{"id":"cd02a555-6fe5-494e-bbec-70cef9774e46","arxiv_id":"1908.05537","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The paper introduces interface-only two-level domain decomposition methods (S2S), proves convergence formulas, and shows spectral coarse spaces are not always asymptotically optimal.","lead":"Domain decomposition solves big equations by splitting the region and iterating between subproblems; this paper moves the correction step onto the interfaces between subregions instead of the whole volume, making each iteration cheaper in memory and time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.8's spectral-radius formula is wrong when n1+n2 is odd and |θ1|≠|θ2|: on tail modes the coarse correction acts as the identity, so the spectral radius is the geometric mean, not the arithmetic maximum.","rationale":"The reader's weakest assumption concerned the diagonalizability hypotheses (H2)-(H3), which is a genuine limitation. My closer read found a sharper and more concrete issue: the odd-parity formula in Theorem 4.8 confuses operator norm with spectral radius. The tail argument does not depend on delicate spectral hypotheses beyond the common orthonormal Fourier basis already assumed, and a 4×4 example verifies the incorrect prediction. This is load-bearing because Theorem 4.8 is one of the paper's principal convergence results for local coarse functions and is advertised as part of the new convergence analysis for the S2S framework; a wrong formula changes predicted iteration counts in any nonsymmetric two-subdomain case with unequal subdomain sizes. The error does not disprove the S2S framework or the perturbation analysis in Theorem 4.6, which rest on independent calculations, and the numerical experiments in Section 6 use symmetric subdomains where the parity error is invisible. The correct statement appears to be ρ(T)=|θ1(m+1)θ2(m+1)|^{(n1+n2)/2} for every parity, because the tail is untouched by the coarse correction. The paper should remain conditional: the framework and the non-optimality results are plausible, but Section 4.3 needs correction and re-scoping before the convergence analysis can be accepted as written.","tokens_in":33635,"tokens_out":43700,"duration_ms":410655,"concrete_test":"Assemble the 4×4 matrices with X=(R^2)^2, G1=diag(0.9,0.81), G2=diag(0.1,0.01), A=I-G, coarse basis e1 in both components, n1=1, n2=0. Let P have columns c1=(1,0,0,0)^T and c2=(0,0,1,0)^T, set R=P^T, A_c=RAP, and T=(I-P A_c^{-1} R A)G. Printing eig(T) yields 0 and ±0.09, so ρ(T)=0.09, whereas the formula in Theorem 4.8 gives max{0.81,0.01}=0.81. Repeating with n1=2, n2=0 confirms the even-parity value 0.0081, which matches the corrected geometric-mean rule for all parities.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 4.3, Theorem 4.8 is internally inconsistent. For the local coarse space Vc=(span{ψ1,...,ψm})^2 and any tail mode k>m, R A kills that mode because R projects onto the first m Fourier coefficients; hence E=I-PA_c^{-1}RA is the identity on the entire tail space. Consequently T=G^{n1+n2} on the tail. The 2×2 block of G on mode k is [[0,θ1(k)],[θ2(k),0]], whose spectral radius is (|θ1(k)θ2(k)|)^{1/2}, so ρ(T)=|θ1(m+1)θ2(m+1)|^{(n1+n2)/2} for all parities. The 'otherwise' formula in Theorem 4.8 gives |θ1θ2|^{(n1+n2-1)/2} max{|θ1|,|θ2|}, which is the operator norm of the single-step block, not its spectral radius. A concrete counterexample with θ1(2)=0.81, θ2(2)=0.01, m=1, n1=1, n2=0: the tail block is [[0,0.81],[0.01,0]] with eigenvalues ±0.09, so ρ=0.09, while the theorem predicts 0.81. This is the unequal-subdomain setting allowed by Examples 3-4; the numerical tests use equal subdomains where θ1=θ2 and cannot reveal the error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces 'spectral two-level substructured' (S2S) domain decomposition methods, in which both the smoothing iteration and the coarse correction are carried out on the interface/skeleton space rather than in the volume. After deriving the substructured Schwarz formulation, the authors write the two-level iteration operator as T = G^{n2}(I - P A_c^{-1} R