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REVIEW 4 major objections 4 minor 33 references

FlexRC: A Flexible Multi-Point Model Order Reduction Method for Many-Port RC Networks

T0 review · 4 major / 4 minor · reviewed 2026-07-08 · grok-4.5

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Pith's one-line read FlexRC builds sparse multi-point reduced models for many-port RC nets that cut reduction and transient sim time

desk verdict FlexRC is a practical multi-point elimination-plus-Arnoldi package for many-port RC MOR; the port-reduction passivity/error story is the real hinge, but the paper still deserves a serious referee. read the letter →

arxiv 2607.05934 v1 pith:7Q3EFQI6 submitted 2026-07-07 eess.SY cs.SY

Yuncheng Xu , Siyuan Yin , Lin Liu , Fan Yang , Xuan Zeng , Chengtao An , Yangfeng Su This is my paper · ORCID
classification eess.SYcs.SY
keywords modelorderreductionRCnetworksmany-portsystemsblockrationalArnoldipassivitymomentmatchingpowergridcircuitsimulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Post-layout circuit simulation needs fast, accurate reduced models of huge resistor-capacitor (RC) networks that have many external ports. Prior high-accuracy elimination methods either lock the expansion frequencies, leave oversized models, or spend too much time on the reduction itself. FlexRC starts from the same sparse elimination of internal nodes, then builds a nonorthogonal projection basis with a modified block rational Arnoldi process so the reduced system stays sparse and banded. The user can pick the expansion frequencies, dial a tolerance that further collapses internal ports, and optionally keep the reduced matrix even sparser. The paper shows that the resulting models match selected moments, remain passive under the controlled port-reduction perturbations, and come with a conservative error bound for that step. On industrial RC nets and IBM power-grid benchmarks the method finishes reduction faster and yields models that simulate faster in the time domain than existing elimination-based competitors.

What carries the argument

A modified block rational Arnoldi process that builds a nonorthogonal projection basis after the initial sparse elimination of internal nodes. The basis maps the many-port RC system onto a sparse banded reduced model whose size and structure are controlled by user-selected expansion points, a port-reduction tolerance, and an optional sparsity parameter.

What would settle it

On a standard industrial RC or IBM power-grid net, measure whether the FlexRC reduced model (with a given port-reduction tolerance) either loses passivity, exceeds the stated conservative error bound, or produces larger reduction or transient-simulation times than the elimination-based baselines the paper compares against.

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Extended reading notes

Core claim

Starting from the same internal-node elimination used by prior high-accuracy methods, FlexRC constructs a nonorthogonal projection basis via a modified block rational Arnoldi process and thereby produces sparse banded reduced-order models of many-port RC networks. The construction admits user-chosen frequency points, a tolerance-driven reduction of the internal subsystem’s ports, and an optional sparsity-control step, while still guaranteeing moment matching, passivity under the induced perturbations, and a conservative error estimate for the port-reduction stage.

Load-bearing premise

The port-reduction perturbations applied to the internal subsystem stay small enough that the claimed passivity properties and the conservative error estimate continue to hold on the industrial RC and IBM power-grid examples, so the reduced models remain accurate enough for transient simulation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes FlexRC, a multi-point model-order reduction method for many-port RC networks. After an elimination step shared with prior high-accuracy methods, it builds a nonorthogonal projection basis via a modified block rational Arnoldi process to obtain sparse banded reduced-order models. The method exposes three user controls: selectable expansion frequencies, a tolerance-driven port-reduction step on the internal subsystem, and an optional sparsity-control strategy. The authors supply a moment-matching analysis, a discussion of passivity under the port-reduction perturbations, and a conservative error estimate for that port reduction. Numerical experiments on industrial RC nets and IBM power-grid benchmarks are used to argue improved reduction time and transient simulation time relative to existing elimination-based techniques.

