{"id":"51090a35-e20a-4b54-85fd-4bb83115960a","arxiv_id":"2505.03057","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For linear systems with quadratic output, H2-optimal reduced models must satisfy mixed-multipoint tangential interpolation conditions on the linear and quadratic transfer functions, and LQO-IRKA computes reduced models that meet them.","lead":"This paper derives interpolation-based first-order optimality conditions for the H2-optimal approximation of linear systems with quadratic outputs, and proposes an iterative algorithm called LQO-IRKA that constructs reduced models satisfying those conditions. It extends the classical Meier-Luenberger framework for linear systems to a larger class of nonlinear-output systems used in vibration and energy applications.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 3.1 uses perturbations that leave the class of LQO systems: the single-residue perturbation for (29b) violates the symmetry constraint (25).","rationale":"The reader's verdict is CONDITIONAL, and I agree that the paper should not be accepted without revision. The reader's weakest_assumption highlighted the simple-pole assumption and the empirical nature of LQO-IRKA convergence; these are legitimate limitations but they are explicitly stated hypotheses or practical observations. The more specific and load-bearing problem is that the proof of Theorem 3.1, the central theoretical claim, uses perturbed transfer functions that do not correspond to admissible LQO systems. The single-residue perturbation in (53) for (29b) violates the symmetry condition (25) that follows from the symmetric quadratic-output matrices. Similarly, the perturbations for (29c) and (29d) use complex data without the conjugate-symmetric updates needed to remain in the real-LQO class. Since the local-optimality inequality (30) is valid only for feasible competitors, the proof as written does not derive the necessary conditions from the actual feasible set. The gaps are likely repair-friendly: symmetric two-index perturbations and real-valued/conjugate-pair counterparts should yield the same conditions, and the numerical success of LQO-IRKA suggests the conditions are correct. Therefore the appropriate action is to request a revision that supplies admissible perturbation arguments, not to reject the work. The paper has independent support: Theorem 3.2 gives a constructive projection framework, the algorithm is reproducible (code at [32]), and the numerical experiments compare favorably against LQO-BT. These support the value of the contribution even while the proof needs tightening. Thus I recommend keeping the CONDITIONAL verdict, with the specific proof gap addressed.","tokens_in":40851,"tokens_out":17562,"duration_ms":170503,"concrete_test":"Re-derive the first-order optimality condition corresponding to (29b) using the admissible symmetric perturbation δm_{j,k}=δm_{k,j}=-ε e^{iθ}ξ (with the conjugate counterpart handled analogously) and verify that the first-order term in the H2 error expansion (52) remains negative and O(ε), yielding the same condition (29b). Also re-check (29c) and (29d) with real-valued perturbations: for (29c) take the real part of the complex residue-direction perturbation, and for (29d) perturb conjugate pole pairs together with conjugate phase shifts. If the same conditions emerge, Theorem 3.1 stands; if different or no condition emerges, the central claim is not established. As a numerical cross-check, on a small order-2 LQO example, compute the directional derivative of ||G-~G||²_H2 along these admissible directions and confirm it vanishes if and only if (29) hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.1 in Appendix B does not establish that the ε-perturbed reduced models used to derive the necessary conditions lie in the admissible class of order-r LQO systems. For condition (29b), the proof perturbs only the (j,k)-th quadratic residue direction m_{j,k} (eq. (53)). But any LQO system satisfies the symmetry m_{j,k}=m_{k,j} (eq. (25)), a consequence of the symmetric quadratic-output matrices M_k. A perturbation that changes only m_{j,k} (with j≠k) violates this constraint, so the resulting ˇG is not of the form (2) and the sub-optimality inequality (30) cannot be invoked. The same class-membership issue affects (29c) and (29d): the perturbed systems use complex residue-direction changes (54) or a single complex pole shift (58) without the conjugate updates required for a real-valued admissible system. Thus, as written, Appendix B does not prove the interpolatory necessary conditions from within the feasible set; it proves a stationarity condition for a larger, unconstrained set of pole-residue data. This is central because Theorem 3.1 is the paper's principal claim. The gaps appear repairable—e.g., using symmetric two-index perturbations for (29b) and real-structure-preserving perturbations for (29c)/(29d)—so the theorem may survive, but the proof requires revision.