theory for restricted additive Schwarz and the inexact-solve framework used for RAS and multiplicative methods."},{"cited_title":"Benzi, A","cited_arxiv_id":null,"evidence_quote":"Supplies the algebraic convergence theory for multiplicative Schwarz and the coloring/damping conditions reprised in Theorems 3 and 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the complementary condition-number-based adaptive precision block-Jacobi approach that motivates choosing a per-subdomain precision."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the row/column rescaling procedure that fits the subdomain systems into the low precision's exponent range before rounding."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the low-precision simulation toolbox used in the numerical experiments, including underflow/overflow modeling."},{"cited_title":"Advanpix LLC., Yokohama, Japan","cited_arxiv_id":null,"evidence_quote":"Supplies the adjustable-digit simulation environment used for the precision scans and larger-N experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the pseudospectral GMRES perturbation bound used to analyze multiprecision Schwarz preconditioners."}],"review_version":1}