Significance. If the analysis and experiments hold, FlexRC would address genuine practical limitations of current elimination-based RC MOR: fixed expansion points, oversized ROMs, and high reduction cost for many-port nets that arise in post-layout simulation. The combination of multi-point flexibility, tolerance-controlled internal port reduction, and optional sparsity control is of clear engineering interest, and the inclusion of moment-matching analysis plus an a-priori port-reduction error estimate is a methodological strength relative to purely heuristic sparsification. Successful, structure-aware reduction of industrial and IBM power-grid RC networks would be a useful contribution to the circuit-simulation community.

major comments (4)
  1. [passivity discussion / port-reduction technique] The central accuracy/passivity claim rests on the tolerance-controlled port-reduction step applied to the internal subsystem after elimination. Congruence with a full-rank basis would preserve positive-semidefiniteness of the conductance and capacitance matrices; the manuscript instead applies 'port-reduction perturbations' and only discusses passivity under those perturbations. The report must make explicit whether those perturbations are structure-preserving (e.g., a congruence or a Schur reduction that retains PSD) or are approximate truncations/droppings of weak couplings. If the latter, exact RC passivity is lost and the subsequent transient claims are no longer underwritten by the congruence argument alone. A precise statement of the perturbation operator and a proof (or counter-example) of PSD retention are load-bearing for the effectiveness claim.
  2. [error estimate for port reduction; numerical experiments] The 'conservative error estimate for port reduction' is cited as supporting accuracy of the reduced models used in transient simulation. The manuscript should state the precise norm and the quantity being bounded (transfer-function residual, state trajectory, port voltages/currents), show that the estimate is independent of the subsequent multi-point projection (or quantify the interaction), and demonstrate on at least one industrial/IBM example that the a-priori bound actually majorizes the observed transient or frequency-response error. Without that comparison, the estimate remains formal and does not underwrite the numerical effectiveness claim.
  3. [moment-matching analysis; modified block rational Arnoldi] Moment-matching analysis is asserted for the modified block rational Arnoldi basis. Because the basis is nonorthogonal and is applied after a perturbed internal subsystem, the standard block-rational Krylov moment-matching argument does not apply verbatim. The manuscript should state which moments (at which expansion points, of which transfer function—original, eliminated, or port-reduced) are matched exactly, and whether matching is lost under the port-reduction perturbations. A short lemma with the precise matching order would make the multi-point claim rigorous rather than heuristic.
  4. [numerical experiments on industrial RC and IBM power-grid examples] The experimental claim of superior reduction time and transient simulation time versus 'existing elimination-based methods' needs a controlled comparison: same expansion points (or a fair multi-point baseline), same error tolerance, same sparsity target, and reporting of both offline reduction cost and online transient cost on identical industrial RC and IBM power-grid instances. If the baselines are single-point or do not exploit port reduction, the speed-up may be attributable to the extra knobs rather than to the algorithmic core. Tables that isolate the contribution of each of the three adjustable components would make the effectiveness claim attributable.
minor comments (4)
  1. [method overview / reduced-model structure] Define the precise meaning of 'sparse banded' for the reduced model (bandwidth as a function of number of expansion points and block size) early, so that the optional sparsity-control strategy can be compared against the default banded structure.
  2. [modified block rational Arnoldi process] Clarify the relationship between the user-specified frequency points and the modified block rational Arnoldi shifts: whether shifts are exactly the user frequencies, how repeated shifts are handled, and how the block size is chosen for many-port inputs.
  3. [tolerance-controlled port-reduction; experimental setup] State the units and normalization of the port-reduction tolerance so that the reported tolerance values in the experiments are reproducible by other groups.
  4. [optional sparsity-control strategy] If the optional sparsity-control strategy can destroy the banded pattern or the moment-matching property, note the trade-off explicitly when the option is introduced.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for a careful and constructive report. The four major comments correctly identify places where the manuscript’s claims on passivity, the port-reduction error estimate, moment matching after the modified Arnoldi step, and the experimental attribution of speed-ups need to be stated more precisely and supported more carefully. We agree with the substance of each point and will revise the paper accordingly: we will give an explicit definition of the port-reduction operator and a clear PSD analysis, tighten the statement of the a-priori error bound and add a numerical majorization check, supply a short lemma that states exactly which moments of which transfer function are matched (and under which conditions matching is retained), and restructure the experimental section so that the contribution of each of the three adjustable components is isolated under controlled baselines. None of these revisions changes the algorithmic core of FlexRC; they make the existing analysis and numerical claims rigorous and attributable. Detailed point-by-point replies follow.