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies H2-optimal model reduction for linear systems with quadratic outputs (LQO systems). The main theoretical result, Theorem 3.1, states that any asymptotically stable order-r LQO reduced model with simple poles that minimizes the H2 error (22) must satisfy the four groups of tangential interpolation conditions (29a)-(29d), generalizing the Meier-Luenberger conditions from LTI systems. Theorem 3.2 gives explicit Petrov-Galerkin bases V and W that enforce these interpolation conditions at prescribed data, and Algorithm 4.1 (LQO-IRKA) iteratively updates the interpolation data using the current reduced model's poles and residue directions. The paper includes a detailed proof of the pole-residue H2 inner product formula (Theorem 2.1), a real-valued basis construction (Lemma 4.1), and numerical experiments on a 1D advection-diffusion problem with a quadratic cost, with code and data released on Zenodo.","tokens_in":41067,"tokens_out":15717,"duration_ms":151846,"significance":"The interpolatory optimality conditions are a natural and non-obvious generalization of the classical H2 interpolation theory, and the projection-based enforcement in Theorem 3.2 gives a constructive route to reduced models satisfying them. The numerical results suggest LQO-IRKA is competitive with balanced truncation for LQO systems and robust to initialization. Strengths include the detailed proofs of Theorem 2.1 and Theorem 3.2, the careful treatment of real-valued bases, the explicit complexity statement (shifted linear solves only), and the availability of reproducible code. The main caveat is the gap in the proof of Theorem 3.1 discussed below; if repaired, the paper would be a solid contribution to the model reduction literature.","major_comments":[{"comment":"The proof of Theorem 3.1 does not establish that the perturbed systems used to derive the necessary conditions lie in the admissible class of real order-r LQO systems. For (29b), equation (53) perturbs only the (j,k)-th quadratic residue direction m_{j,k}; when j≠k this violates the symmetry constraint m_{j,k}=m_{k,j} of equation (25), so ˇG is not of the form (2) and the sub-optimality inequality (30) cannot be invoked. For (29c), equation (54) perturbs only b_k, and for (29d), equation (58) shifts only λ_k; in both cases, for complex-conjugate pole pairs the necessary conjugate update of b_{k+1} or λ_{k+1} is omitted, so the perturbed system is complex-valued rather than a real admissible LQO system. These are not merely formalities: the contradiction argument proves stationarity over a larger unconstrained pole-residue class, which does not imply stationarity over the constrained LQO set. The theorem is likely repairable by using symmetric two-index perturbations for (29b) and real-structure-preserving perturbations for (29c)-(29d), but the proof of the central claim must be revised.","section":"Appendix B / Theorem 3.1"}],"minor_comments":[{"comment":"In the sentence following equation (37), \"to prove (33b)\" should read \"to prove (33c)\", since the argument establishes the left-tangential Lagrange condition.","section":"Proof of Theorem 3.2"},{"comment":"The block matrix Q stated as (1/√2)[1 1; i −i] does not correctly relate W and W_p; the transformation should be (1/2)[1 −i; 1 i] up to column scaling, and \"orthogonal\" should be \"unitary\".","section":"Lemma 4.1"},{"comment":"The numerical computation of the relative H2 error in (47) is not described; the authors should state whether a full eigendecomposition of the order-3000 system or a Lyapunov solve was used.","section":"Section 5.2"},{"comment":"Only a single benchmark is tested, and no comparison is made with the Wilson/gramian-based H2-optimal method of [33] or the Riemannian BFGS method of [44]; such a comparison would strengthen the practical claims.","section":"Section 5.3"},{"comment":"The notation \\(\\overline{G_1(s)}\\) for conjugation of coefficients only is confusing; suggest defining it explicitly or using a different symbol.","section":"Section 2.3 / Eq. (17)"},{"comment":"The absence of a convergence proof for LQO-IRKA is acknowledged, but the authors should state explicitly that convergence and attainment of (29) are empirical observations rather than guaranteed properties.","section":"Section 4.2.2"}],"recommendation":"major_revision","confidential_remarks":"The proof gap in Appendix B is genuine but appears fixable; I recommend major revision rather than rejection. The paper's central contribution is valuable and the numerical results are reproducible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This is a genuine extension of IRKA to LQO systems: Theorem 3.1 gives new interpolation-based first-order necessary conditions for H2-optimal LQO model reduction, and the projection framework in Theorem 3.2 plus the LQO-IRKA algorithm are natural and usable. The paper ships code and data, and Theorem 2.1's H2 norm/inner product formulas are worth having.\n\nThe main soft spot is the proof of Theorem 3.1 in Appendix B. The perturbation for (29b) modifies only the single residue direction m_{j,k}, which violates the symmetry condition m_{j,k}=m_{k,j} of (25) that every LQO system must satisfy. So the perturbed ˇG is not in the admissible class, and the sub-optimality inequality (30) does not apply. The perturbations for (29c) and (29d) have the same problem: they move a single pole or residue direction without the conjugate partner, so the resulting transfer functions don't correspond to a real-valued LQO system. This is a central gap because it means the theorem is not proved from within the feasible set. It is, however, repairable: symmetric two-index perturbations for (29b), real-structure-preserving perturbations for the others, and the contradiction should still go through. So I take the theorem as likely true but not yet proved.