A)G^{n1} and analyze it under the hypothesis that A and G share a complete eigenbasis (H2-H4). They prove that with the spectral coarse space the spectral radius is |λ_{m+1}|, study one-dimensional perturbations of the coarse space (Theorem 4.6), treat local coarse spaces built from common eigenfunctions of the two subdomain blocks (Theorem 4.8), and propose PCA- and neural-network-based construction of the coarse space. Numerical experiments on Poisson and high-contrast diffusion problems compare S2S variants with the volume SHEM method. The main claims are that S2S matches standard convergence at lower memory/communication cost and that spectral coarse spaces need not be asymptotically optimal.","tokens_in":34003,"tokens_out":18282,"duration_ms":173539,"significance":"The paper's conceptual contribution is genuinely useful: it gives a single framework that connects substructured Schwarz smoothers with Galerkin coarse correction, and it provides explicit, checkable formulas for the two-level spectral radius in the diagonalizable case. Theorem 4.6 and Corollary 4.7 are the strongest part of the paper; they give precise, falsifiable conditions under which the spectral coarse space is suboptimal or even causes divergence, and the numerical experiment in Section 6.3 confirms the predicted value ρ(T)=1.2322. The PCA and DNN constructions are well motivated, and the implementation tricks in the appendix are practical. The paper does not ship code or machine-checked proofs, but the algebraic derivations are explicit enough to be reproduced. If the error in Theorem 4.8 and the gap in Theorem 4.4 are repaired, this would be a solid contribution to the domain-decomposition literature.","major_comments":[{"comment":"The formula for the 'otherwise' case is not the spectral radius and is false when θ1(m+1) and θ2(m+1) differ and n1+n2 is odd. For k>m the coarse correction acts as the identity on the k-th mode, because R projects onto the first m Fourier coefficients; hence T equals G^{n1+n2} on the tail space, for every parity. The 2×2 block of G on mode k is [[0,θ1(k)],[θ2(k),0]], whose spectral radius is (|θ1(k)θ2(k)|)^{1/2}. For example, take m=1, n1=1, n2=0, θ1(2)=0.81, θ2(2)=0.01; the tail block has eigenvalues ±0.09, so ρ(T)=0.09, while Theorem 4.8 predicts max{0.81,0.01}=0.81. The error enters in the sentence 'a similar direct calculation leads to ||T^n||_op = |π^{n(n1+n2)}(m+1)|': for odd powers the max-norm of G^n is not the n-th power of the single-step max-norm, so the parity-dependent expression is an operator norm, not a spectral radius. The correct statement is ρ(T)=|θ1(m+1)θ2(m+1)|^{(n1+n2)/2} for all parities. Because the numerical tests in Section 6 use equal subdomains (θ1=θ2), they cannot detect the discrepancy.","section":"Theorem 4.8, Section 4.3"},{"comment":"The proof uses the identity [I-P A_c^{-1} R A]^* = [I- A P A_c^{-1} R] after 'Since A and G are self adjoint'. This identity requires P=R^* with respect to the inner product in which the adjoint is taken. In the manuscript, P and R are defined via ⟨·,·⟩ in (3.1), whereas the theorem assumes {ψ_k} is orthonormal with respect to a possibly different inner product ⟨⟨·,·⟩⟩ and takes adjoints with respect to ⟨⟨·,·⟩⟩. If ⟨⟨·,·⟩⟩≠⟨·,·⟩, the adjoint of R under ⟨⟨·,·⟩⟩ is not the map P, and the claimed simplification ~T=diag(0,Λ_m) does not follow. The statement can be repaired by assuming that the same inner product is used for P,R and for the orthonormal eigenbasis, or by redefining P as the ⟨⟨·,·⟩⟩-adjoint of R. Without such a repair, Theorem 4.4 is unsupported, although the spectral-radius statement of Theorem 4.3 remains valid.","section":"Theorem 4.4, Section 4.2"}],"minor_comments":[{"comment":"In the γ=0 case the squared denominator is written as (λ̃2 ε² + λ̃2_1)²; this appears to be a typo for (λ̃2 ε² + λ̃1)².","section":"Equation (4.12)"},{"comment":"The matrix P_l = U_l Σ_l V_l^T has size N_s×q (if U_l is N_s×l and V_l is q×l), while G^r X is N_s×N_s, so the norm comparison is dimensionally inconsistent; please clarify how P_l is embedded in N_s×N_s before stating the bound.","section":"Lemma 