{"id":"2241c546-6ede-44ac-a86d-85ec6e0a6484","arxiv_id":"1908.00820","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"An adaptive pole-matching framework, with branch-and-bound optimization, predictor-corrector steps, adaptive refinement, and optional regression, builds parametric reduced-order models by interpolating local ROM poles.","lead":"This paper introduces an adaptive method for building parametric reduced-order models by matching and interpolating the poles of locally built reduced models across a parameter range. It combines a branch-and-bound pole-matching algorithm with a predictor-corrector and adaptive refinement framework, and reports improved accuracy and storage savings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm III.1 is a 2-opt local search, not a branch-and-bound global solver; pairwise local minima of the linear assignment objective (14) exist, so the claimed guarantee of correct pole-matching is unsupported.","rationale":"The reader's weakest-assumption analysis identifies exactly the right soft spot: Theorem III.1 justifies only a pruning rule for failed swaps, not global optimality of the returned permutation. My reading of the algorithm confirms that it is a greedy 2-opt local search: it starts from the identity permutation, tries all single transpositions, accepts the first strictly improving one, and stops when no permitted swap improves the objective. The objective (14) is a linear assignment problem, and for such problems pairwise local optimality does not imply global optimality in general. This is not just a missing proof; the algorithmic claim is incorrect as stated, and a counterexample can in principle be exhibited. Because pole matching is the foundation of the interpolation and regression steps, a wrong match can silently corrupt the pROM; Fig. 3 demonstrates this sensitivity for regression. The paper does contain useful empirical evidence, and its own discussion in Section VI acknowledges that regression requires correct matching of dominant poles. The adaptive refinement strategy can catch some mismatches through the test at the midpoint, but it does not repair the false claim that Algorithm III.1 solves the combinatorial optimization problem. The appropriate disposition is therefore the same conditional acceptance the reader recommended: the numerical method may be useful, but the paper must either provide a genuine global-optimality argument for the special structure of (14), or honestly relabel Algorithm III.1 as a heuristic and adjust the claims accordingly. No change to the reader's verdict is needed.","tokens_in":13358,"tokens_out":7344,"duration_ms":82854,"concrete_test":"Implement Algorithm III.1 exactly as written and compare its output against exhaustive enumeration over all permutations for (14), using only complex-conjugate-pole data (set ns=0) with nd=4, 5, and 6. Generate 10^4 random instances of D1 and D2 with well-separated rows, and also run instances seeded from the pole data of the two numerical examples. If any instance returns a permutation with fd strictly larger than the exhaustive minimum, the claim that Algorithm III.1 solves (14) is refuted. A targeted version is to search specifically for a 3-cycle local minimum: construct D1 and D2 such that the identity permutation beats every single swap but the cycle (1 2 3) has lower cost; if Algorithm III.1 terminates at identity, the concern is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The branch-and-bound claim for the pole-matching optimization (14) is the load-bearing link in the framework: every interpolation and regression step assumes the poles are matched correctly. Algorithm III.1, however, is not a global optimization method for (14). The objective fd(vd) = ||D1Wd - M(vd,D2)Wd||_F^2 is a linear assignment problem: it decomposes as a sum over rows, so swapping two entries of vd changes only two assignment costs. The algorithm only ever performs single transpositions of the current permutation, accepts a transposition only if it strictly decreases fd, and terminates when no allowed transposition improves the objective. Theorem III.1 proves only a pruning rule: if a particular swap increases fd, then the swapped configuration cannot be optimal. It says nothing about permutations reachable only by sequences of swaps, such as cycles. For linear assignment problems, a permutation that is locally optimal under all single transpositions need not be globally optimal; a 3-cycle can have strictly lower cost than the identity even when every pairwise swap increases cost. Thus Algorithm III.1 can return a strictly suboptimal pole matching. The paper's own Section VI and Fig. 3 show that a wrong matching corrupts the regression globally, and Section III explicitly describes Scenario 3 where the optimization 'can be misleading.' Since the paper claims to 'solve' (14) and to 'guarantee correct pole-matching,' and since the adaptive refinement step is designed around this matching, the central correctness claim rests on a heuristic whose global optimality is neither proved nor generally true. The intended small-perturbation regime may make the initial guess correct in practice, but the algorithm cannot certify that, and no recovery mechanism is guaranteed if it is wrong.