\n\nMilder issues: LQO-IRKA's convergence is reported empirically only; the numerical section uses one benchmark and omits comparisons with the existing H2-optimal LQO methods in [33] and [44]; and the computation of the reported H2 errors isn't fully specified. These are fixable.\n\nBottom line: this is a paper for the model-reduction community, and it deserves peer review. The central contribution is novel and the flaws are concrete and addressable. I'd send it to referees with a note to scrutinize Appendix B and to ask for a fuller numerical comparison.","headline":"Genuinely extends H2-optimal interpolation theory to LQO systems, but the main proof has a repairable yet central class-membership gap.","tokens_in":41677,"tokens_out":5753,"would_cite":true,"duration_ms":54989,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C20","41A05","49K15","65J05","65F99","93A15","93C10","93C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"An $\\mathcal{H}_2$-optimal reduced linear quadratic-output model must tangentially interpolate the full system's linear and quadratic transfer functions at the mirror images of its own poles.","keywords":["model reduction","H2-optimality","linear quadratic-output systems","tangential interpolation","multivariate rational interpolation","iterative rational Krylov algorithm","Petrov-Galerkin projection","Volterra kernels"],"falsifier":"One could settle Theorem 3.1 by computing, for a small LQO example, a global $\\mathcal{H}_2$-optimal order-$r$ reduced model with simple poles and evaluating whether (29a)-(29d) fail at its mirrored poles; any violation would refute the necessity claim. Alternatively, a stable full-order LQO system whose LQO-IRKA iterates converge to a model that still violates one of the four conditions would show the algorithm does not deliver the claimed optimality certificate.","tokens_in":40598,"feed_emoji":"📐","tokens_out":6447,"duration_ms":52259,"temperature":0.7,"pith_summary":"This paper derives first-order necessary conditions for a reduced model of a linear quadratic-output (LQO) system, meaning linear state dynamics with an output made of a linear term plus a quadratic term $M(x\\otimes x)$, to minimize the $\\mathcal{H}_2$ approximation error. The conditions say that at the mirror images of the reduced model's poles, the reduced linear and quadratic transfer functions must tangentially match the full-order ones, and certain weighted sums of the two must be matched in both value and derivative. These mixed-multipoint tangential interpolation conditions generalize the classical Meier-Luenberger conditions for linear systems. The paper also shows how to enforce all conditions by Petrov-Galerkin projection and gives an iterative rational Krylov algorithm, LQO-IRKA, that converges to models satisfying them. If correct, this turns $\\mathcal{H}_2$-optimal quadratic-output model reduction into a multivariate rational interpolation problem that can be solved with shifted linear solves.","feed_headline":"H2-optimal surrogates must match quadratic outputs at mirrored poles","feed_subtitle":"New interpolation conditions turn quadratic-output model reduction into a multivariate rational approximation problem.","key_machinery":"The load-bearing object is the pole-residue expansion of the reduced transfer functions, $\\tilde G_1(s)=\\sum_j c_j b_j^T/(s-\\lambda_j)$ and $\\tilde G_2(s_1,s_2)=\\sum_{j,k} m_{j,k}(b_j\\otimes b_k)^T/((s_1-\\lambda_j)(s_2-\\lambda_k))$. Theorem 2.1 uses these expansions to express the $\\mathcal{H}_2$ inner product and norm of an LQO system as finite evaluations of $G_1$ and $G_2$ at the mirrored poles $-\\lambda_j$. Lemma 2.1's symmetry identities for $G_2$, inherited from the commutation matrix, let the authors group the left-tangential and derivative conditions into the compact form (29c)-(29d). Theorem 3.2 then constructs projection bases $V,W$ whose columns are the shifted solves of (31) and (32), so that Petrov-Galerkin projection enforces all $3r+r^2$ interpolation conditions simultaneously.","core_discovery":"On the paper's own terms, the central discovery is Theorem 3.1: if an asymptotically stable order-$r$ LQO system $\\tilde G$ with simple poles minimizes the squared $\\mathcal{H}_2$ error against a full-order LQO system $G$, then the transfer functions $\\tilde G_1$ and $\\tilde G_2$ satisfy the four tangential interpolation conditions (29a)-(29d). In particular, $\\tilde G_1$ and $\\tilde G_2$ individually right-interpolate $G_1$ and $G_2$ at $-\\lambda_k$ along residue directions, and linear combinations of the two transfer functions are interpolated in the left-tangential Lagrange and bi-tangential Hermite senses at all pairs of mirrored poles. This gives the LQO analogue of the Meier-Luenberger characterization and, because the proof only uses Hardy-space membership of the full-order functions, the paper notes the conditions extend beyond the specific LQO realization.","pith_inferences":["The proof of Theorem 3.1 does not require the full-order model to have LQO structure, so the same interpolation conditions