{"id":"fb4dc946-0169-4cfe-987b-d90a4aa87564","arxiv_id":"2412.08891","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Reduced-basis projections of parametric eigenvalue problems are proven to approximate eigenvalues and eigenspaces including repeated eigenvalue cases, with error bounds verified on 1D to 3D finite element examples.","lead":"This paper proves error bounds for a computational shortcut that solves parametric eigenvalue problems, such as quantum energy-level calculations, using small reduced models. The bounds cover repeated eigenvalues and are demonstrated on finite element problems from one to three dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The WLOG reduction to A ≻ 0 is not tracked: for indefinite A, Theorem 4.6's upper bound is false (a 2x2 counterexample), so negative-eigenvalue examples lie outside the stated guarantees.","rationale":"Agreeing with the reader's identification: the positivity assumption is indeed the most load-bearing. The proof of Theorem 4.6 explicitly uses the A-inner product and Lemma 4.4 (non-expansiveness of P_A), both of which require A SPD. The \"without loss of generality\" claim is not a harmless normalization because shifting A changes the constants κ_k and the spectral interval that enters Theorem 4.7. My 2x2 example shows the resulting theorem as stated is false for indefinite A, so this is not merely a cosmetic gap. I nevertheless recommend keeping the reader's CONDITIONAL verdict rather than REJECT: the SPD case is proved correctly, the counterexample only affects the claimed reduction to definiteness, and the numerical evidence in the SPD examples (Sections 5.1, 5.2, 5.5) is consistent with the bounds. The negative-eigenvalue examples (Sections 5.3 and 5.4) are presented as demonstrations of ROM accuracy, not as verification of the bounds; they should simply be described as outside the theorem's current hypotheses until the shifted bounds are derived. Since the reader already made the verdict CONDITIONAL, my stress-test leaves that verdict unchanged.","tokens_in":16595,"tokens_out":11321,"duration_ms":115250,"concrete_test":"Run the 2x2 check: A = diag(−1, 2), M = I, Q = (1,1)^T/√2. Compute \\tilde λ_1 = Q^T A Q = 1/2 and κ_1 = ∥e_1∥/∥P_A e_1∥ = 1/√2; compare 1/2 ≤ κ_1^2 λ_1 = −1/2. The inequality fails. Then repeat with A + 2I and verify that the valid bound is 1/2 ≤ κ_{A+2I,1}^2(λ_1 + 2) − 2 = 10.5. If the first comparison fails, Theorem 4.6's WLOG statement is false and the theorems must be restricted to A ≻ 0 or restated with shifted constants. A useful companion check is to rerun the Section 5.3 Gaussian-well example at a parameter with λ_1 < 0 and test whether \\tilde λ_1 ≤ κ_1^2 λ_1 holds; however, the 2x2 case already settles the point analytically.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing point is the assertion in Section 4 that A ∈ S^n_{++} can be assumed \"without loss of generality\" by replacing A with A + tM, t > −λ_1. The proofs of Theorems 4.6 and 4.7 need ∥·∥_A to be a norm and P_A non-expansive in it, which holds only for the shifted operator A + tM, not for the original A. The bounds are not shift-invariant: under A → A + tM both λ_k and \\tilde λ_k increase by t, but κ_k changes, and the upper bound becomes \\tilde λ_k ≤ κ_{A+tM,k}^2(λ_k + t) − t, not κ_k^2 λ_k. Hence the stated theorem is false for indefinite A. Concretely, take A = diag(−1, 2), M = I, Q = (1,1)^T/√2, r = 1. Then \\tilde λ_1 = 1/2, κ_1 = 1/√2, so the claimed upper bound κ_1^2 λ_1 = −1/2 is violated by 1/2 ≤ −1/2. Shifting by t = 2 gives the correct but different bound 1/2 ≤ 10.5. Theorem 4.7 has the same defect: for ν_j < 0, τ_j is negative and 1 + τ_j need not be a valid factor. Since Sections 5.3 and 5.4 solve problems with negative eigenvalues, those examples are not covered by the stated bounds. This is a scope error rather than a flaw in the SPD analysis: restricting the theorems to A ≻ 0, or restating them with shifted constants, would repair it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies reduced-basis (subspace) approximation of parametric symmetric generalized eigenvalue problems Ax=lambda Mx. The main theoretical results are Theorem 4.6, which gives the two-sided bound lambda_k <= tilde_lambda_k <= kappa_k^2 lambda_k for the k-th projected eigenvalue, and Theorem 4.7, which bounds the M-norm distance from any vector in the true eigenspace E_j to the projected eigenspace by (1+tau_j) times the oblique-projection