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an adaptive parametric reduced-order modeling (PMOR) method based on interpolating the poles and residues of local reduced-order models (ROMs). Pole matching between ROMs at nearby parameter values is formulated as a combinatorial optimization problem, which the authors claim to solve with a branch-and-bound algorithm (Section III-B). The adaptive framework (Algorithm IV.1) combines a predictor-corrector strategy, an adaptive refinement strategy using a test-ROM at interval midpoints, and an optional polynomial-regression post-processing step. Numerical experiments on an academic 1008-state example and a branchline-coupler microwave model report relative errors around 10^-4 and reduced storage.","tokens_in":13689,"tokens_out":12705,"duration_ms":127674,"significance":"If the central claims hold, the method would be a useful contribution: it is compatible with any MOR method, uses a pole-residue realization that removes coordinate-system ambiguity, and the adaptive refinement loop validates interpolated ROMs against freshly computed ROMs, which is an external and non-circular check. The numerical results are encouraging and the storage reduction via regression is clearly demonstrated. However, the correctness of the entire framework rests on the pole-matching optimization, and the paper does not establish that Algorithm III.1 solves that optimization globally. The two examples are also limited to a single parameter, so the broader advantages stated in the abstract are not yet demonstrated.","major_comments":[{"comment":"Algorithm III.1 is presented as a branch-and-bound solver for the global optimization problem (14), but it is actually a greedy local search: it starts from the identity permutation, accepts a new permutation only when a single transposition strictly decreases the objective, and terminates when no allowed transposition improves it. The objective in (14) decomposes as a sum of independent row costs and is a linear assignment problem. For such problems, a permutation that is locally optimal under all single transpositions need not be globally optimal; a 3-cycle can have strictly lower cost than the identity even when every transposition increases the cost. Theorem III.1 only proves that the specific swapped configuration tested is not optimal, and it gives no bound that would rule out such cycles. Therefore the statement that Algorithm III.1 'solves' (14) or 'guarantees correct pole-matching' is unsupported. This is load-bearing: Section VI and Fig. 3 show that a wrong matching can globally corrupt the regression, and the adaptive refinement strategy does not cure this when mismatched poles have similar positions and residues.","section":"Section III-B, Theorem III.1 and Algorithm III.1"},{"comment":"The pruning bookkeeping with the matrix F does not follow from Theorem III.1. In the theorem, a failed swap of the i-th and j-th entries at the current permutation vd forbids any assignment with vd_i = B and vd_j = A, where A = vd_i and B = vd_j before the swap. The update F(vd_i, vd_j) = 0, however, forbids the condition vd_{vd_i} = vd_j and vd_{vd_j} = vd_i, which is a different condition unless the current permutation happens to satisfy vd_i = i and vd_j = j. Consequently, the algorithm may prune permutations that Theorem III.1 does not rule out, or fail to prune permutations that it does rule out. This undermines the claimed branch-and-bound efficiency and correctness even for a single swap.","section":"Section III-B, F-matrix update rule (lines 10-11 and Eq. (22))"}],"minor_comments":[{"comment":"The phrase 'brutal-force method' should be 'brute-force method'.","section":"Introduction, Section I"},{"comment":"In the definition of the relative error, the integrand in the numerator appears as H(ω,p) minus H(ω,p), which is identically zero; presumably the second factor should be the pROM transfer function, e.g., \\hat H(ω,p).","section":"Section V, relative error definition"},{"comment":"The conference title is given as 'Numerical Electromagnetic and Metaphysics Modeling and Optimization'; it should be 