should characterize $\\mathcal{H}_2$-optimal rational approximants for systems with second-order or delay dynamics; whether projection can construct them is left open by the paper.","No convergence proof for LQO-IRKA is given; a natural test is whether iterates always reach a stationary point from arbitrary initial data, and whether the simple-pole restriction ever excludes the true minimizer.","The mixed conditions couple every pair of poles through the $m_{k,\\ell}$ residues, so a large-scale implementation may need to exploit sparsity or low-rank structure in the sums over $\\ell$; the paper does not address this cost.","A data-driven analogue could identify the interpolation data directly from frequency samples of $G_1$ and $G_2$, bypassing the state-space realization entirely."],"forward_implications":["Every $\\mathcal{H}_2$-optimal reduced LQO model with simple poles is a tangential interpolant, so the quest for optimal reduced models can be reposed as choosing poles and residue directions that satisfy (29a)-(29d).","Petrov-Galerkin projection with the bases (31) and (32) enforces all the optimality conditions at once, guaranteeing that any converged LQO-IRKA iterate meets the first-order necessary conditions.","LQO-IRKA requires only shifted linear system solves and matrix-vector products, making the framework applicable to large-scale systems where balancing-based LQO methods need costly Lyapunov solves.","When the quadratic output term is zero, conditions (29) reduce to the classical linear Meier-Luenberger interpolation conditions, so the paper's results contain the linear theory as a special case.","Because the $\\mathcal{H}_2$ error bounds the time-domain $L^\\infty$ output error via (20), satisfying these optimality conditions also controls worst-case output accuracy over time for finite-energy inputs."],"supporting_citations":[{"why":"Meier-Luenberger's original SISO $\\mathcal{H}_2$ optimality result that this paper generalizes to LQO systems.","marker":"[28]"},{"why":"Supplies the MIMO tangential interpolation conditions and the IRKA iteration that LQO-IRKA extends.","marker":"[24]"},{"why":"Defines LQO Gramians, the system $\\mathcal{H}_2$ norm, and balanced truncation, providing the error metric and baseline method used in the numerical comparisons.","marker":"[9]"},{"why":"Establishes the tangential interpolation framework for quadratic-bilinear outputs, including the right-tangential conditions that Theorem 3.2 extends.","marker":"[17]"},{"why":"Provides the prior Wilson/Gramian-based $\\mathcal{H}_2$ optimality framework for LQO systems, which the interpolatory conditions complement.","marker":"[33]"},{"why":"Gives the linear-system $\\mathcal{H}_2$ norm and inner product formulae that Theorem 2.1 generalizes.","marker":"[2]"},{"why":"Introduces multipoint Volterra series interpolation for bilinear systems, the closest earlier analogue of the mixed-multipoint conditions.","marker":"[19]"}],"fun_headline_variants":["Quadratic-output H2 reduction: interpolate at mirrored poles","LQO-IRKA: H2-optimal surrogates via tangential interpolation","Meier-Luenberger extends to quadratic-output systems","New optimality conditions for quadratic-output model reduction","Tangential interpolation solves H2-optimal LQO reduction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the $\\mathcal{H}_2$-optimal reduced model can be chosen with simple poles and that small pole/residue perturbations stay within the class of order-$r$ asymptotically stable LQO systems; the paper does not prove that a minimizer with simple poles always exists, and LQO-IRKA's convergence is reported empirically rather than proved.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic-output H2 reduction: interpolate at mirrored poles","LQO-IRKA: H2-optimal surrogates via tangential interpolation","Meier-Luenberger extends to quadratic-output systems","New optimality conditions for quadratic-output model reduction","Tangential interpolation solves H2-optimal LQO reduction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1235,"prompt_tokens":975,"completion_tokens":260,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":175}},"tokens_in":591,"tokens_out":260,"duration_ms":2473,"temperature":1.0,"reasoning_tokens":175,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:01:33.806849+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One could settle Theorem 3.1 by computing, for a small LQO example, a global $\\mathcal{H}_2$-optimal order-$r$ reduced model with simple poles and evaluating whether (29a)-(29d) fail at its mirrored poles; any violation would refute the necessity claim. Alternatively, a stable full-order LQO system whose LQO-IRKA iterates converge to a model that still violates one of the four conditions would show the algorithm does not deliver the claimed optimality certificate.","supporting_citations":[{"cited_title":"Approximation of linear c onstant systems","cited_arxiv_id":null,"evidence_quote":"Meier-Luenberger's original SISO $\\mathcal{H}_2$ optimality result that this paper generalizes to LQO systems."},{"cited_title":"Antoulas, and Christop her Beattie","cited_arxiv_id":null,"evidence_quote":"Supplies the MIMO tangential interpolation conditions and the IRKA iteration that LQO-IRKA extends."}],"review_version":1}