{"id":"7be4ea95-b752-4bf1-b967-8394c199896a","arxiv_id":"2510.14479","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A state-space reduction in linear-quadratic parabolic optimal control automatically induces an equivalent control-space reduction, with certified error bounds and a convergent adaptive algorithm.","lead":"The paper shows that when you approximate the state in an optimal control problem by a low-dimensional subspace, the optimal control automatically inherits a matching low-dimensional structure. This gives cheaper solves with the same optimizer, plus error certificates and an adaptive algorithm, demonstrated on a parabolic PDE example.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 13's convergence proof assumes that strictly increasing POD ranks imply a dense union of reduced spaces; this inference is invalid, so convergence to the full-order optimum is not established.","rationale":"The paper's cleanest contributions — the equivalence Lemma 2 and the residual-based bounds in Corollary 5 — are mathematically sound and numerically supported. Table 1 confirms the equivalence, and Table 2 confirms the error bounds. The reader's CONDITIONAL verdict is appropriate. The most load-bearing weakness is the convergence proof of Algorithm 1: the rank-increase argument does not imply density of the union of POD spaces, so the invocation of Lemma 11 is unsupported. This is not a disagreement with the consensus; it is an internal gap in the proof. There are also independent, printed errors in the proof of Theorem 7 — the Lagrangian in (22) is missing the ∂_t y term, and the trapezoidal-rule justification is garbled — but those do not affect the equivalence or the control error bounds used by Algorithm 1. The recommended verdict remains CONDITIONAL: the algorithmic idea is likely salvageable and the numerics are favorable, but the stated convergence theorem and the value-error representation need correction before the certification claims are accepted as proven.","tokens_in":19300,"tokens_out":24935,"duration_ms":209530,"concrete_test":"Settle the density step analytically: consider the family V^{r_k} = span{e_2, ..., e_{k+1}} in ℓ², which satisfies the rank-increase condition of Lemma 10 but whose union is not dense. Check whether the snapshot recurrence S_{k+1} = S_k ∪ {y(u_k), p(y(u_k))} with u_k optimal for V^{r_k} can generate such a family under assumptions (A1)-(A2). If yes, exhibit the instance with u_k not converging to the FOM optimum; if no, identify the additional property of the recurrence that forces closure(∪_k V^{r_k}) = V, and replace the sentence 'we construct an orthonormal basis of V' with a proof of that property.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the convergence of Algorithm 1, Theorem 13. In the proof, Lemma 10 is used correctly to show that every non-terminating iteration strictly increases r_k = dim V^{r_k}. The proof then states: 'By iterating this procedure, we construct an orthonormal basis of V and Lemma 11 ensures convergence, since r_k → ∞.' This is the load-bearing step, and it is not justified. A strictly increasing sequence of finite-dimensional subspaces need not have a dense union: in ℓ², V^{r_k} = span{e_2, ..., e_{k+1}} has r_k → ∞ but its union is contained in the proper closed subspace {x : x_1 = 0}. Lemma 11 requires V^r to be spanned by an orthonormal basis of all of V, not merely of a proper subspace W. Lemma 12 only excludes the case y(u_k), p(y(u_k)) ∈ V^{r_k}; it does not exclude the possibility that all iterates and snapshots lie in a fixed proper W. Thus the proof of Theorem 13 does not establish convergence to the FOM minimizer. The numerical experiments (Section 5.2) use energy-truncated POD with r_k ≤ \\bar r_{S_k}, which Remark 14 explicitly places outside the assumptions of the proof, so the numerics do not close this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies unconstrained linear-quadratic optimal control problems with parabolic PDE constraints and infinite-dimensional control/state spaces. Its main contributions are: (i) a proof that a Galerkin/POD reduction of the state space induces a reduced control space via the first-order optimality condition, so that the state-reduced OCP and the combined state/control-reduced OCP have the same minimizer (Lemma 2); (ii) lower and upper a posteriori error bounds for the optimal control with respect to an arbitrary control (Corollary 5); (iii) an error representation for the optimal objective value (Theorem 7); and (iv) an adaptive POD-based algorithm (Algorithm 1) with a claimed convergence theorem (Theorem 13). Numerical experiments on a 2D heat equation illustrate the equivalence, the sharpness of the estimators, and computational speedups, with code provided.","tokens_in":19624,"tokens_out":15126,"duration_ms":128801,"significance":"If the main results hold, the paper makes a useful contribution: control-space reduction is obtained for free from state-space reduction, which simplifies both