5.1, Section 5.1"},{"comment":"The high-contrast many-subdomain runs are outside the diagonalizability and monotone-eigenvalue hypotheses (H2)-(H3) under which Theorems 4.3-4.8 are proved; the text should state explicitly that the robustness observed there is an empirical finding, not a consequence of the theorems. The observation in Section 6.3 that the PCA method can diverge for small r further underscores this point.","section":"Section 6.3"},{"comment":"The phrase 'S2S-EHM' appears in Table 6.3, while the text uses 'S2S-HEM'; please unify the nomenclature.","section":"Table 6.3"},{"comment":"The inequality |λ_k|≤|λ_{k-1}| is not defined for k=1; state the monotonicity condition for k≥2.","section":"Hypothesis (H3)"}],"recommendation":"major_revision","confidential_remarks":"The main reason for major_revision rather than reject is that the faulty Theorem 4.8 has a simple, verifiable correction, and the remaining analysis—especially Theorem 4.6 and the framework itself—appears sound. The authors should also double-check whether any subsequent statement in Section 4.3 or Section 6 relies on the parity formula; as printed, the theorem is a clear counterexample to the paper's convergence claims for unequal subdomains."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe core idea is solid: define both the smoother and the coarse correction on the interface/skeleton, and you get a genuine memory saving without hurting iteration counts. The perturbation analysis in Theorem 4.6 is the real gem — a closed-form rational spectral radius for a one-dimensional perturbed coarse space, with clean conditions for when the spectral coarse space is not optimal. The numerical work is extensive and mostly supports the claims, including the divergence example and the smoothing repair.\n\nBut there are two problems in the theory that a referee will need to see addressed.\n\nFirst, the proof of Theorem 4.4 silently uses the adjoint identity [I - P A_c^{-1} R A]^* = [I - A P A_c^{-1} R]. This requires R^* = P in the inner product used for the adjoint. The theorem's own setup allows the eigenvector inner product ⟨⟨·,·⟩⟩ to differ from the one used to define P and R. If they differ, the identity doesn't follow. It is a fixable gap — just state that R and P are adjoints in the analysis inner product — but as written the proof isn't complete.\n\nSecond, Theorem 4.8 states a wrong formula when n1+n2 is odd and the local eigenvalues are not equal in modulus. On tail modes the coarse correction is the identity, so T = G^{n1+n2}. The block of G on mode k has eigenvalues ±√(θ1θ2), so the spectral radius is |θ1θ2|^{(n1+n2)/2} for every parity. The theorem's 'otherwise' branch gives |θ1θ2|^{(n1+n2-1)/2} max{|θ1|,|θ2|}, which is the single-step operator norm, not the spectral radius. Counterexample: m=1, n1=1, n2=0, θ1(2)=0.81, θ2(2)=0.01. The theorem predicts 0.81; the true spectral radius is 0.09. Since the numerics use equal subdomains (θ1=θ2), the error is invisible in the paper.\n\nThe diagonalizability assumptions (H2)-(H3) are restrictive and leave the high-contrast many-subdomain experiments only partially explained; the authors are fairly candid about this. No code or data is included, which is a pity.\n\nWho gets value: researchers in domain decomposition and coarse space construction. The main framework and Theorem 4.6 deserve a serious referee; the paper should go to review, but the referee should request a corrected Theorem 4.8 and a completed proof of Theorem 4.4.","headline":"A genuinely useful substructured two-level DD framework with a real error in Theorem 4.8 and a proof gap in Theorem 4.4; the main idea still holds up.","tokens_in":34556,"tokens_out":6738,"would_cite":true,"duration_ms":60794,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N55","65F10","65N22","35J57"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper moves two-level domain decomposition onto subdomain interfaces, proves convergence at lower memory cost, and shows that the standard spectral coarse space is not always the fastest choice.","keywords":["domain decomposition methods","Schwarz