residual of that vector. The proofs use the min-max principle, an oblique projector P_A onto the reduced basis, and an assumed full-rank condition on P_A Phi^(r). The paper also presents numerical experiments for Laplacian, harmonic-oscillator, Gaussian-well, diatomic-well, and heterogeneous-diffusion problems in one to three dimensions, including a case with repeated eigenvalues.","tokens_in":16933,"tokens_out":9070,"duration_ms":90971,"significance":"If the analysis is understood as restricted to A positive definite, the paper delivers a clean and useful result: guaranteed one-sided eigenvalue bounds and near-best eigenspace approximation for symmetric generalized eigenproblems without requiring simple eigenvalues. This is a genuine improvement over earlier reduced-basis eigenvalue bounds that are limited to the smallest or simple eigenpairs, and the numerical study covers substantial three-dimensional problems with parameter-dependent coefficients and degenerate eigenspaces. The bounds are genuine mathematical inequalities with no curve-fitted constants; the numerical experiments illustrate rather than fit the theory. The main caveat is that the stated theorems are not valid for indefinite A, and the paper's own numerical examples with negative eigenvalues lie outside the stated hypotheses; this is a fixable scope error, not a flaw in the positive-definite analysis.","major_comments":[{"comment":"The assertion that A can be assumed positive definite 'without loss of generality' by replacing A with A+tM is not valid for the stated results. Lemma 4.4 and the proofs of Theorems 4.6 and 4.7 use the A-induced norm and the non-expansiveness of P_A, which require A to be positive definite. Under the shift A -> A+tM, the oblique projector P_A, the constants kappa_k, and the spectral interval in Theorem 4.7 all change, and the inequality for the original pencil becomes tilde_lambda_k(A) <= kappa_{A+tM,k}^2(lambda_k(A)+t) - t rather than kappa_k^2 lambda_k(A). The stated bound is therefore false for indefinite A: for A=diag(-1,2), M=I, Q=(1,1)^T/sqrt(2), r=1, one has tilde_lambda_1=1/2 and kappa_1=1/sqrt(2), while kappa_1^2 lambda_1=-1/2, violating tilde_lambda_1 <= kappa_1^2 lambda_1. Since Sections 5.3 and 5.4 report examples with negative eigenvalues, those numerical results are not covered by the stated theorems. The theorems should be restricted to A positive definite, or restated with shifted constants and shifted bounds, and the numerical claims should be adjusted accordingly.","section":"Section 4, paragraph preceding Lemma 4.4"},{"comment":"The eigenspace bound in Theorem 4.7 also inherits the indefiniteness problem. When nu_j < 0, the quantity tau_j = nu_j / min_{k in S_j^c} |nu_j - tilde_lambda_k| is negative, and the step 'by a triangle inequality' yielding ||(I - tilde_P^{S_j}_M P_A)x||_M <= (1+tau_j)||(I-P_A)x||_M is not a valid inequality; the triangle inequality gives (1+|tau_j|), not (1+tau_j). The proof is correct only when all relevant eigenvalues are positive, so that nu_j > 0 and tau_j >= 0. This is another manifestation of the same WLOG defect, but it affects the main eigenspace approximation result independently and should be fixed by either assuming A positive definite or by writing the factor as 1+|tau_j| under appropriate conditions.","section":"Theorem 4.7 and its proof"}],"minor_comments":[{"comment":"There is a typo: 'Moerover' should be 'Moreover'.","section":"Section 4, paragraph before Theorem 4.6"},{"comment":"The notation 'Gr_m(R^n)' should be 'Gr_k(R^n)' when referring to the k-dimensional subspace R(tilde_Phi^(k)).","section":"Theorem 4.6 proof"},{"comment":"The text states that the coarse training set leads to a basis of column size r=16, but then says Figure 6 illustrates bounds for '1 <= k <= r=36'; this should be r=16 for the coarse case.","section":"Section 5.2, coarse-sampling paragraph"},{"comment":"The caption repeats the word 'parametric' ('parametric parametric diatomic well potential'); please remove the duplicate.","section":"Section 5.4, Figure 16 caption"},{"comment":"The term 'density' for the training sets D_train={1,3}x{0,1} and D_train={1,2,3}x{0,0.5,1} is not defined; please clarify what density means in this context.","section":"Section 5.2, training-set description"},{"comment":"The numerical 'verification' of the upper bounds in Theorem 4.6 and Theorem 4.7 computes kappa_k and tau_j from the full-order eigenvectors whose error is being bounded (see