analysis and implementation. The lower a posteriori bound in Corollary 5 is a genuine addition over the usual upper-only estimates, and the numerical verification is careful and reproducible. However, two load-bearing parts of the manuscript are not sound as written: the proof of Theorem 7 contains line-level errors, and the convergence proof of Theorem 13 has a serious logical gap. The equivalence result (Lemma 2) and the error estimator (Corollary 5) appear sound and are well supported by the numerics, so the paper's core idea is promising, but the stated theorems need substantial revision.","major_comments":[{"comment":"The Lagrangian in (22) omits the time-derivative term ⟨∂_t y, p⟩; with the displayed L, the stationarity condition L'(\\bar x)=0 is not equivalent to the optimality system (3b),(6),(7). Moreover, the proof asserts \\hat J(\\bar u)-\\hat J(\\hat u_r)=L(\\bar x)-L(\\hat x^r). This is false as printed: L(\\hat x^r) uses the reduced state \\bar y_r, whereas \\hat J(\\hat u_r) is evaluated at the full-order state y(\\hat u_r). The correct statement would involve the reduced objective \\hat J^r(\\hat u_r), and the proof would need a Lagrangian that encodes the reduced state equation. As it stands, the proof does not establish the error representation (25).","section":"Sec. 3.2, Eq. (22) and proof of Theorem 7"},{"comment":"The decisive step in the proof is the sentence: 'By iterating this procedure, we construct an orthonormal basis of V and Lemma 11 ensures convergence, since r_k → ∞.' This is not justified. Lemma 10 only guarantees that dim V^{r_k} strictly increases. A nested sequence of finite-dimensional subspaces with strictly increasing dimensions need not have a dense union; e.g., in V=ℓ², V^{r_k}=span{e_2,...,e_{k+1}} has r_k→∞ but the union is contained in the proper closed subspace {x_1=0}. Lemma 11 requires an orthonormal basis of all of V. The earlier statement that \\bar r_{S_{k+1}}=∞ implies u_{k+1}=\\bar u is equally unjustified, since an infinite-dimensional POD subspace need not be all of V. An additional argument is needed to prove that the union of the POD spaces is dense, or the theorem must be weakened or placed under extra assumptions (e.g., a controllability/persistence condition). Th","section":"Sec. 4.2, proof of Theorem 13"}],"minor_comments":[{"comment":"The proof assumes 'W.l.o.g.' that exactly one snapshot in S_+\\S satisfies (32). This is not a harmless reduction unless the argument is repeated for each such snapshot; otherwise the sum over S_+\\S must be handled directly. The proof should be rewritten to treat the general case.","section":"Sec. 4.1, Lemma 10"},{"comment":"The proof refers to [25, Theorem 3.11] and states that replacing finite-dimensional controls by infinite-dimensional U 'does not change the proof.' This is not obvious, since the control space appears in the cost, the optimality condition, and the compactness arguments. Please provide a self-contained argument or a more precise reference.","section":"Sec. 4.2, Lemma 11"},{"comment":"The numerical experiments use energy-truncated POD with r_k ≤ \\bar r_{S_k}, which Remark 14 explicitly places outside the assumptions of Theorem 13. The paper should state clearly that the numerical verification of convergence is for the modified algorithm, not for the theorem as proved.","section":"Sec. 5.2 and Remark 14"},{"comment":"The column headings are ambiguous: 'k' is not defined in the table, and the two error-estimator columns (overline and underline Delta) are easily confused. Please add a caption defining k, e_u, and the estimator notation.","section":"Table 2"},{"comment":"The proof of Theorem 7 switches between \\hat J and \\hat J^r without comment. Since these are different objects (FOM reduced cost vs. state-reduced cost), the notation should be fixed throughout Section 3.2.","section":"Sec. 3.2, notation"}],"recommendation":"major_revision","confidential_remarks":"The paper has two solid, verified components (Lemma 2 and Corollary 5) and a useful numerical study. The main obstacle is Theorem 13: the proof conflates increasing subspace dimension with density, and the statement 'r_k→∞ implies an orthonormal basis of V' is plainly invalid. I would not reject the paper outright, because the convergence claim may be repairable with a genuine density argument or additional assumptions, but the revision must contain a real fix, not a rewording. The Theorem 7 proof also needs correction independent of the convergence issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the central observation is correct and practically useful. Galerkin state reduction induces a control space through the optimality condition, and Lemma 2 — the control-and-state-reduced problem has the same minimizer as the state-reduced one — is a simple, right, and useful thing to lean on. The lower and upper a posteriori bounds in Corollary 5 are a clean repackaging of the standard residual estimate for an arbitrary control, and the numerics are honest: code is provided, the equivalence is confirmed to machine precision, and the speedups (up to 34x) are plausible because the inner problem is cheap and the FOM solves feed both the estimator and the snapshots.