methods","substructured methods","two-level methods","coarse spaces","spectral coarse space","deep neural networks","elliptic equations"],"falsifier":"In the two-channel jumping-coefficient experiment of Section 6.3, the paper reports $\\lambda_1 = 0.999$, $\\lambda_2 = -0.9989$, $\\lambda_3 = -0.99863$, $\\lambda_4 = 0.99861$, $\\lambda_5 = 0.2392$ and predicts the two-level spectral radius $\\rho(T) = 1.2322$ for the coarse space $\\operatorname{span}\\{\\psi_1 + 0.01\\psi_5\\}$. Computing $\\rho(T)$ directly for that coarse space, and tracing $\\rho(T)$ for the family $\\operatorname{span}\\{\\psi_1 + \\varepsilon \\psi_2\\}$ as $\\varepsilon$ varies, would settle both the claim that a perturbed coarse space can reach $|\\lambda_3| < |\\lambda_2|$ and the claim that a poorly chosen perturbation makes the iteration diverge.","tokens_in":33402,"feed_emoji":"🧩","tokens_out":20029,"duration_ms":162874,"temperature":0.7,"pith_summary":"Classical two-level domain decomposition methods build coarse spaces from functions defined over the whole computational domain, which makes the correction step costly in memory and time. This paper introduces a new class, the S2S ('spectral two-level substructured') methods, in which both the smoothing iteration and the coarse correction act only on the interfaces between subdomains, so a coarse function is an array of size $O(1/h^2)$ rather than $O(1/h^3)$ in three dimensions. The analysis proves that with the natural spectral coarse space — the span of the slowest-decaying eigenfunctions of the interface iteration operator — the two-level iteration converges at a rate equal to $|\\lambda_{m+1}|$, the modulus of the next eigenvalue. It also proves that this standard choice is not always the fastest: perturbing the coarse space can strictly reduce the spectral radius when eigenvalues have opposite signs or eigenvectors are non-orthogonal, and a badly chosen perturbation can make the method diverge. A sympathetic reader should care because the framework lowers the cost of two-level elliptic solvers and corrects a widely used rule of thumb about spectral coarse spaces.","feed_headline":"Move two-level solvers fully onto subdomain interfaces","feed_subtitle":"S2S matches volume convergence at lower memory, and spectral coarse spaces are not always optimal.","key_machinery":"The load-bearing object is the two-level iteration operator $T := G^{n_2}(I - P A_c^{-1} R A) G^{n_1}$ on the interface space $\\mathcal{H}$, where $G$ is the substructured Schwarz smoother and $I - P A_c^{-1} R A$ is the coarse-correction projection onto the coarse space $V_c$. Under the hypothesis that $A$ and $G$ share a complete eigenbasis with moduli $|\\lambda_k|$ non-increasing, $T$ admits a block lower-triangular infinite-matrix representation whose diagonal block is $\\operatorname{diag}(\\lambda_{m+1}, \\lambda_{m+2}, \\dots)$; this representation is what yields $\\rho(T) = |\\lambda_{m+1}|$ for the spectral coarse space. The finer mechanism is the closed formula $\\lambda(\\varepsilon, \\gamma)$, giving the nontrivial eigenvalue of $T$ for the perturbed coarse space $\\operatorname{span}\\{\\psi_1 + \\varepsilon \\psi_2\\}$ as a function of the perturbation $\\varepsilon$ and the overlap $\\gamma = \\langle \\psi_1, \\psi_2 \\rangle$; its monotonicity, zeros, and vertical asymptote determine exactly when the spectral choice is suboptimal and when the method diverges.","core_discovery":"The paper establishes that a two-level domain decomposition iteration can be formulated and analyzed entirely on subdomain interfaces, with iteration operator $T := G^{n_2}(I - P A_c^{-1} R A) G^{n_1}$ acting on interface functions. For the spectral coarse space $V_c = \\operatorname{span}\\{\\psi_1, \\dots, \\psi_m\\}$, built from the $m$ dominant eigenfunctions of the one-level operator $G$, the analysis yields $\\rho(T) = |\\lambda_{m+1}| < 1$ whenever $A$ and $G$ share a complete eigenbasis (Theorems 4.3 and 4.5). The central conceptual result, Theorem 4.6, is that this spectral coarse space is not asymptotically optimal in general: for the one-dimensional coarse space $\\operatorname{span}\\{\\psi_1 + \\varepsilon \\psi_2\\}$, a nonzero perturbation $\\varepsilon$ strictly reduces the spectral radius when the two eigenvalues have opposite signs or the eigenvectors are non-orthogonal, while a poorly chosen perturbation can make the iteration diverge. The paper adds a remedy — replacing $V_c$ by $G^r V_c$, the coarse space smoothed by $r$ applications of the iteration operator — which restores convergence and improves the representation of slow modes (Corollary 4.7). On the numerical side, it proposes a PCA/SVD procedure and a deep neural network optimization of $\\rho(T)$ for building coarse spaces, and validates the method on Poisson and high-contrast diffusion problems.","pith_inferences":["The paper leaves implicit the higher-dimensional version of its one-dimensional result: for $m > 1$ the minimizer of $\\rho(T)$ should generically not be the span of the top $m$ eigenvectors, so learned coarse spaces (DNN, PCA) can be strictly better than the spectral space, not merely cheaper to build.","The divergence mechanism of Theorem 4.6 suggests a safeguard the authors do not state: for any black-box or learned coarse space, one can certify safety during construction by monitoring $\\rho(T)$ or by substituting $G^r V_c$ for $V_c$, making coarse-space generation self-validating.","Because substructured coarse functions are one spatial dimension smaller than volume functions, the PCA and DNN generation procedures become tractable in three-dimensional regimes where their volume analogues are prohibitive; a multilevel S2S, with each level itself substructured, would compound the memory savings."],"forward_implications":["For elliptic problems in two and three dimensions, S2S methods match the convergence rate of volume two-level methods while storing restriction and prolongation operators of size $O(1/h^2)$ instead of $O(1/h^3)$, and the equivalent implementations S2S-B1 and S2S-B2 avoid the extra residual computation so each steady-state iteration costs a single smoothing application.","With the spectral coarse space, the asymptotic convergence factor is exactly the modulus of the $(m+1)$-st eigenvalue of the interface smoother, $\\rho(T) = |\\lambda_{m+1}| < 1$, whenever the smoother and the operator share a complete eigenbasis, covering self-adjoint operators and finite-dimensional non-symmetric operators (Theorems 4.3 and 4.5).","When the dominant eigenvalues have opposite signs or the eigenvectors are non-orthogonal, a non-spectral coarse space can strictly outperform the spectral one; the numerical experiments confirm this, with the DNN-built coarse space converging fastest in the two-subdomain Poisson setting.","A coarse space that poorly represents the dominant modes can make the two-level iteration diverge even though the one-level iteration converges; replacing the coarse space by $G^r V_c$, its image under $r$ smoothing steps, restores convergence and improves the approximation of slow modes (Corollary 4.7).","For local, per-subdomain coarse functions the convergence factor is governed by products of the local eigenvalues: $\\rho(T) = |\\theta_1(m+1)\\, \\theta_2(m+1)|^{(n_1+n_2)/2}$ when $n_1$ and $n_2$ share parity, with the other parity inserting one extra factor $\\max\\{|\\theta_1(m+1)|, |\\theta_2(m+1)|\\}$ (Theorem 4.8)."],"supporting_citations":[{"why":"Supplies the baseline result that for symmetric positive definite problems the spectral coarse space minimizes the energy norm of the two-level operator; Theorem 4.6 extends and qualifies this conclusion.","marker":"[55]"},{"why":"Provides the two-level iteration framework and the formula for T as $G^{n_2}(I - P A_c^{-1} R A) G^{n_1}$ that the S2S convergence analysis is built on.","marker":"[12]"},{"why":"Introduces the idea of learning prolongation and restriction operators by optimizing a deep multigrid network, which the S2S-DNN coarse-space construction follows.","marker":"[44]"},{"why":"Defines the harmonically enriched multiscale (SHEM) coarse space whose interface functions seed the S2S-HEM variant and whose volume implementation is the numerical