Appendix A). This is a consistency check rather than an independent a posteriori verification, and the text should state this limitation explicitly.","section":"Section 5 and Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper is basically sound for the positive-definite case; the indefinite-case WLOG statement is the only substantive technical defect, and it is fixable by restricting the theorems or by carefully tracking shifted constants. I would not reject. The numerical verification of the bounds using exact eigenvectors should be softened in the revision. The paper is within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth reading. The core contribution, Theorems 4.6 and 4.7, is a clean and largely correct a priori error analysis for reduced-basis approximation of symmetric generalized eigenvalue problems that does not require simple eigenvalues. The two-sided bound on the k-th eigenvalue and the eigenspace error estimate in terms of the oblique projection residual are real additions to the literature, which mostly handles simple eigenvalues or a posteriori certification. The proofs for positive definite A are straightforward and check out, and the numerical verification in the SPD examples (Sections 5.1, 5.2, and the repeated-eigenvalue case in 5.5) is informative.\n\nThe soft spot is the one flagged in the stress test. Section 4 says A can be assumed positive definite without loss of generality by replacing A with A + tM, but that shift changes the oblique projector P_A, the constants kappa_k, tau_j, and the spectral intervals. The proofs of Theorems 4.6 and 4.7 genuinely rely on the A-norm being a norm and P_A being non-expansive, which only holds for the shifted operator. As stated, Theorem 4.6 is false for indefinite A; the 2x2 counterexample with A = diag(-1,2) is valid. Sections 5.3 and 5.4 solve problems with negative eigenvalues, so those examples fall outside the stated guarantees. This is a scope error, not a fatal collapse: restricting the theorems to A positive definite, or restating them with shifted constants, repairs it. The paper should do that.\n\nTwo minor points. The abstract and conclusion claim ROM solves the problems \"thousands times faster,\" but no timings are reported; that claim should be removed or backed with data. Also, the verification of the bounds computes kappa_k and tau_j from the very full-order eigenvectors whose error is being bounded, which makes the numerical \"verification\" weaker than it looks, though the bounds themselves are rigorous inequalities and no constant is fitted.\n\nWho gets value from this: researchers working on reduced-order modeling of parametric eigenvalue problems, especially when eigenvalues are repeated or clustered. The paper deserves a serious referee; the flaw is real but localized and fixable. I would send it out, and I would ask the authors to either sharpen the theorems to the SPD case or properly track the shift for indefinite problems. It is not a desk reject.\n\nFor the record, I agree with the reader's conditional verdict but think the paper is closer to accept-with-revisions than reject.","headline":"Genuinely useful SPD reduced-basis eigenvalue bounds with a real, but repairable, scope error: the WLOG shift to positive definite A is not tracked, so the stated theorems do not cover the indefinite examples in Sections 5.3 and 5.4.","tokens_in":17490,"tokens_out":1207,"would_cite":true,"duration_ms":14220,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N25","65F15","65N30","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"For symmetric generalized eigenvalue problems, projecting onto a reduced basis yields eigenvalue estimates provably bounded above by a computable factor times the true eigenvalue, and eigenspace errors are bounded too.","keywords":["reduced order model","eigenvalue problems","subspace approximation","a priori error estimates","non-simple eigenvalues","min-max principle","parametrized eigenvalue problems","oblique projection"],"falsifier":"Compute $\\kappa_k$ and $\\tilde{\\lambda}_k$ by the Appendix A procedure for a positive-definite pair $(A,M)$ at a test parameter where the reduced basis is deliberately poor (e.g., one snapshot drawn from a two-dimensional parameter domain); if $\\tilde{\\lambda}_k > \\kappa_k^2 \\lambda_k$ at any test point, Theorem 4.6 is false. For the unresolved indefinite case, repeat with an indefinite $A$: a violation there would show the shift-to-positive-definite