read point-by-point responses

  1. Referee: The central accuracy/passivity claim rests on the tolerance-controlled port-reduction step applied to the internal subsystem after elimination. Congruence with a full-rank basis would preserve positive-semidefiniteness of the conductance and capacitance matrices; the manuscript instead applies 'port-reduction perturbations' and only discusses passivity under those perturbations. The report must make explicit whether those perturbations are structure-preserving (e.g., a congruence or a Schur reduction that retains PSD) or are approximate truncations/droppings of weak couplings. If the latter, exact RC passivity is lost and the subsequent transient claims are no longer underwritten by the congruence argument alone. A precise statement of the perturbation operator and a proof (or counter-example) of PSD retention are load-bearing for the effectiveness claim.

    Authors: We agree that the present discussion is not sufficiently precise and that the nature of the port-reduction operator is load-bearing. In the revised manuscript we will (i) define the port-reduction operator explicitly as a tolerance-driven truncation of weak internal couplings after the elimination step (i.e., an approximate dropping, not a pure congruence or exact Schur reduction on the retained ports), (ii) state clearly that exact positive-semidefiniteness of the conductance/capacitance blocks is therefore not automatically inherited from congruence, and (iii) replace the informal passivity discussion by a short analysis that bounds the size of the symmetric part of the perturbation and gives conditions under which the perturbed internal matrices remain positive semidefinite (or, when they do not, how a simple diagonal compensation restores PSD while preserving the same first-order error level). We will also note that the subsequent multi-point projection is still a congruence, so any residual passivity violation can only originate from the port-reduction step itself. These changes make the passivity claim accurate rather than overstated; the numerical effectiveness claims will then rest on the controlled error estimate (see next point) rather than on an unqualified congruence argument. revision: yes

  2. Referee: The 'conservative error estimate for port reduction' is cited as supporting accuracy of the reduced models used in transient simulation. The manuscript should state the precise norm and the quantity being bounded (transfer-function residual, state trajectory, port voltages/currents), show that the estimate is independent of the subsequent multi-point projection (or quantify the interaction), and demonstrate on at least one industrial/IBM example that the a-priori bound actually majorizes the observed transient or frequency-response error. Without that comparison, the estimate remains formal and does not underwrite the numerical effectiveness claim.

    Authors: The referee is correct that the current estimate is stated too loosely to underwrite the numerical claims. In the revision we will: (1) specify that the bound is an a-priori estimate on the H2 (or induced L2) residual of the port-to-port transfer function of the internal subsystem after elimination, measured in the Frobenius norm of the discarded coupling blocks scaled by the chosen tolerance; (2) prove that the bound depends only on the eliminated/port-reduced system and is therefore independent of the subsequent multi-point Arnoldi projection (the projection error is controlled separately by the usual residual of the rational Krylov process); and (3) add a dedicated numerical check on at least one industrial RC net and one IBM power-grid instance that plots the a-priori bound against the observed frequency-response and transient port-voltage errors, confirming that the bound majorizes the measured error. Where the bound is conservative we will say so explicitly. These additions turn the estimate from a formal remark into a verifiable accuracy certificate for the port-reduction step used in the experiments. revision: yes

  3. Referee: Moment-matching analysis is asserted for the modified block rational Arnoldi basis. Because the basis is nonorthogonal and is applied after a perturbed internal subsystem, the standard block-rational Krylov moment-matching argument does not apply verbatim. The manuscript should state which moments (at which expansion points, of which transfer function—original, eliminated, or port-reduced) are matched exactly, and whether matching is lost under the port-reduction perturbations. A short lemma with the precise matching order would make the multi-point claim rigorous rather than heuristic.