'Numerical Electromagnetic and Multiphysics Modeling and Optimization'.","section":"References [14] and [15]"},{"comment":"The paper confines the adaptive framework to a single scalar parameter (Section IV), yet the abstract claims 'constant size of the parametric reduced-order model with respect to the number of parameters' and 'capability to deal with complicated parameter dependency'; these broader claims are not supported by the numerical results in this manuscript.","section":"Section IV-C and abstract"}],"recommendation":"major_revision","confidential_remarks":"The core numerical framework appears promising and the adaptive refinement with an external ROM_T,t check is a genuine strength. The main obstacle is the unsupported claim that Algorithm III.1 globally solves (14); the authors should either provide a correct global optimality proof (and fix the F-matrix bookkeeping) or reposition the algorithm as a heuristic and adjust the 'guarantee' language throughout the abstract, introduction, and conclusions. The regression claims in Section IV-C should also be qualified accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid engineering paper, but the headline claim about branch-and-bound is inflated. Algorithm III.1 is a 2-opt local search with a pruning rule, not a global solver, and the paper's own framework only partially compensates.\n\nWhat's actually new: the adaptive framework — predictor-corrector, adaptive refinement, regression post-processing — is a real contribution. It turns the authors' earlier pole-residue interpolation idea into a practical method that is MOR-agnostic, keeps pROM size constant, and handles complicated parameter dependence. The numerical results are convincing: relative error drops from ~1e-2 to ~1e-4 on two examples, and regression cuts storage. They also deserve credit for clearly discussing when matching is hard (Scenario 3) and why regression is fragile.\n\nThe soft spot is the solver. The objective (14) is a linear assignment problem. Algorithm III.1 only tries single transpositions from the current permutation, accepts improving swaps, and terminates at a local optimum. Theorem III.1 is a valid pruning rule for a specific failed transposition, but it doesn't establish global optimality, and pairwise local minima exist for linear assignment. So the claim that the algorithm 'solves' (14) and 'guarantees correct pole-matching' is not supported. This matters because wrong matching corrupts regression globally, as the paper's own Fig. 3 shows. The adaptive refinement does catch some of these errors by testing at midpoints and inserting new ROMs, so in practice the method may still be fine; but the guarantee is absent. Minor issues: the MOR method for the first example is unspecified, and no code is provided, so replication takes effort.\n\nWho it's for: people building parametric ROMs by interpolating local models, especially in circuit/electromagnetics. A serious referee should engage with this; the unsupported global optimality claim should be fixed, either by renaming it a heuristic and adding a fallback, or by proving a property for the specific perturbation regime. I'd accept it for review.","headline":"Useful adaptive PMOR framework, but the 'branch-and-bound' pole matcher is really a 2-opt heuristic, so the global optimality claim doesn't hold up.","tokens_in":14244,"tokens_out":8178,"would_cite":true,"duration_ms":80248,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B11","65D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Pole matching makes reduced parametric models 100x more accurate","keywords":["parametric model order reduction","pole matching","pole-residue realization","branch and bound","adaptive refinement","predictor-corrector","ROM interpolation","regression"],"falsifier":"Construct two small ROMs whose correct pole matching is known by construction to require two swaps, with the first swap increasing the weighted distance; run Algorithm III.1 from the natural ordering. If it returns the wrong permutation, the central matching guarantee fails. A practical version is to take a diagonal parameterized system with two pairs of poles whose dominance crosses, build ROMs at two parameter values, compute the known optimal matching by exhaustive search, and compare it with the branch-and-bound result.","tokens_in":13167,"feed_emoji":"📉","tokens_out":6738,"duration_ms":64588,"temperature":0.7,"pith_summary":"This paper proposes a way to build parametric reduced-order models (pROMs) by interpolating the poles and residues of local reduced-order models computed at selected parameter values. The central difficulty is knowing which pole in one ROM corresponds to which pole in