step does not transfer the bound.","tokens_in":16393,"feed_emoji":"📐","tokens_out":9104,"duration_ms":89773,"temperature":0.7,"pith_summary":"This paper establishes rigorous a priori error estimates for subspace (reduced-basis) approximations of parametric symmetric generalized eigenvalue problems $A(\\mu)\\varphi = \\lambda M(\\mu)\\varphi$. The central result is that the $k$-th projected eigenvalue $\\tilde{\\lambda}_k$ never falls below the true eigenvalue $\\lambda_k$ and is bounded above by $\\kappa_k^2 \\lambda_k$, where $\\kappa_k$ is a computable measure of how well the reduced subspace captures the first $k$ eigenvectors. For eigenspaces, the paper shows that the $M$-norm error of a projected eigenspace is controlled by the residual of an oblique projection onto the reduced subspace, multiplied by a spectral-gap factor. These bounds hold without requiring simple eigenvalues, so they cover repeated eigenvalues that arise naturally in heterogeneous diffusion and Schrödinger-type problems. If correct, the estimates give practitioners a certificate that a small projected system can replace a large full-order solve.","feed_headline":"Projected eigenvalues provably stay inside a computable band","feed_subtitle":"Reduced-basis approximations of symmetric eigenproblems now carry rigorous error bounds, including repeated eigenvalues.","key_machinery":"The load-bearing object is the oblique projector $P_A = Q(Q^\\top A Q)^{-1} Q^\\top A$, which maps $\\mathbb{R}^n$ onto the reduced basis space along the $A$-orthogonal complement. Lemma 4.4 proves $P_A$ is non-expansive in the $A$-norm, $\\|P_A x\\|_A \\le \\|x\\|_A$, and this single inequality converts the Rayleigh quotient of a projected eigenvector into a comparison with the true eigenvalue. The two constants that carry the quantitative control are $\\kappa_k$, a generalized Rayleigh quotient measuring the distortion of the first $k$ eigenvectors under $P_A$, and $\\tau_j$, a spectral-gap factor defined from the distance between $\\nu_j$ and the projected eigenvalues of other eigenspaces. The min-max theorem (Theorem 4.2), built on the generalized Rayleigh quotient $R_{A,M}(x)=x^\\top A x / x^\\top M x$, supplies the lower bound and orders the eigenvalues.","core_discovery":"The paper's central claim is that the Rayleigh-Ritz (subspace) projection of a symmetric generalized eigenproblem inherits guaranteed one-sided error control. Theorem 4.6 states that for $1\\le k\\le r$, the projected eigenvalue satisfies $\\lambda_k \\le \\tilde{\\lambda}_k \\le \\kappa_k^2 \\lambda_k$, where $\\kappa_k$ is the supremum over the true eigenvector subspace of the ratio $\\|y\\|_M / \\|P_A y\\|_M$, and $P_A = Q(Q^\\top A Q)^{-1}Q^\\top A$ is the oblique projector onto the reduced basis. Theorem 4.7 states that for any vector $x$ in the eigenspace $E_j$ of a distinct eigenvalue $\\nu_j$, the $M$-norm distance from $x$ to the projected eigenspace is at most $(1+\\tau_j)\\|(I-P_A)x\\|_M$, with $\\tau_j$ a scaled inverse spectral gap. Because the analysis works with whole eigenspaces rather than individual eigenvectors, it covers non-simple eigenvalues, where eigenvectors themselves are ambiguous but the eigenspace is well defined.","pith_inferences":["The bounds suggest a direct adaptive greedy criterion: use $\\kappa_k$ (or the residual $\\|(I-P_A)x\\|_M$) as a cheap error indicator to choose the next snapshot parameter; the paper does not propose such a greedy algorithm.","Because Theorem 4.7 is stated for entire eigenspaces, it likely transfers to block Rayleigh-Ritz approximations of several eigenvectors at once; testing this on the double-eigenvalue example of Section 5.5 would be a direct check.","When an eigenvector rotates sharply with the parameter, $\\kappa_k$ grows and the upper bound widens, which explains the coarse-sampling failure in the harmonic oscillator example and suggests that snapshot sets must straddle the rotation region.","An open direction implicit in the paper is the indefinite case: the shift argument is asserted but not proved, so extending Theorems 4.6 and 4.7 to problems whose $A$ has negative eigenvalues (as in the Gaussian and diatomic well examples) requires a separate argument."],"forward_implications":["At any tested parameter, the reduced-basis eigenvalue $\\tilde{\\lambda}_k$ is a guaranteed upper bound of $\\lambda_k$ and is never more than $\\kappa_k^2$ times it, so accuracy can be certified without re-solving the full