\n\nThe soft spots are real but localizable. The proof of Theorem 13 does not establish convergence as written. It shows r_k strictly increases and then asserts that iterating constructs an orthonormal basis of V. That inference is invalid: strictly increasing finite-dimensional subspaces can all lie in a proper closed subspace. To get density of the union of POD spaces you need an argument that the snapshots eventually span V. Lemma 12 only rules out both snapshots lying in the current W^r; it doesn't exclude all iterates and snapshots living in a fixed proper subspace. The numerics don't close this gap because Section 5.2 uses energy-truncated ranks, which Remark 14 explicitly puts outside the proof's assumptions. I suspect the algorithm genuinely converges for parabolic problems like the one tested, but the proof needs another idea.\n\nThere are also line-level errors in the proof of Theorem 7. The Lagrangian in (22) is missing the ∂_t y term, so the printed identity \\hat J(\\hat u_r)=L(\\hat x_r) doesn't hold as stated. These look fixable.\n\nWho this is for: people doing adaptive POD for linear-quadratic PDE-constrained optimal control, especially in online/realtime settings. The equivalence is a worthwhile result on its own; the certified adaptive algorithm is the intended contribution. I'd send this to a serious referee, explicitly asking them to check the convergence proof. The right outcome is major revision, not rejection.","headline":"Control/state reduction equivalence is clean and useful; the adaptive convergence proof has a real density gap, and Theorem 7's printed proof has fixable errors.","tokens_in":20085,"tokens_out":3904,"would_cite":true,"duration_ms":31361,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49K20","49M05","49M41","65G20","93C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For unconstrained linear-quadratic optimal control problems governed by time-varying parabolic PDEs, reducing the state space automatically induces a reduced structure in the optimal control, making the control-and-state-reduced problem equ","keywords":["linear-quadratic optimal control","parabolic PDEs","adaptive model-order reduction","proper orthogonal decomposition","control space reduction","a posteriori error estimates","variational discretization"],"falsifier":"Run the adaptive algorithm on a parabolic optimal control problem whose true optimal state and adjoint generate snapshots that all lie in a fixed finite-dimensional subspace not reachable from the initial guess (e.g., a target state orthogonal to the span of all initially generated snapshots); if the algorithm terminates with a non-zero gradient while the POD rank stops increasing, the density premise behind Theorem 13 fails. Alternatively, construct a nested sequence of finite-dimensional POD spaces with strictly increasing dimensions whose union is a proper closed subspace of V and check whe","tokens_in":19187,"feed_emoji":"🎯","tokens_out":5449,"duration_ms":43498,"temperature":0.7,"pith_summary":"The paper establishes that, for unconstrained linear-quadratic optimal control of parabolic PDEs, you never need to construct a separate reduced space for the control: the first-order optimality condition places the reduced optimal control in a space inherited from the reduced adjoint basis, so the control-and-state-reduced problem has exactly the same minimizer as the state-only-reduced problem. This equivalence, proven in Lemma 2, lets the authors import variational discretization into adaptive model-order reduction without introducing any extra approximation error. They supply two-sided a posteriori error bounds for the optimal control, an error representation for the objective value in which the control-reduction term vanishes, and an adaptive POD algorithm that provably converges to the full-order optimal control. If correct, this means a cheaper combined reduction is as accurate as state reduction alone, and numerical experiments show speed-ups up to roughly 34× over a full-order gradient method.","feed_headline":"Reduced states automatically shrink the optimal control space","feed_subtitle":"For parabolic optimal control, control- and state-reduced models match state-only accuracy at up to 34× speed-up.","key_machinery":"The central mechanism is the variational-discretization identity: the first-order optimality condition for the state-reduced problem forces the optimal control into the finite-dimensional space U_r generated by the images of the reduced adjoint basis under the control operator B'. This identity (Lemma 2) makes the control- and state-reduced OCP equivalent to the solely state-reduced OCP. The convergence argument rests on Lemma 10, which shows that adding a snapshot with a genuinely new direction strictly increases the maximal POD