another; the paper formulates this as a combinatorial matching problem and solves it with a branch-and-bound algorithm. Around that solver it builds an adaptive framework: predictor-corrector steps propose new parameter points, adaptive refinement verifies interpolation accuracy, and optional regression compresses the stored repository. If the framework works as claimed, pROMs can be assembled from ROMs produced by any reduction method, with a model size that does not grow with the number of parameters and with relative errors improved from about $10^{-2}$ to $10^{-4}$ in the reported examples.","feed_headline":"Pole matching makes reduced parametric models 100x more accurate","feed_subtitle":"Branch-and-bound pole matching plus adaptive refinement cuts error from 10^-2 to 10^-4 while shrinking storage.","key_machinery":"The central object is the pole-residue realization of a reduced-order model: every pole (real or conjugate-complex pair) is stored with its residue, so a ROM is represented by a small matrix of pole positions and residue weights. The argument runs through the pole-matching optimization, which minimizes a weighted Frobenius distance between pole-residue matrices over row permutations, decoupled into independent problems for complex and real poles; the branch-and-bound algorithm (Algorithm III.1) uses the swap-pruning Theorem III.1 to avoid exhaustive enumeration. The adaptive framework then rides on two mechanisms: a predictor-corrector rule that chooses the reference ROM for matching, and a midpoint error check comparing an interpolated ROM against a freshly computed ROM that triggers refinement. Regression is the compression mechanism that stores polynomial coefficients instead of the full repository.","core_discovery":"The paper's claim is that pole-residue data are the right quantities to interpolate when building a parametric reduced-order model, provided the poles are first matched correctly and the repository ROMs are placed adaptively. All local ROMs are transformed into a pole-residue realization, where the transfer function is a sum of terms each determined by a pole position and a residue; the paper then treats matching as a weighted least-squares assignment of rows between two ROMs and solves it with a branch-and-bound search whose pruning rule is supplied by Theorem III.1. Because matching is reliable only for perturbed poles, the framework uses a predictor-corrector strategy: a predicted ROM is extrapolated from already-built repository ROMs and used as the reference for matching the candidate ROM, and an adaptive refinement loop inserts an extra ROM at the midpoint whenever the interpolated ROM fails an error check against a freshly computed ROM. An optional regression phase replaces the stored repository by entrywise polynomials in the parameter. The numerical experiments report that the full framework reduces relative error from the $10^{-2}$ magnitude to $10^{-4}$ and reduces storage relative to the interpolated repository.","pith_inferences":["A robust global solver for the pole-matching assignment, rather than a local swap search, would let the same framework interpolate ROMs whose pole orderings differ strongly, extending it to heterogeneous ROM sources in industrial practice.","The midpoint error check could be developed into an a posteriori error estimator: if the interpolated-ROM error correlates with the distance between neighboring repository parameters, the same loop could certify accuracy over the whole parameter interval.","For non-normal systems, where small parameter changes can move poles abruptly, a natural extension is to match pseudospectral regions or dominant invariant subspaces instead of individual poles; the paper itself notes pole sensitivity as a limitation."],"forward_implications":["Any model-order reduction method can feed the framework, since only accurate local ROMs are required and the state-space form of the full-order model is never needed globally.","The pROM size stays constant as the number of parameters grows, because only pole and residue entries are stored and interpolated, not a global projection subspace.","Complicated parameter dependence, including pole crossing, can be captured locally by adaptive step selection without assuming affine parameter dependence.","Storage can be cut from $N(4n_d+2n_s)$ repository entries to $(q+1)(4n_d+2n_s)$ polynomial coefficients when the regression post-processing is used.","Stability is easy to monitor because the interpolated model is built directly from pole positions; linear interpolation in the left half-plane preserves stability."],"supporting_citations":[{"why":"Introduces the pole-residue realization and the