    Authors: We accept this criticism. The present text asserts moment matching without isolating the effect of nonorthogonality and of the port-reduction perturbation. The revised manuscript will contain a short lemma that states: (a) in the absence of port reduction, the modified block rational Arnoldi basis (even though nonorthogonal) still produces a reduced model that matches the block moments of the eliminated system’s transfer function at each user-chosen expansion point up to the order determined by the number of Arnoldi steps per point; (b) when the tolerance-controlled port-reduction is applied first, exact moment matching holds for the port-reduced (perturbed) transfer function, while the moments of the original/eliminated transfer function are matched only up to an additive residual controlled by the same port-reduction tolerance that appears in the error estimate; (c) the matching statement is therefore with respect to the system that is actually projected, and any loss of matching relative to the unperturbed system is quantified rather than ignored. The proof follows the standard block-rational Krylov argument once the nonorthogonal basis is written in the appropriate oblique-projection form; we will include the short derivation. This makes the multi-point claim rigorous. revision: yes

  4. Referee: The experimental claim of superior reduction time and transient simulation time versus 'existing elimination-based methods' needs a controlled comparison: same expansion points (or a fair multi-point baseline), same error tolerance, same sparsity target, and reporting of both offline reduction cost and online transient cost on identical industrial RC and IBM power-grid instances. If the baselines are single-point or do not exploit port reduction, the speed-up may be attributable to the extra knobs rather than to the algorithmic core. Tables that isolate the contribution of each of the three adjustable components would make the effectiveness claim attributable.

    Authors: We agree that the current experimental section does not fully isolate the three adjustable components and that some of the reported speed-ups could be attributed to the extra knobs rather than to the algorithmic core alone. In the revision we will restructure the numerical section as follows. (1) All methods will be run on the same industrial RC and IBM power-grid instances, with identical error tolerances and, where applicable, the same sparsity target. (2) We will include a fair multi-point baseline (elimination followed by a standard multi-point block rational Arnoldi without our port-reduction or sparsity-control steps) so that the benefit of the modified process itself is visible. (3) We will add ablation-style tables that turn on, one at a time, (i) user-chosen multi-point expansions, (ii) tolerance-controlled internal port reduction, and (iii) the optional sparsity-control strategy, reporting offline reduction time, ROM size/sparsity, and online transient simulation time for each configuration. (4) Single-point elimination-based methods will remain as reference points, but will no longer be the sole baseline. These changes make the effectiveness claim attributable to each component and remove the ambiguity the referee correctly identified. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FlexRC is an algorithmic MOR method whose claims rest on external numerical benchmarks, not on self-referential fits or definitional reductions.

full rationale

This is a standard model-order-reduction methods paper. The claimed derivation chain is: (1) start from the same elimination step used by prior high-accuracy RC MOR methods; (2) build a nonorthogonal projection basis via a modified block rational Arnoldi process; (3) optionally apply tolerance-controlled internal port reduction and sparsity control; (4) analyze passivity under the resulting perturbations, moment matching, and a conservative port-reduction error estimate; (5) demonstrate effectiveness on industrial RC nets and IBM power-grid examples via reduction time and transient simulation time. None of the six circularity patterns applies. There is no self-definitional loop (X is not defined as Y and then used to “derive” Y). There is no fitted parameter renamed as a prediction of a closely related quantity. Load-bearing support is external numerical comparison, not a self-citation uniqueness theorem or an ansatz smuggled in solely by author self-citation. Moment-matching and passivity arguments are standard congruence/projection analysis for RC systems; the port-reduction step is presented with an explicit (conservative) error estimate rather than as a tautology. Success is judged against external benchmarks (industrial RC, IBM power grids), so the paper is self-contained against outside data. Minor residual risk that baseline choices or tolerances are self-serving is ordinary methods-paper uncertainty, not circularity under the stated criteria. Score 0; steps empty.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

Abstract-only ledger. Free parameters are the user-facing knobs the method exposes (frequency points, port-reduction tolerance, sparsity-control settings). Axioms are standard MOR/circuit-theory background the method relies on. No new physical entities are invented; FlexRC is an algorithmic construction.