system.","Repeated or clustered eigenvalues are covered: the eigenspace bound applies to the entire degenerate space, not to a chosen eigenvector, so one does not need to resolve the ambiguity of eigenvectors within an eigenspace.","The constants $\\kappa_k$ and $\\tau_j$ are computable from snapshots and the reduced solution, so the error estimates can be evaluated in practice and not just stated asymptotically.","Denser snapshot sampling directly reduces the growth of $\\kappa_k$, as demonstrated in the harmonic oscillator example where refining the training set from four to nine parameter points cut the interpolation error from order $10^0$ to below $10^{-6}$.","The framework applies to parametric boundary conditions and coefficient fields from finite-element discretizations of Laplace, Schrödinger, and heterogeneous diffusion problems, including multi-dimensional parameter domains."],"supporting_citations":[{"why":"Supplies the symmetric generalized eigenvalue decomposition and the spectral/min-max background used throughout Section 4.","marker":"[40]"},{"why":"Earlier reduced-basis output bounds for symmetric positive definite eigenvalue problems that this paper extends beyond the first eigenpair.","marker":"[27]"},{"why":"Provides the certified greedy strategy and a posteriori error estimates for parametrized elliptic eigenvalue problems that motivate the new a priori bounds.","marker":"[30]"},{"why":"Earlier simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems, the direct predecessor for approximating several smallest eigenvalues at once.","marker":"[38]"}],"fun_headline_variants":["Subspace eigenvalue bounds now cover repeated eigenvalues","Rayleigh-Ritz error control for symmetric eigenproblems","Reduced-basis eigenspaces get rigorous one-sided bounds","New error estimates for subspace eigenvalue approximations","Provable band for projected eigenvalues, even non-simple"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $A$ can be taken positive definite without loss of generality by shifting it to $A+tM$; the proofs of Theorems 4.6 and 4.7 use the $A$-norm and the oblique projector $P_A$, and the paper does not establish the stated bounds for indefinite $A$, even though two of its own numerical examples involve negative eigenvalues.","fun_headline_variants_meta":{"raw":{"variants":["Subspace eigenvalue bounds now cover repeated eigenvalues","Rayleigh-Ritz error control for symmetric eigenproblems","Reduced-basis eigenspaces get rigorous one-sided bounds","New error estimates for subspace eigenvalue approximations","Provable band for projected eigenvalues, even non-simple"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1286,"prompt_tokens":886,"completion_tokens":400,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":326}},"tokens_in":502,"tokens_out":400,"duration_ms":4482,"temperature":1.0,"reasoning_tokens":326,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:27:59.622672+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\kappa_k$ and $\\tilde{\\lambda}_k$ by the Appendix A procedure for a positive-definite pair $(A,M)$ at a test parameter where the reduced basis is deliberately poor (e.g., one snapshot drawn from a two-dimensional parameter domain); if $\\tilde{\\lambda}_k > \\kappa_k^2 \\lambda_k$ at any test point, Theorem 4.6 is false. For the unresolved indefinite case, repeat with an indefinite $A$: a violation there would show the shift-to-positive-definite step does not transfer the bound.","supporting_citations":[{"cited_title":"SIAM, 1998","cited_arxiv_id":null,"evidence_quote":"Supplies the symmetric generalized eigenvalue decomposition and the spectral/min-max background used throughout Section 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier reduced-basis output bounds for symmetric positive definite eigenvalue problems that this paper extends beyond the first eigenpair."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the certified greedy strategy and a posteriori error estimates for parametrized elliptic eigenvalue problems that motivate the new a priori bounds."},{"cited_title":"Simultaneous reduced basis approximation of parameterized 22 elliptic eigenvalue problems.ESAIM: Mathematical Modelling and Numerical Analysis, 51(2):443–465, 2017","cited_arxiv_id":null,"evidence_quote":"Earlier simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems, the direct predecessor for approximating several smallest eigenvalues at once."}],"review_version":1}