pole-matching idea that this paper extends.","marker":"[14]"},{"why":"Shows why interpolating matrices or frequency responses of local ROMs fails in general, motivating pole-based interpolation.","marker":"[9]"},{"why":"Demonstrates that direct interpolation of local state-space matrices fails because ROMs live in different coordinate systems.","marker":"[10]"},{"why":"Conference version whose adaptive pole-matching framework is expanded here with refinement and regression.","marker":"[15]"},{"why":"Supplies the branch-and-bound principles used by the pole-matching solver.","marker":"[17]"},{"why":"Provides the parametric FOM benchmark adapted to include nonlinear parameter dependence and pole crossing.","marker":"[19]"},{"why":"Provides the branchline coupler benchmark used in the microwave example.","marker":"[22]"},{"why":"Supplies the Krylov method used to build the repository ROMs in the numerical tests.","marker":"[23]"}],"fun_headline_variants":["Pole-matching method slashes parametric ROM error 100x","Adaptive pole matching shrinks reduced-model error to 1e-4","Branch-and-bound pole matching cuts ROM error and storage","Parametric ROMs get 100x accuracy via adaptive pole matching","Pole-matching reduces parametric model error by two orders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the branch-and-bound search, which only tries single swaps and stops when no swap lowers the matching cost, returns the correct pole correspondence; if the true correspondence requires a sequence of swaps where some intermediate swap looks worse, the matching can be wrong and the interpolation corrupted.","fun_headline_variants_meta":{"raw":{"variants":["Pole-matching method slashes parametric ROM error 100x","Adaptive pole matching shrinks reduced-model error to 1e-4","Branch-and-bound pole matching cuts ROM error and storage","Parametric ROMs get 100x accuracy via adaptive pole matching","Pole-matching reduces parametric model error by two orders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000878,"raw_usage":{"total_tokens":3807,"prompt_tokens":963,"completion_tokens":2844,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":2756}},"tokens_in":579,"tokens_out":2844,"duration_ms":19140,"temperature":1.0,"reasoning_tokens":2756,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:31:42.155337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two small ROMs whose correct pole matching is known by construction to require two swaps, with the first swap increasing the weighted distance; run Algorithm III.1 from the natural ordering. If it returns the wrong permutation, the central matching guarantee fails. A practical version is to take a diagonal parameterized system with two pairs of poles whose dominance crosses, build ROMs at two parameter values, compute the known optimal matching by exhaustive search, and compare it with the branch-and-bound result.","supporting_citations":[{"cited_title":"Interpolation of reduced-order models based on modal analysis,","cited_arxiv_id":null,"evidence_quote":"Introduces the pole-residue realization and the pole-matching idea that this paper extends."},{"cited_title":"Reduced-order modelling of paramet- ric systems via interpolation of heterogeneous surrogates,","cited_arxiv_id":null,"evidence_quote":"Shows why interpolating matrices or frequency responses of local ROMs fails in general, motivating pole-based interpolation."},{"cited_title":"Parametric model order reduction by matrix interpolation,","cited_arxiv_id":null,"evidence_quote":"Demonstrates that direct interpolation of local state-space matrices fails because ROMs live in different coordinate systems."},{"cited_title":"An adaptive method for interpolating reduced-order models based on matching and continuation of poles,","cited_arxiv_id":null,"evidence_quote":"Conference version whose adaptive pole-matching framework is expanded here with refinement and regression."},{"cited_title":"Branch and bound algorithms — principles and examples,","cited_arxiv_id":null,"evidence_quote":"Supplies the branch-and-bound principles used by the pole-matching solver."},{"cited_title":"Data-driven parametrized model reduction in the Loewner framework,","cited_arxiv_id":null,"evidence_quote":"Provides the parametric FOM benchmark adapted to include nonlinear parameter dependence and pole crossing."},{"cited_title":"Fast evaluation of time-harmonic Maxwell’s equations using the reduced basis method,","cited_arxiv_id":null,"evidence_quote":"Provides the branchline coupler benchmark used in the microwave example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Krylov method used to build the repository ROMs in the numerical tests."}],"review_version":1}