free parameters (3)
  • user-specified frequency points
    Multi-point MOR quality depends on the chosen expansion frequencies; they are free design choices of the user/method, not derived from first principles.
  • port-reduction tolerance
    Tolerance that controls how aggressively internal ports are dropped; directly trades accuracy for size and is not fixed by theory in the abstract.
  • sparsity-control strategy parameters
    Optional sparsity knobs that further shape the banded reduced model; values are method/user choices.
assumptions (4)
  • domain assumption RC networks admit an elimination-based reduction step that preserves the relevant port behavior for subsequent projection.
    Abstract states FlexRC 'starts from the same elimination step as previous methods'; that shared step is taken as given domain machinery.
  • domain assumption A modified block rational Arnoldi process can build a nonorthogonal projection basis that yields a sparse banded reduced model with moment-matching properties.
    Core algorithmic claim rests on standard Krylov/Arnoldi MOR theory plus the authors' modification; full justification is not in the abstract.
  • domain assumption Passivity of the reduced model can be discussed/controlled under the perturbations introduced by internal port reduction.
    Abstract claims a passivity discussion under port-reduction perturbations; this is a domain property that must hold for the reduced models to be usable in circuit simulation.
  • standard math Standard linear algebra and moment-matching theory for rational Krylov methods.
    Underlying math for block rational Arnoldi and moment matching is classical.

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Cite this review

Pith. "Pith review of FlexRC: A Flexible Multi-Point Model Order Reduction Method for Many-Port RC Networks." pith.science (2026). https://pith.science/paper/7Q3EFQI6

@misc{pith2026260705934,
  author       = {Pith},
  title        = {Pith review of: FlexRC: A Flexible Multi-Point Model Order Reduction Method for Many-Port RC Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Q3EFQI6}},
  note         = {Machine review of arXiv:2607.05934}
}
read the original abstract

Efficient model order reduction for many-port resistor-capacitor (RC) networks is essential in post-layout circuit simulation. Existing high-accuracy elimination-based methods have certain limitations, such as fixed frequency points, large reduced-order models, or high reduction cost. This paper proposes FlexRC, a flexible multi-point model order reduction method for many-port RC networks. FlexRC starts from the same elimination step as previous methods, and then constructs a nonorthogonal projection basis by a modified block rational Arnoldi process to generate a sparse banded reduced model. FlexRC features three adjustable components: user-specified frequency points, a tolerance-controlled port-reduction technique for the internal subsystem, and an optional sparsity-control strategy. We discuss passivity under port-reduction perturbations, analyze moment matching, and provide a conservative error estimate for port reduction. Numerical experiments on industrial RC examples and IBM power-grid examples demonstrate the effectiveness of FlexRC in terms of reduction time and transient simulation time.

Figures

Figures reproduced from arXiv: 2607.05934 by the authors.

Figure 1
Figure 1. Transient response and signed error for the first output of [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Sparsity patterns of G˜ + C˜ for the two-point reduced models of AAADC_net64. The left panel shows the default FlexRC model with 300682 nonzeros, and the right panel shows FlexRC-SC with 47692 nonzeros. VI. CONCLUSION In this paper, we proposed FlexRC, a flexible multi-point model order reduction method for large-scale RC networks with many ports. FlexRC allows the user to specify the frequency points to improve acc… view at source ↗
Figure 3
Figure 3. Sparsity patterns of G˜ + C˜ for the two-point reduced models of AAADC_net76. The left panel shows the default FlexRC model with 297312 nonzeros, and the right panel shows FlexRC-SC with 48603 nonzeros. APPENDIX A PROOFS A. Proof of Proposition III.1 Proof. From the construction of V2, we have Vˆ 2D = (Gi + s2Ci) −1 Bi , V2 = Vˆ 2T −1 2 . Hence (Gi + s2Ci) V2 = BiD−1T −1 2 . Thus colspan {(Gi + s2Ci) V2} ⊆ colspan {… view at source ↗

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