rank, and on Lemma 11, the convergence of the reduced solutions to the full-order solution as the reduced spaces grow to V.","core_discovery":"The paper proves that a Galerkin reduction of the state space V_r ⊂ V induces, through the optimality condition ū = −1/β B' p̄, a reduced control space U_r = span{J_U^{-1} B' v_i} such that the minimizer of the control-and-state-reduced OCP equals that of the state-reduced OCP (Lemma 2). It further proves that the POD-based adaptive algorithm, which enriches V_r with the current state and adjoint snapshots, produces a sequence of reduced controls converging to the full-order optimal control, with rigorous lower and upper a posteriori bounds on the control error and a representation of the objective-value error in which the control-reduction contribution vanishes.","pith_inferences":["The equivalence suggests that control constraints would break the pure inheritance argument; handling bound constraints would require a separate projection step, since the reduced control must be forced into the admissible set.","The rank-growth proof (Lemma 10) ensures the dimensions increase, but not that the union of the POD spaces is dense in V; if snapshots accumulate in a proper closed subspace, the algorithm could stagnate at a non-optimal control. A testable extension would explicitly check density, e.g., by tracking the decay of the snapshot residual in the ambient space.","The energy-truncated POD used in the numerical experiments (r_k ≤ r̄_{S_k}) lies outside the formal convergence proof; a natural next step is to extend the proof to energy truncation or to construct counterexamples where truncation stalls the algorithm.","The error-representation identity (Theorem 7) suggests a goal-oriented enrichment strategy: use the residual terms in (25) to select which state directions to add rather than pure snapshot accumulation, potentially accelerating convergence."],"forward_implications":["Because the control space is inherited, the combined reduced problem has the same minimizer as state-only reduction; control reduction introduces no additional approximation error.","The two-sided a posteriori estimates (Corollary 5) apply to any candidate control, so error certification does not require solving the reduced problem to optimality.","The adaptive POD Algorithm 1 converges: for ε = 0 the iterates converge to the FOM optimal control; for ε > 0 it terminates in finite steps with a certified error (Theorem 13).","The objective-value error representation (Theorem 7) isolates the state-reduction contribution, since the control-reduction term vanishes.","Numerically, the combined reduction achieves speed-ups up to 34× over the full-order model and outperforms state-only reduction, while the error estimators bracket the true error."],"fun_headline_variants":["State reduction automatically shrinks the control space","Provable control-space reduction from state-space reduction","Reduced states imply reduced optimal control space, provably","Control space reduction follows from state reduction, proven","Optimal control space shrinks with state reduction, verified"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The convergence proof assumes that every non-terminating iteration adds a snapshot with a genuinely new direction, so that the POD spaces eventually span all of V; the proof establishes only that the dimensions strictly increase, not that their union is dense, and the numerical experiments additionally use energy-truncated POD that falls outside the proof's assumptions.","fun_headline_variants_meta":{"raw":{"variants":["State reduction automatically shrinks the control space","Provable control-space reduction from state-space reduction","Reduced states imply reduced optimal control space, provably","Control space reduction follows from state reduction, proven","Optimal control space shrinks with state reduction, verified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000148,"raw_usage":{"total_tokens":980,"prompt_tokens":649,"completion_tokens":331,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":393,"completion_tokens_details":{"reasoning_tokens":257}},"tokens_in":393,"tokens_out":331,"duration_ms":3566,"temperature":1.0,"reasoning_tokens":257,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:35:38.490969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the adaptive algorithm on a parabolic optimal control problem whose true optimal state and adjoint generate snapshots that all lie in a fixed finite-dimensional subspace not reachable from the initial guess (e.g., a target state orthogonal to the span of all initially generated snapshots); if the algorithm terminates with a non-zero gradient while the POD rank stops increasing, the density premise behind Theorem 13 fails. Alternatively, construct a nested sequence of finite-dimensional POD spaces with strictly increasing dimensions whose union is a proper closed subspace of V and check whe","supporting_citations":[],"review_version":1}