baseline.","marker":"[31]"},{"why":"Presents the SHEM coarse space for restricted additive Schwarz and the heuristic that spectral spaces correct badly converging modes, which the paper refines.","marker":"[30]"},{"why":"Supplies the stochastic trace estimator used to replace $\\rho(T)$ minimization with the sample-average loss minimized by the neural network.","marker":"[43]"},{"why":"Provides the sine-Fourier diagonalization of two-subdomain Schwarz operators that grounds Example 3 and the local-coarse-space analysis of Theorem 4.8.","marker":"[27]"},{"why":"Supplies the known convergence of the parallel Schwarz method in substructured form, justifying the one-level smoother on which S2S is built.","marker":"[14]"}],"fun_headline_variants":["Interface-only two-level solvers beat volume methods","Spectral coarse spaces not always optimal in two-level DD","Two-level DD on interfaces only: spectral coarse spaces suboptimal","Interface-only two-level methods: cheap, but spectral coarse spaces suboptimal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that the smoother and the problem operator share one complete set of eigenvectors, with the slower modes also the larger ones; when that shared basis does not exist, as for overlapping Schwarz methods with strongly jumping coefficients, the predicted rate $\\rho(T) = |\\lambda_{m+1}|$ is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Interface-only two-level solvers beat volume methods","Spectral coarse spaces not always optimal in two-level DD","Two-level DD on interfaces only: spectral coarse spaces suboptimal","Interface-only two-level methods: cheap, but spectral coarse spaces suboptimal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000727,"raw_usage":{"total_tokens":3344,"prompt_tokens":1117,"completion_tokens":2227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":733,"completion_tokens_details":{"reasoning_tokens":2158}},"tokens_in":733,"tokens_out":2227,"duration_ms":15565,"temperature":1.0,"reasoning_tokens":2158,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:12:10.236576+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the two-channel jumping-coefficient experiment of Section 6.3, the paper reports $\\lambda_1 = 0.999$, $\\lambda_2 = -0.9989$, $\\lambda_3 = -0.99863$, $\\lambda_4 = 0.99861$, $\\lambda_5 = 0.2392$ and predicts the two-level spectral radius $\\rho(T) = 1.2322$ for the coarse space $\\operatorname{span}\\{\\psi_1 + 0.01\\psi_5\\}$. Computing $\\rho(T)$ directly for that coarse space, and tracing $\\rho(T)$ for the family $\\operatorname{span}\\{\\psi_1 + \\varepsilon \\psi_2\\}$ as $\\varepsilon$ varies, would settle both the claim that a perturbed coarse space can reach $|\\lambda_3| < |\\lambda_2|$ and the claim that a poorly chosen perturbation makes the iteration diverge.","supporting_citations":[{"cited_title":"Xu and L","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline result that for symmetric positive definite problems the spectral coarse space minimizes the energy norm of the two-level operator; Theorem 4.6 extends and qualifies this conclusion."},{"cited_title":"Ciaramella, M","cited_arxiv_id":null,"evidence_quote":"Provides the two-level iteration framework and the formula for T as $G^{n_2}(I - P A_c^{-1} R A) G^{n_1}$ that the S2S convergence analysis is built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the SHEM coarse space for restricted additive Schwarz and the heuristic that spectral spaces correct badly converging modes, which the paper refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic trace estimator used to replace $\\rho(T)$ minimization with the sample-average loss minimized by the neural network."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the sine-Fourier diagonalization of two-subdomain Schwarz operators that grounds Example 3 and the local-coarse-space analysis of Theorem 4.8."},{"cited_title":"Ciaramella, M","cited_arxiv_id":null,"evidence_quote":"Supplies the known convergence of the parallel Schwarz method in substructured form, justifying the one-level smoother on which S2S is built."}],"review_version":1}